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1、91 FTM $1 A W Finite ElementStress Analysisof a GeneticSpur Gear Tooth by: EugeneA. Tennyson,The Universityof Tennessee American Gear ManufacturersAssociation III TECHNICALPAPER Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/111111100
2、1, User=Wing, Bernie Not for Resale, 04/18/2007 11:59:44 MDTNo reproduction or networking permitted without license from IHS -,-,- Finite Element Stress Analysis of a Generic Spur Gear Eugene A. Tennyson The University of Tennessee, Space Institute TheStatements andopinionscontainedhereinare thoseof
3、theauthorandshouldnotbeconstruedasanofficial actionor opinionof the American Gear ManufacturersAssociation. ABSTRACT: The prediction of bending stresses in a gear tooth, resulting fxom an externally appliedtorque, requires special consideration when designing spur gearsystems. The tooth geometry is
4、such that excessrisers exist which must be accounted for. In addition,variablesaffectingtheexactloadpointon thetoothand thedirectionofthe applied loadare critical. Aninteractivepreprocessorisdevelopedwhichgeneratesall theinformation, includingadetailedtoothprofile, necessaryto perform afiniteelement
5、bending stressanalysisofthe gearsystem. Tovalidatetheprocedure, a test group of spur gears is identified and analyzed. The results are then compared to those obtained via the American Gear ManufacturersAssociation(AGMA)standards. The comparisonrevealed thefiniteelementstressesto be slightlymore cons
6、ervative than the corresponding AGMA standard stresses. A generalized stress equationand geometry factor, based on the finite element approach,are also introduced. This paper is intended only as a proof of concept. Copyright 1991 American Gear ManufacturersAssociation 1500King Street, Suite201 Alexa
7、ndria,Virginia,22314 October,1991 ISBN: 1-55589-615-4 Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:59:44 MDTNo reproduction or networking permitted without license from IHS
8、 -,-,- IA FiniteElementStress Analysisof a Generic Spur Gear Tooth EugeneA. Tennyson“ TheUniversityof TennesseeSpaceInstitute,Tullahoma,Tennessee37388 The predictionof bendingstresses in a gear tooth, resultingfroman externallyappliedtorque, requiresspecialconsideration whendesigning spurgearsystems
9、.Thetoothgeometryis suchthatstress risers existwhichmustbe accountedfor.In addition,variablesaffectingthe exactload pointon the toothandthedirectionoftheappliedloadarecritical.Aninteractive preprocessor is developedwhich generatesall theinformation, includingadetailedtoothprofile,necessary to perfor
10、ma finiteelement bendingstress analysisofthe gearsystem.Tovalidatetheprocedure,a test groupof spur gears is identifiedandanalyzed.The results are then comparedto those obtainedvia the AmericanGear Manufacturers Association(AGMA) standards. The comparisonrevealedthefiniteelementstressesto be slightly
11、moreconservative thanthe corresponding AGMAstandardstresses.A generalizedstress equationandgeometry factor,basedon thefiniteelementapproach,are also introduced.This paper is intendedonlyas a proofofconcept. Introductionrepresents the transmitted tangential load and F the The selection and design of
12、spur gear systems istoothface width. AlthoughEq. (1) represented primarily guided by the anticipated bending stressessignificant progress at the time, it is not very accu- in the loaded tooth. Indeed, excessive bending stressrate in view of todays computationalcapabilities. in the fillet region is o
13、ften the cause of gear failure. It is thereforeimportantto have the means to_e reliably predict these stresses,w_ Wilfred Lewis1,in 1893,proposed a bending stress formula, which for the first time, took into accountLood Pof_4“_n the form of the tooth. To this day, this formulai If-ehc_-e._ remainsth
14、e basis for most gear design. The gear tooth is assumed to be a cantilever beam of uniform cross section, rigidly fixed at the base as shown in Figure 1. The “theoretical weakest section“ AB is thenlocatedby inscribing a parabolawithinthe tooth outline. The parabola should be tangent to the fillets
15、on either side, and its vertex is at the pointFig.1 lewis Parabola of UniformStrength where the line of action crosses the center line. The bending stress a/: at AB is then determined by theIn 1942, DolanandBroghamer 2 conducteda following Lewis equationphotoelasticstudy of stresses in gear teeth wh
16、ich providedmuch detailedinformationaboutthe 6Wtnature of stress distributions in the neighborhood of L = hFt(1)fillets. This study also yielded a new stress concen- tration factor K which, when applied to the Lewis in which t is the length of AB and h is the distanceequation, results in a more real
