CALCULATION OF LOAD CAPACITY OF GEAR TEETH BY SIGG

CALCULATION OF LOAD CAPACITY OF GEAR TEETHBYH. SIGGChief Design Engineer

I N D E XIntroduction1. The formulae by ISO2. The MAAG method to predictthe scoring resistance3. The AGMA approach; Service factor

Calculation of load capacity of gear teethIntroduc ionThe load capacity of two meshing gears is limitedby three main factors:1. Bending strength of the teeth.2. Surface durability of the tooth flanks.3. Scoring resistance of the tooth flanksThere are two main methods known today to predictstrength and durability with good accuracy:1. Calculation by ISO (International Standard Organisation)2- Calculation by AGMA (American Gear ManufacturersAssociation)The ISO-standards are used predominantly by Europeangear manufacturers.The formulae given· in the MAAGHandbook are based on these.In the United States gears are calculated by the AGMAstandard.Basically However the two methods are not too much different.there are some discrepancies, especially wherepermissible tooth loads are concerned; e.g. surfacehardened gears are not yet as widely used in the U.S.A.as in Europe.It is probably for this reason that therelation between the allowable stresses for surface hardened and through-hardened teeth is smaller in theAGMA-standards than in the ISO recommendation.The main formulae by ISO are listed in this paper.the discussion of the various factors i s outside theBut

- 2 scope of this paper;we refer to a pUblication byVDI (1).The fin a l ISO-st a ndard has not yet been published.It is e xpect e d that it will be available by1974.AGr-1A has introduced the so-called "Service Factor".This factor and its calculation are discussed in thispaper.The third criteria, the calculation of the scoringresistance, has gained in importance in recent years.Especially in surface hardened, high speed/high powergearing it is th e scoring limit together with thetooth bending strength which determine the size ofthe gear.However, neither AG MA nor ISO have so far provideda formula to predict the tooth loading where scoringmust be expected.In this paper a calculationm thodis described which is based on Prof. Blok's flashtempera tu re theory and which has been in use by MAAG formany years.

1.:;;-ZU""CHThe formulae by ISOOn pages 5 and 6 the formulae are listed for thecalculatior. of tooth root strength and surfacedurability.A graph shows the relation betweenthe various stresses and the meaning of the safetyfactor.The nominal stress (fff ; ()Hf) is based only onF rerethe nominal powe r to be transmitted by the eear, on thetooth size (module) and on the tooth geometry .It is the stress which exists under ideal conditionsand can be calculated with good accuracy.The actual stress (SF; 0H) is arrived at bymultiplying the nominal stress by the variousK-factors which consider load distribution betweenthe teeth and across the face width, dynamic loadsdue to tooth errors and overloads inflicted bythe driving and driven machines.The limit stre sse s (oF lim; 0H lim) have beenfound by fatigue testing of various materials,through-hardened and surface-hardened ir. differentways.The ben d ing fatigue strength of a standardtooth form e.g. has been measured on the pulsatingmachine.The durability or resistance againstpitting has been established by testing actualgears in a back-to-back test arrangement or byrolle tests etc.The max. allowable stresses (oFPD.}JPmax;1maxare based on the limit stresses corrected by)various factors which have an influence on thestr es s limits.In tooth bending the size factorKmust be considered for cases where the moduleFX

is considerably larger than the module ofthe test ge ar. Similar factors are introduced indetermining HP: The lube oil has anmaxinfluence, also the peripheral speed at thepitch circle, the surface roughness of thetooth flanks and the hardness ratio of thepinion and gear material.The safety factor is simply derived by dividingthe max. allowable stress by the actual stress.

