1、06FTM11On Tooth Failure Analysis in Small-Teeth-NumberGearing: An Analytical Approachby: Dr. S.P. Radzevich, Eaton | Automotive, Innovation CenterTECHNICAL PAPERAmerican Gear Manufacturers AssociationOn Tooth Failure Analysis in Small-Teeth-NumberGearing: An Analytical ApproachDr. Stephen P. Radzevi
2、ch, Eaton | Automotive, Innovation CenterThe statements and opinions contained herein are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.AbstractThe paper pertains to analytical study of the major reasons that cause toot
3、h failure in gearing having smallteethnumber. Below,gearswithsmallteethnumberarereferredtoSTN-gears. Fortheanalysisofgeartoothfailure, tooth contact stresses, and the combined contact/shear stresses (CS-stresses) are investigating.Thestudyisbasedonthein-depthanalysisofthegeartoothloadingandonaccount
4、ingofvariationintimeofthetoothloadingandofothergearparametersinvariousphasesoftoothmeshing. TheearlierdevelopedbytheauthorDG/K-method1ofsurfacegenerationisusedinthestudy. Forthecomputationofthecontactloadsof parallel-axis gears, a novel load distribution model is proposed in the paper, as well as a
5、novel analyticalmethod for computation (a) of contact stresses, and (b) of combined CS-stresses in STN-gearing isdiscussed. The CS-stresses are caused by simultaneous impact of (a) contact stresses together with (b)stresses caused by the pinion/gear tooth profile sliding. To use the developed method
6、 of computation,minimum of preliminary assumptions are required. While developed for use in STN-gearing, the method isalso recommended for the accurate computation of stresses in gearing having regular teeth number. UsingcommercialsoftwareMathCAD/Scientific,acomputercodesforcomputationofbothofcontac
7、tstresses,andof the combined CS-stresses are worked out. The reported results of the research could be used forextendingoftheexistedstandardAGMA908-B-89totheareaofgeardrives comprisedof STN-gears,prised of gears having teeth number less than Ng= 12. The developed method for the computation isready t
8、o put in practice.Copyright 2006American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314October, 2006ISBN: 1-55589-893-91The DG/K-method is based on fundamental results obtained in differential geometry of surfaces, and on kinematics of multi-parametricmotio
9、nofarigidbodyintheE3space. Theinterestedreadermaywishtogofordetailstothemonograph:Radzevich,S.P.,FundamentalsofSurfaceGeneration,Monograph,Kiev,Rastan,2001,592p. CopyofthemonographisavailablefromTheLibraryofCongress.1On Tooth Failure Analysis in Small-Teeth-NumberGearing: An Analytical ApproachDr. S
10、tephen P. Radzevich, Eaton | Automotive, Innovation CenterIntroductionGears fail by pitting and wear as well as by toothbreakage. Gears having small teeth number areconsidering in this paper.Often, thoseinvolutegears arereferredtoasSTN-gears, base diameter db.gof which exceeds thegear limit diameter
11、 dl.g. When the inequality db.gdl.gobserves, thennot the entireactive toothprofileis shaped properly following the involute profile.The AGMA standard 1 encompasses involutegears having the minimal teeth number Ng= 12. Itcould be supposed that according to 1, the STN-gearsarethosehavingteethnumberNgd
12、fobserves for STN-gears.An example of application of small-teeth-number-involute gears (further STN-gears) is depicted inFig. 1. A gearset oftheTruetracdifferential (Fig.1)is comprised of two side-gears (having Ng= 15/28teeth number, normal pressure angle n=30, andpitch helix angle =38), which are e
13、ngaged inmesh with four to six pinions (having Np=5/7teethnumber) 2, 3.Gears of the automotive differentials are a goodexample of STN-gears.Because of small teeth number, many features oftheSTN-gearingobservewhenoperating. Thisre-sults inthat conventional approach for computationcontactstressesisnot
14、applicableforcomputationofSTN-gearing. This is due to the known approachwas developed for gearing with ordinary teethnumber.Figure 1. An example of application of helical STN-gears 2, 3.2Literary SurveyThestressescausedbythepressurebetweenelas-tic bodies are of importance in connection with thedesig
15、nand/orinvestigationofSTN-gears. Hertz4was the first who developed the mathematicaltheory for the surface stresses and deformationsproduced by pressure between curved bodies, andthe results of his analysis are supported by experi-ment. Formulasbasedonthistheorygivethemaxi-mum compressive stresses, w