17、istic value for the from C to AB in Figure 1. Furthermore,Wtbendingstress at the toothfillet. Thefactor K reflects the combined effect of the tangential compo- nentWtand the radial componentWrof the external load W on the tooth. As an example, the *GraduateStudent,Dept. of Mechanical Engineering.str
18、ess concentration factor K for a 20 pressure angle stub tooth is given by Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:59:44 MDTNo reproduction or networking permitted with
19、out license from IHS -,-,- It_o.15f t1 0.45Model Definition K = 0.18 + I.rfJI.hJ(2)ToothProfile One of the most widely accepted gear tooth pro- where .rfrepresents the fillet radius of the tooth asfiles is defined by the involute curve which provides shown in Figure 1.a naturalline of contactfor two
20、 matinggears. The AmericanGearManufacturersAssociationBefore a finite element grid can be generated, it is (AGMA) incorporated both the Lewis equation andnecessary to derive a set of cartesian coordinates the Dolan and Broghamer stress concentration factor(z;i, YIi ) for a generic point on the tooth
21、 profile. K into its standard.The AGMA bending stress for a statically loaded standard addendum spur gear withz85 00 helix angle is given by“_. 2.80 _votute AGMA=FJ(3)“g zTs _oid /_2.70 whererAGMAis the bending stress number, Rp the pitch radius of the gear, and Pd the diametral pitch.2.65 The Geome
22、try Factor Ydepends on the Lewis Formz60 Factor Y as well as on the stress concentration factor III /_ In practice then, Eq. (3) is often used to deter-0.o437o._75 o.m2 mine the maximum static bending stress in a spur1/2Tooth Thickness (In) gear tooth subjected to the usual operating loads. An alter
23、native to the AGMA standard approach toFig.2 Involuteand Trochoid Curves computing bending stresses in gears is providedby the finite element method. The possibility of usingOne of the more popular methods for cutting the finite element method is very appealing, espe-involute teeth is the hobbing pr
24、ocess 4-5.This process cially tothe designer whois only occasionallyconsists of traversing a straight sided rack across a responsible for gear selection and application. Inrotating gear billet. As the rack moves across the addition, it is expected that, when properly imple-rotating gear an involute
25、is generated on the gear mented, the finite element approach willyield highlytooth blank. Inaddition, a fillet is cut in the root of accurate results,the tooth in the form of a trochoid curve as depicted In the course of the present study an interactivein Figure 2. Equations for both these curves ca
26、n be preprocessor was developed which queriesthe userderived. Assuming a base radius Rb is given, it can for minimal information concerning the spur gearbe seen from Figure 3a that the profile angle _bpi system at hand. The preprocessor outputs all thecorresponding to a generic point on the involute
27、 at information needed by a standard finite elementradius ri is given by program to perform a stress analysis of the gear tooth.In order to validate the procedure, atest_pi=sin-1(R_/ri)(4) groupof spurgear systemswas identified and analyzed using thefinite elementmethod. The resulting bending stress
28、es are compared with thoseThe involute angle0i isthen defined as obtained via the AGMA standard Eq. (3). In addi-O i = t_ntbpi- dppi(5) tion, a new generalized bending stress equation and geometry factor are proposed, based on the finite elementresults.In the next section, the modelAlso, from Figure
29、 3b it can be seen that definitionfor a generic spur gear tooth will beti discussed.Oa= 0i +_r i(6) 2 Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:59:44 MDTNo reproduction
30、or networking permitted without license from IHS -,-,- A Yli = ri cos Obi(1O) Thecuttingedge of thehobrackgeneratesthe shapeof the root fillet. This shape,which is called the trochoid,is boundedby the dedendumcircle and the involutecurve as shown in Figure 4a. Fig. 3a The Involute Curve m F?4aThe Tr
31、_hOidTiUl_e UsingEqs.(4-6), it is now possibleto determine thecrdinates(xliYli)fagenericpintnthe_/_Ibz involute.First,for a given tooth thicknesstp at the pitchradius,thetooththicknessti at any other radiusri can be found as ti=2ri_Rp+Oa-Oil(7)Fig. 4b Hob Rack Geometry Inorder to define the cartesia
32、n coordinates (XTi, YTi) whereOa is definedby Eq. (6) for ri = Rf.In orderof a generic point on the trochoidcurve, it is neces- to obtainthe coordinates(xli,Yfi)about the toothsary to definethe distance L in Figure 4b. L is the centerlinethe angle0bi is defineddistancefrom the point Z to the hob too
33、th center line. The point Z is the center of the hob tip radius R h- Obi= Oa- 0i = Op+tp_0i(8)DefiningT has the hob tooththickness,and Tp as 2Rpthe tooththickness at the pitch radius, it is seen that so that x,ii = risin Obi(9)Th = _r/Pd - Tp(11) Copyright American Gear Manufacturers Association Pro