- 5 1.1Calculation of tooth root stren gt h ( ISO-Proposal)yF limKFx FP maxSF6FKFo(KFf3KvKI;/; './.-;:/ . /;/// ././/',./. /·'/ /I . // / ././.,'".Nominal tooth root st r ess :mnActual tooth root stress:Max. allowable toot h root stress:6.FP max --(5F limSafety factor:SF 0FP max----6" FI X(tooth fr a cture)6Fref

- 6 1.2Calculati o n of su rfa ce dur a bility(ISO-Pr o po s al)(Hertzi a n s tress e s)6H limKL ZRCHZKHXv- ,0' HP maxSH6HKF,.,Kp 'I 'Kv(5KI.H r ef///""J/"---INo minal t o oth surf a ce s tr e ss:(5H ref 11Wtdlu 1uZH.ZActual t o oth surfac e str e ss:Max. allowabl e tooth surface stre s s:Safety fac t or': 6 HP max(Pi t tin g )M ZE.

- 7 Symbols:tmndlu'0 'Il0HNorrna 1 modulemm -.(1).J-}Pinion pitch diameterSpeed ratioTooth form factor;YFYErl 'Il·rlTooth load per mm of face vlid thkp/mmWl/ YsStress concentration factorZHTooth flank form factorZMZE.Material factor p .r1Longitudinal load distributior. factorContact ratio factorOverload factorDynamic factorvKFo(KLZvZR'CHC'(Transver s e load distribution factorKHo( XSize factor Lubrication factorSpeed factorCRSurface roughness factorHardness ratio factorLube oil factor (additive s )Lube oil factor (viscosity)CLCSurface treatment factors0.·rlEE(1)orirl.J-}(/)H0.J-}OF lim H limTB limcokp/mmkp/rruTI c22Tooth s trength, bendingSurface durabilityFlash temperature limitSFSafety factor, tooth bendingSHSafety factor,tooth surfaceSBSafety factor,scoringC) :,.;Contact ratioHelix factorKrKU).J-}page 123Y(3Cil2: AG-Hb

-0-2. The MAAG method to predict the scoring resistanceThe formulae are based on the theory by Prof. Blok onthe flash temperature which occurs at the contact pointsof two tooth flanks when going through mesh. Thistemperature is a criteria in predicting the danger ofscoring (2).The highest temperatures occur at the tips of all thepinion and gear teeth where the sliding speed is largest.Experience also shows that it is at these pOints wherescoring takes place first. If the flash temperature exceedsa certain level the oil film between the tooth flanksevaporates and looses its capability to carry load. Therefore the factor "lube oil" has a big influence on thescor ing limit.As a point of reference the limit flash temperature hasbeen determined from experience and tests for a straightmineml oil v.;rithout EP-additives, having a viscosity of04 E / 500C (30 cst /500C).Furthermore, this limit temperature is valid tor newlygrowld gears wi th a surface roughness of ""1 fm (25 finch).TB lim 1400CIf for a partic ular case any of the above factors differfrom the values given, the limit flash temperatureTB l'lrnis corrected accordingly by the respective C-factorto obtain the max. allowable flash temperature TBP max.Calculation of the nominal flash temperature -----.------At the tooth tip of the pinionAt the tooth tip of the gearATB ref 2 A . f2

- 9 -vAEd:- cos wkg/mmTooth load per mm of face widthContact ratioWorking pressure angle (transverse)vm/sPitch line velocityammCentre distanceThe factors fl and f2 are taken from the graphs Fig. 3and 4. They are plotted as a function of the speed ratio uand a factor kl and k2 resp.,k1 (u u1) ." (1 tgc{tv )tgo( alvlhereby: coseX"a lJ whereby:tgoCtwt·g 2 d bl 1idal--cos 0 2asee fig. 2The actual flash temperature is obtained by multiplyingthe nominal flash temper a tures T1 and TBref 2 withBrefthe same K-factors which were applied to calculate theactual surface stress.