16、hich occur at thecenter of the tooth surfaces contact ellipse. Unfor-tunately, they allow computation neither the maxi-mum shear stresses, which occur in the interiors ofthe compressed teeth, nor the maximum tensilestresses,whichoccurattheboundaryofthecontactarea and are normal thereto.Hertz based h
17、is work on the assumption that thecontact area was small compared with the radii ofcurvature of the contacting bodies. Because of thevery small area involved in what initially approxi-mates a point or line contact, contact stresses foreven light loads are very high; but as the formulasshow,stressesd
18、onotincreaseinproportionofload-ing. Furthermore, because of the facts that thestress is highly localized and triaxial, the actualstress intensity can be very high without producingapparentdamage. TomakeuseoftheHertzformu-las for purposes of design or safe-load determina-tion, it is necessary to know
19、 the relationshipbetween theoretical stresses and likelihood offailure, whether from excessive deformation orfracture.Bothsurfaceandsubsurfacestresses werestudiedbyBelayev 5. Timoshenko6 wasamongthefirstwho used the Hertzs method for investigation ofstrength of gear teeth.Tons of research had been c
20、arried out in the fieldsince the first publications on the topic 4, 5, 6,7, 8, 9, 10, 11, 12, 13, and others. Manyefforts have been taken aiming use of FEM (FiniteElement Method) for solving the problem of com-putationofcontactstressesingeartooth. However,most of research pertained to computation of
21、 justcontact stresses in gearing with relatively big num-ber of teeth. Norelativemotions (sliding)of thepin-ion and of the gear teeth are put into account.If the motion involved is a true rolling without anyslip,thenunderconditionsofslowmotionthestressconditions are comparable with those produced by
22、staticloading. Ifthereisevenaslightamountofslip,however,theconditionsareverymuchmoresevereand failures likely to occur through mechanicalwear. Theonlyguidesofartoproperdesignagainstwear is real or simulated service testing. Whenthemotion involved is at high speed and produces cy-clic loading, as in
23、gear train, fatigue is an importantconsideration.The practical design of gears that sustain directbearing is based so far largely on experience sincethisaloneaffordsaguideastowhether,atanygivenload and number of stress cycles, there is enoughdeformation or surface damage to interfere withproper func
24、tioning.Statement of the Problem:Todevelopananalyticalmethodforcomputation(a)oftoothcontact stresses,and(b)of toothcombinedCS-stresses for gearing having small teethnumber.For computation of the contact loads of parallel-axis STM-gears, a novel load distribution model isdeveloped (Fig. 2). It is of
25、critical importance tostress here that the proposed novel model for thecomputation of the contact loads is valid for thecases when the neighboring contact lines do notoverlap each other. The scenario when the neigh-boring contact lines overlap each other deservesbeen considered in a separate paper.T
26、heconsiderationbelow relates tothecomputationof contact stresses for STN-gearing of all of de-signs, say for cylindrical gears, bevel and hypoidgears, worm gears etc. However, due to limitedlength of the paper, the computation of contactstresses just in helical STM-gearing is consideredasanexample.
27、Thereportedresultsoftheresearchare also recommended for the computation of con-tact stresses for gearing having regular teethnumber.Major Features of STM-GearingGearingwithsmall number of teeth significantly dif-fers from gearing with regular teeth number. Themajordifferencesbetweenthegearingofbothk
28、indsare briefly summarized below.3Figure 2. Normal load per unit length wNof asingle line of contact (The load wNis appliedin the direction within the plane of actionperpendicularly to the line of contact).Features of tooth geometry. Because of smallteethnumber,STM-gearingisfeaturingaconsider-ableva
29、riationofcurvaturewithintheactivelengthofthe line of action. As an illustrative example, thevariation of radii of normal curvatures of a pair ofspur involute gears, as well as the variation of thenormal curvatures themselves are shown in Fig. 3.Toothprofilecurvatureof thegear gandof thepin-ionpchang
30、elinearly withinthe activelength of theline of action. Changes of normal curvature of thegear kgand of the pinion kpfollow hyperbolic rela-tionship. Relativenormalcurvaturekrelisminimalata point that corresponds to the middle of the centerdistanceC,anditincreasestowardsaxisofboththepinionandthegear.