34、vided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:59:44 MDTNo reproduction or networking permitted without license from IHS -,-,- bz= b- Rh(12)where andtan/3i -dzZi=-RP_iein_Ti+bzcs_i(18) -dYzibzsila_Ti + Rprlicosrli L=2T“ -bztand
35、_ffr!(13) eos_ppFinally, transferringthe trochoid coordinates to the gear tooth coordinate system via the angleA in where _pp is referred to as the standard pressureFigure 4a, it follows that angle, or the tooth pressure angle, andis defined by Ex. (4) for ri =Rp.Furthermore, b is thededen-ZFi=YTisi
36、nA-zTiCOSA(19) alumradiusand be is shown in Figure 4b. As the gear blank rotates over an angle ni,the_lFi=YTiCOsA+ZTisinA(20) hob traverses a distance niRp.The exacttrochoidal curve can then be generated by letting ,7 i varyfromwhere 0 to 7r/2. Thecoordinates(Zgi,7Zi) of the tro- choid center Z are
37、found asA= _PdL(21) 2RpRp _gi=(Rpr_i)eos_i-(Rp-bz)ein_i(14) Equations (9-10) and (19-20) can now be used to determine an exact profile for a generic spur gear Yzi = (RP - bz)cos_i + (R/ _i )ein_i(15)tooth. Equation (18) must be numericaltysolved for _i between 0 and _r/2. In addition, the involute c
38、oordi- Next, thecoordinates(gTi_YTi)of the correspon-nates must be evaluated from the base curve where 0i ding pointon the trochoidcan be obtainedbyis 0, to the top of the land radius where ri is equal adding to (zg i, Ygi) the coordinates of the hob tipto Rp. And finally, the intersection of the in
39、volute radius Rh .and the trochoid must be found in order to deter- vrmine the exact transition point. The analyst is now in a position to generate a finite element grid for a xrgeneric tooth. To apply a given load to the tooth, it iis necessary to determinethe coordinatesof the applicationpoint of
40、the load. x7 ToothLoad ,Spur gears develop critical bending stress when loaded at the highest point where a pair of teeth are still in contact. This point is called the Highest Point of Single Tooth Contact(I-IPSTC). This point is Yzused as the load applicationpoint only for high quality gears; the
41、toothtip is used for standard quality gears. Referringto Figure6, the normal operatingpressure anglen_,for a gear set with Fig.5 TroehoidCoordinatespitch radii Rpa and Rpp is given by Letting fl be the angle formed by a line normal to_n_= (Rpc +Rpe)sind,pp-(Rpp+a)2_(Rpe_b)21/2 the trochoid generated
42、 by the point Z and the YT axis, as shown in Figure 5, it is seen that(22) To determine the HPSTC, consider again Figure 6, ZTi = zgi+.R h cc_s _i(16)fromwhichit followsthat YTi-Ygi - Rh sinl_i(17)Rl= Rp +a(23) Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA
43、 Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:59:44 MDTNo reproduction or networking permitted without license from IHS -,-,- A _o = ee-1 (R_/RI)(24)finite elementmodel are taken as shownin Figure7. The boundarybetweenthe toothand the rest of the Equations(23) a
44、nd (24) are valid for boththe geargear is pinned.This allows for the necessaryfreedom G and the pinion P. Furthermore,of movementwhen bendingloads are applied. In this study the tooth stress problemis considered AO=ER_p+R_p_2Rt_pRtpeOS(,oG_eb1_p)l/2to be a plane stress problem.This presumesthat thel
45、oad is uniformly distributedalong the width of the (25)tooth and that the front and back faces are allowed to freely expand. 2_rRbp AC =(26)Results Np Stresses OC=AC-AO(27)The principlestressesfor eachstaticallyloaded toothweredeterminedusingaquadraticfinite Opelementmeshas shownin Figure7. Thismesh
46、 _prvedt be adequati_:_ : Fig. 6 Geometry ofMeshingTeeth Therefore,the locationRe of the HPSTCfor thec gear is found to be Fig. 7 Finite Element Mesh and Boundary Conditions R=ttp, cTU=ER_G+OU2_2R_,aOCeos(Cjoa_tbp_,)1/2for all cases in termsof convergencerequirements. (28)A totalof twenty gear sets
47、were used in this study. Finite Element ModelThesetsconsistedoffivespecificgears,each Whenthetoothmodelaccuratelyreflectsthematchedwith three pinionsand a hypotheticaltip geometry,a finite elementstress analysis can yieldloading.For each gear and pinionset the HPSTC highly accurate results. By using
48、 the analyticalequa-was determinedandincorporatedintothefinite tionsfor the involuteand the trochoid,it is possibleelementmodelin orderto producemoreaccurate to generatean accurateprofileof the tooth.Fromresults.Table1 lists the specificationsfor the partic- this,a finiteelementgridfor thetoothcan beular set of gears used. establishas shown in Figure7.The actual maximumprinciplestress occursat the Next,theHPSTCis locatedanalytically.If thetoothfillet as expected.Due to the geometryof the exactlocationof theHPSTCdoesnotcooincidetooth,the maximumcompressivestress occurs at the withnodeloca