1、93FTM4Stress Analysis ofSpiral Bevel GearsA Novel Approach to Tooth Modellingby: Ch. Rama Mohana Rao and G. MuthuveerappanIndian Institute of TechnologyAmerican Gear Manufacturers Associatione TECHNICAL PAPERStress Analysis of Spiral Bevel GearsA Novel Approach to Tooth ModellingCh. Rama Mohana Rao
2、and G. MuthuveerappanIndian Institute of TechnologyThestatementsandopinionscontainedhereinare thoseof the authorandshouldnotbe construedas anofficial action oropinion of the American GearManufacturers Association.ABSTRACT:Spiralbevel gears find extensiveapplicationin highperformancetransmissions suc
3、hashelicopter drivesbecause theircurved teethprovide smoother andquieteroperation than straightbevel gearteeth. In this paper, a geometricalapproachhas been proposed to generatethe tooth surfacecoordinatesof spiralbevel gears. The procedure is versatile and can beadapted to any type of spLralbevel g
4、ear with appropriate modifications. Various types of spiral bevel gears such aslogarithms, circular cut and zerol types are analyzed in this paper.A newprocedure for a theoretical determinationof exact tooth load contact line on the surface of the spiral bevel geartooth, which is vital for estimatin
5、g the stresses, has been developed. The generzliTedmodel developed uses threedimensional finite dement method with eight noded isoparametricbrick elements.Copyright 1993American Gear ManufacturersAssociation1500 King Street, Suite 201Alexandria, Virginia,22314October, 1993ISBN: 1-55589-597-2STRESSAN
6、ALYSIS OF SPIRAL BEVELGEARS A NOVELAPPROACHTOTOOTH MODELLINGCh. Rama Mohana Rao, Research ScholarG. Muthuveerappanj Assistant ProfessorMachine Elements LaboratoryDepartment of Mechanical EngineeringIndian Institute of TechnologyMadras, IndiaI. INTRODUCTIONSpiral bevel gear as a power approximation o
7、f logarithmic spiral, wastransmitting element finds extensive developed by Gleason works to facilitateapplications in engineering industries. For relative ease of manufacture.example, they are widely used inautomobiles for rear axle drives and in Huston et al 1-3 have studied thehelicopters for roto
8、r transmissions. The geometrical characteristics and thesuitability of these gears is that their distortions along the centre line of asmoothness in operation and large load circular cut spiral bevel gear. But thecarrying capacity at high rotational study concentrated mainly on crown (flat)speeds, g
9、ears. Litvin et al 4 have proposed amethod for the generation o2 spiral bevelThe tooth forms of spiral bevel gears gears based on parallel motion of aare curved in the form of spirals unlike straight line that slides along two matingthe straight bevel gears. Several spirals ellipses. An approach bas
10、ed on differentialare currently used in the design of spiral geometry is explained in the reference 5.bevel gears such as logarithmic spiral, thecircular cut spiral, and the involute The pitch surface of a bevel gear is aspiral. The logarithmic spiral has the cone and ralling pitch cones have spheri
11、caladvantage of providing a constant angle motion. Hence, the exact representation ofbetween tooth centre line and radial line, a bevel gear tooth is possible only on theat all points along the centre line. The surface of a sphere and is difficult tologarithmic spiral bevel gear provides visualize.
12、Therefore, the Tredgoldsuniform geometrical characteristics for the approximation which reduces the problem totooth profile in the transverse plane of one of spur gears is used in which thethe gears. The circular cut spiral, an tooth is represented as a plane surface.In the present work, spiral beve
13、l gear 3. GEOMETRIC MODELLING OF SPIRAL BEVEL GEARtooth model is generated using Tredgolds TOOTHapproximation. Since a detailed study isnot available about the contact line, which In this work the geometric modelling ofis more vital for the stress analysis using logarithmic spiral is explained first
14、,FEM, a geometrical method is developed to followed by circular cut and zerol typesgenerate the same on the surface of the with suitable modifications.tooth. The stress analysis was conducted ondifferent types of spiral bevel gears such 3.1 Logarithmic Spiral :as logarithmic, circular cut and zerolt
15、ypes using three dimensional finite The logarithmic spiral has the advantageelement method, of providing a constant spiral angle at any2. LIST OF SYMBOLS point along the spiral curve and is shownin Fig. l. The equation for logarithmica Addendum spiral is written asd DedendumE Modulus of elasticity r
16、 = ri e_(e-e,_) (i)F Face widthi Speed ratio where ri = inner radius of the spiralm Module r = radius at any point onN Numberof teeth the spiralp Circularpitch 8 = radial angle at any pointPb Basepitch on the spiral about thePN Normal base pitch centre linePx Axial pitch 81N= radial angle made by th
17、eS Thickness of gear tooth inner radius with theSpiralangle centerlinePitchangle _ = cotu Poissons ratiop Material density A logarithmic spiral is developed fore Angle of rotation the given spiral angle B between the inner2.1 Attached Words : radius ri and the outer radius rO in aplane sector and th
18、en the sector is rolledB Referring radii along back cone planeF Referring radii along front cone plane to form a cone with O as apex and 2_oasincluded angle, where _o is the pitch angle2.2 Suffixes : of the gear. Each spiral starts from one ofi Refers a section along the face width the nodal points
19、at the first section ofof the tooth the tooth at the toe side. The inner radiusj Refers a section in the particular of the spiral is the distance of the pointsection measured from the apex of the front coneo Along pitch circle and the spiral ends at the back cone on theb Along base circle heel side.