31、 Forfurtheranalysis,it iscriticalto point out here that maximal value of the relativecurvature kmaxrelobserves at the point of the line ofaction that corresponds to outside diameter of thegear do.g, and the limit diameter of the pinion dl.p.Similarly, maximum of relative curvature kmaxrelob-serves f
32、rom theoppositesideof theactivelengthofthe line of action, at the point that corresponds tooutside diameter of the pinion do.p, and the limit di-ameter of the gear dl.g. However, the inequalitykmaxrel kmaxrelis always valid the equalitykmaxrel= kmaxrelobservesjustincaseswhenNg=Np.Figure 3. Variation
33、 of the major toothcurvatures for spur STN-gearing havinginvolute tooth profile.As it can be seen from Fig. 3, the teeth flank curva-tures change progressively while the point of con-tact moves from Pgto Ppon the line of action, therelative radius of curvature changing correspond-ingly. The semi-cir
34、cle constructed with the line ofactionNgNpasdiametercanbeshowntorepresent,to an appropriate scale, the termg+ prel4below, the square rootg+ pis designatedas a. This is the term, which determines thechange in surface stress at the point of contactwhilst it moves from Pgto Ppon the line of action.Near
35、 the point Ng, the product a relapproacheszeroandhencethesurfacestressapproachesinfin-ity, and it will be deduced that the surface stress atthetipof thewheel, point Pp, is usually greater thanthat atthetipof thepinion, pointPg. Variationoftherelative curvature relitself is constructed on thepremises
36、 of change of a relwithin the segmentPgPp. TheinterestedreadermaywishtogotoFig.4for details of construction.An intermediate remark. It is of importance topoint out the readers attention here tothe pointTRat the line of action at which the normal relativecurvature krelreaches its minimal value(krel=
37、kminrel), and, correspondingly, the normalrelative radius of curvature relreaches its maxi-mum (rel= maxrel) (Fig. 3). The point TRdividesthe entire line of action NgNpon two equal portionsTRNgand TRNp, and therefore, the equalityTRNg= TRNpis valid. The following equations forcomputation of coordina
38、tes of the TRare derivedrT.p=12d2b.p+ C2sin2trT.g=12d2b.g+ C2sin2t andcosvg=r2T.g+ C2 r2T.p2 rT.gCandcosvp=r2T.p+ C2 r2T.g2 rT.pC(1)Eg=rT.gcos vgand Ep=rT.pcos vpIf twoidentical gears areinmesh, thenthepointTRcoincides with the pitch point P.Equations (1) could be expressed in terms of pa-rametersof
39、designofthegearandofthepinion. Forthe readers convenience, derivation of Eqs. (1) ispresented in Appendix A.Whenthetoothratiosignificantlyexceeds theunity,then the point TRis located out of the active part ofthelineofaction. Thismightbethemajorreasonforwhy this point has not attracted attention of g
40、earprofessionals earlier.Actual location of the point TRcould be of criticalimportance for STN-gearing. Because of relativecurvature of the gear and of the pinion tooth sur-faces is minimal exactly at the point TR, it is highlydesired this point being located within the activeprofiles of thetooth su
41、rfaces. Moreover, it locationclosetothepitchpoint is desired. This wouldresultin reduction of contact stress.ThedesiredlocationofthepointTRiswithintheac-tive portion of the line of action (Fig. 3). Equations(1) yield analytical determinations of conditionsunder which this requirement is satisfied. I
42、nequali-ties (rT 0.5 do.gand rT 0.5 do.p) describes thedesired location of the point TRwithin the activeportionofthelineofaction. Theseinequalitiescouldbe expressed in terms of parameters of design ofthe STN-gear drive.Equations (1) reveal that two points TRexist for agiven gear train. One of them i
43、s for a certain direc-tion of rotation of the gear and of the pinion, andanother one is for the opposite rotation of thebodies.Evidently,thepointTRexistsforanyandalldesignsofgeardrives:For(a)helicalgears,(b)bevelgears,(c) hypoid gears, (d) spiroid gears, etc. For spatialgearing, a three-dimensional
44、TR-curve observesinstead of the TRpoint. Generally speaking, ananalytical representation of the TR-curve could bederived on the basis of equality that is equivalent toTRNg= TRNp.Coordinates oftheTRpoints couldbeemployedforthepurposeofderivationof ananalytical criteriontodistinguish STN-gearing from
45、ordinary gearing.Similarly, a graphical interpretation of variation ofothergeometricalparameters ofthegearandofthepinion within the active length of the line of actioncan be constructed.The plane of action for a STN helical gear drive isshown in Fig. 5. Here, Lpcdesignates length of thepathofcontact
46、intransversesection,pb.xistheaxialbasepitch,pb.tisthetransversebasepitch,andFisface width.5Figure 4. For the determining of the function rel.6Figure 5. For the determining of the function rel.7Without goingintodetails, anillustrativeexampleofcomparison of a STN-gearing with gearing havingregular tee
47、th number is presented in Fig. 6. Here,the STN-pinion having teeth number Np= 6 (Table1) is compared with a gear having teeth numberNg= 60 while other parameters of it design remainthe same.Table 1. Design Parameters of theSTN-PinionNumber of teeth 6Normal pressure angle 30Helix angle 40.5526,LHPitc
48、h diameter 0.7897Diametral pitch (nor-mal)10/12.59Circular tooth thick-ness0.2024-0.2004Outside diameter 1.022-1.018Root diameter 0.668-0.653Face width 1.406-1.402The interested reader may wish to go to 14 and15 for details of the comparison.Similar graphs (see Fig. 3 and Fig. 4) could beconstructed
49、 for helical STM-gearing having invo-lute tooth profile as well.Variation of the resultant force Fwithin the activeportion of the line of action is one of the major rea-sons for vibration and gear noise excitation.Thecomparisonillustratessignificantdifferencebe-tween the STN-pinion and pinion having regularteeth number (No acceleration/deceleration of therotating pinion/gear is considered in the compari-son. For the case of rotation with