20、t Transverse sectionn Normal section Corresponding to any point P in a bevelgear tooth, two cones can be assumed with( There are other notations which are not apexes 0 and O(Fig.2) and are called frontmentioned above, but are defined at cone and back cone respectively. Theappropriate places in the t
21、ext) notations used to represent the variousdistances of a point on a bevel gear tooth, A right handed logarithmic spiral bevelbased on a front cone, a common central gear tooth is modelled in the present work.plane and a back cone are also shown in theFig.2. These terms are used frequently in In a
22、logarithmic spiral, the spiralthe description of the generation procedure angle is constant on the plane of a frontof spiral bevel gear tooth in the following cone but it is assumed to be varying withthe back cone radius at every section. Theparagraphs.spiral angle _j at back cone radius radiusAssum
23、ing the apex of the front cone RBj is given asas the origin for the coordinate system,the profile of the tooth at the first tan Bj = (RBj / RBo) tan _o (2)section of the tooth ( section at the toewhere 8o is the spiral angle measured atside ) is generated following Tredgoldsthe pitch line of the gea
24、r.approximation 6. The generatedcoordinates of the first section fall on aThrough each point in the first section,plane of the back cone at the toe and area spiral is generated on the surface of theused to generate the entire surface profilealong the tooth length, respective front cone. The calculat
25、edspiral angle _j at the point where thespiral starts, is assumed to remain thei q same from the toe to the heel side of the0 J tooth along the generated logarithmicspiral. The initial radius of the spiral is+ the distance of the point from the apex O.The radius of the spiral at each subsequentsecti
26、on is calculated as follows withFig.l Logarithmic spiral reference to the Fig.3.-back conepoints in the first section, the followingj_f_ _ values are defined_o = an-l(Nl/N2 ) (3)0aEx l l lj ) / Z lj (4)for i=l ; j=l,nlRFIj = /XIj2 + YIj 2 + ZIj 2 (5)Fig.2 Front and back cones in a bevel gear for j=l
27、,nlRFlo = mI NI/(2 sin _o) (6)Several such sections are considered at RFio = RFIo + (i-l) F /(n-l) (7)equidistant positions along the face width for i=l,nof the tooth and the respective coordinates RFij = RFIj RFio / RFIo (8)to describe these subsequent sections are for i=l,n ; j=l,nlgenerated from
28、those of the first section.3where n denotes the total number of (BCin) J = tan-l(Xlj/Ylj ) (15)sections along the face width and n1denotes the total number of nodal points inone particular section. The suffix j=orefers to the points along the pitch line. ,/_o%In a spiral bevel gear, each sectionalon
29、g the length of the tooth is out ofphase relative to the previous section andso each point in a section can beconsidered to be rotated by a definiteangle from the corresponding point in the xfirst section. The angle of rotation 8Cij 0of a generated point at the ith section of ELEVATION OF BEVEL GEAR
30、 TOOTHthe spiral with respect to the axis of thegear is calculated using the followingequation for the logarithmic spiral(Fig.4).RF_ilog (RFij/RFIj ) _- .8Fij = (9) _cot _j 0 01RCij = RFij sin 6j (I0) TOP VIEW W R.T.THE SECTION 001eCij = RFij eFij / RCij (II) Fig.3 Half section of bevel gearwhere 8
31、is positive along clockwisedirection for the case of right handlogarithmic spiral.8Cij is the angle of rotation of a point /_AI_.thin the 1 section from the correspondinginitial point, about the axis of the gear. _%_The coordinates of these points are _ _%1_I_ “3calculatedasfollows 0 I0,!“0/= cos OC
32、 +(OC. )Yij Reij ij _,_ j (13)Az 6 B2 = eF ij (measured on the plane of the oone)Zij = RCij / tan(_j) (14)B20 02 = _ j (measured with the axis of the gearwhere eCin is the initial angle of thepoint at the first section about thevertical axis of the tooth and it is Fig.4 Spiral AIB2 represented on th
33、edefined by surface of the front coneOne spiral is generated from each point The horizontal and vertical cutter settingsin the first section until the heel side is 7 are given byreached and the coordinates of the spiral H = Rm (21)at all the sections are calculated using V = Rc (22)Eqns.(12-14). A c
34、omputer program has beendeveloped to calculate the coordinates of The polar angle e is given bythe spiral bevel gear tooth using thismethod. A computer plot of a single gear e = 2 Tan -I V _ _ H2 + V2 - D2 (23)tooth model, and a full gear model obtained H + Uby this program are shown in Fig.5.2 R 2r
35、 + m3.2 Circular - Cut Spiral : Where U = (24)2rFig.6 shows the top view of circular-cut The single tooth model and full view oftooth centre line on pitch plane with gear the zerol bevel gear developed are shown incenter O and cutter center C. The cutter fig. 9.radius is Rc. The mid spiral angle _mi
36、Smeasured at the mean gear radius point Pm“The horizontal and vertical cuttersettings 7 are given byH = Rm - Rc sin _m (16)V = Rc cos _m (17)The polar angle e is given by 1= 2 Tan -I V _ _ H2 + V2 - D2 (18)H + Uwhere a) Finiteelementmodel b) Fullgearmodelr2 + R2m - 2 Rm R sin _mc Fig.5 Model of loga
37、rithmic spiral bevel gearU = (19)2rThe variation of spiral angle along ?circular-cut spiral tooth center line isgivenby sin _ = (r2- _ + 2 Rm Rc sin _m)2 r Rc (20) _pr _ _The single tooth model and a full view 0 _7-_-_-r4_of the circular cut spiral bevel geardeveloped are shown in Fig.7.C(H,V)Zerol
38、bevel gear can be considered as a Fig.6 Top view of circular-cut tooth centrespecial case of circular cut spiral bevel line on pitch planegears with zero mid spiral angle(Fig. B).4aa4_0 Noofnodes it may be due to the fact thatparameters.spiral angle of zerol gear is very smalland it varies from 0 to
39、 5 degrees ini. The maximum root stress decreases asbetween the toe and the heel sides.the pitch cone angle increases(Fig.18).12Moreover, it may be noted that the spiral 300angle of zerol gear is considerably less “iwhen compared to that of logarithmic andcircular-out spiral bevel gears.2.00t t fJ 4
40、50 degTable. 3 Gear Parameters and Material _ I50 _38.66degProperties Assumed E ooeo 55.69dog( eases29.74dog. 26.56deg1,00 ,J,lModule 4 mm 000 500 70.00 75.00 2000Distance along face w,dth(mm)Pressure angle 20 Fig.20 Variation of maximum root stress alongNumber of teeth on 24 face width for differen
41、t pitch angles.the pinion3.50Number of teeth on 24,30,36,42 and 48 _ C,rcularcutspiralthe gear |.300Facewidth 20mmShaft angle 90 _ 250Pitch angle 263429443341“ _20038039 “, and 45 Torque 6.9 kgf m _ t,_ ;5:06 deegg E 150 e_ae 35.69 degeeee_ 29.745 degkgflmm2“ _ 26.56 degYoungs modulus 21000Poisson s
42、 ratio 0.3 _00 . ,.,._0.00 5.00 ! 0.00 15.00 20:00Material density 7.85 gm/cc Distance along face width(mm)Fig.21 Vorlotlon of maximum root stress alongface width for different pitch angles.4.00 4.00.“ zeroIi ,_. t _3.50 _ “JB5.606ddee_g_3.50 “-“_-“e I, _3.00_*3.002.5oE=2.50 tt ! ! _ logarithmic _ .
43、 ,. .E eeooo circular E 2.00 33 69 deg 29.74 de9:_ “-“ 26.56 deg2.00 ._.,.,.,.,. 1.5020.0025.00 so.oo35.0040.0045.0050.00 o.oo 5.86 iG.bb i it is marginally higher with 13 F.L.Litvin, Theory of Gearing,NASA RP-1212, 1989.reference to zerol bevel gear. 14 O.C. Zienkiewicz, The FiniteElement Method, Mc Graw Hill Book Company,UK.(4) With reference to the three types ofgears considered, it has been noted thatthe location of the maximum root stress isobserved to lie in between the middle andheel regions; however in the case of zerol14