-.Lv-Similar to the strength and durability calculations,we can derive a S a fety F a ctor SB for scoring:(Blok/MAAG- fla sh temp. theory)TBP m a x - - - r - - - - IIITB limC"? CLCR CsTBP maxSBTB Kr Kv///:/ /',/ /'TB ref. . / /-./".-/./""" .Nominal flash temperature:TBrefA .f1PinionTooth-TipActual flash tempe rature:A .f2GearTooth-tip

Max. allowabie fl a sh temperature:TBP maxSafety factor(scoring)T BP maxTBComments:As already mentioned, the calculation of the scoringlimit is relatively new, but from practical experiencea good deal is known today in this field. By comparingthese results with the calculated figures, we found thatthe flash temper a ture theory is sound and a goodinstrument to predict the scoring limit. Still:. thevarious C-factors listed above are not yet known adequately. Until further facts are available we haveabandoned the idea of calculating a Safety factor. Forthe time being we simply calculate the nominal flashtemperature TB ref multiplied by the overloadf ctorKrFor each gear app lication we know from practical experiencewhat flash temperatures are permissible to assure a soundsafety against scoring.\'1i th growing villowledge and experience \'Je sha 11 eventuallybe able to calculate with good accuracy a safety factorSB as laid out on page 10.

3.The AG MA-Approach; Service FactorAGMA has issued two standards for the rating of singleand double helical gear teeth:1) Rating the strength of gear teeth: AGMA 221.022JRating the surface durability of gear teeth: AGMA 211.02These two standards contain the basic formulae for the calculation of the tooth bending stress and surface stress(Hertzian stress). They also give allowable stresses forvarious materials and hardness. These standards aregenerally applicable throughout the gear industry as faras helical teeth are concerned.For a number of special gear applications standards areavailable which reflect the individual design practicein that particular field, e.g.Rolling mill gears:323.01Speed reducers and increasers:420.03High speed gear units:421. 06etc.These individual design practices are all based on thestandards 211.02 and 221.02.3.1The Service FactorThe service factor is best described in the AGf"lA StandardPractice for High Speed Helical and Heringbone Gear Units,AGMA 21.06. It is a well defined factor and relates the

so called "rated ho rsepo'V'ler" to "Service horsepower".For each mesh there are three factors to be calculated:1. Service factor for tooth bendin g strength, onefor the pinion- and one for the gear-tooth:horsepower rating strength)service horsepower2. Service factor for surface streng th:pCSF: achorsepower ratingjdurabilityscservice horsepower,IPThis factor is the same for pinion and gear because thesurface stresses at the contact point of the two flanksare always equal.The lowest value of the three is the Service Factor ofthe gear.The horsepower rating:; P t and Pare the max. allowableaacpowers to be transmitted hy the gear, based on toothstrength and durability respectiveley. The strengthrating is calculated using the formula of AGMA 221.02,the durability rating is derived from AG!11A 211.02. AllK- and C-factors are taken as unity except for:- Lo a d distributi on factors Kmand CmThese factors are derived fromStrength:Durability:KmCm-AGMA 221. 02Table 2AGMA 211. 02Fig.4,first red curve

1.-Dynamic factors Kvand Cvv-AGMA 221. 02Fig. 6 curve 2-AGr-w. 211. 02Fig. 6 curve 3Strength:KDurability:Cv. ,:; The horsepower rating depends further on:Gear dimensionsSpeed ra tioTooth geometry and moduleMaterial and surface hardnessBy definition, the service horsepower is equal to themaximum continuous horsepower capacity of the primemover. Therefore, it would be wrong if a buyer wouldorder a gear for a somewhat higher horsepower thanneeded, "just to be on the safe side". This safety istaken care off by the service factor AGMA 421.66,table 3, recommends service factors for various gearapplications.It is important to note that the Service Factor doesnot include gear tooth accuracy attribution factors Kmand Cand the dynamic factorsmK. vall The load dis-only depend on face width)and Cvare only a f\illctionof peripheral speed. In actual fact, however, bothfactors are influenced to a great extent by tootherrors Therefore, the AGMA Service Factor is no criterion atall for the quality of a gear; it only gives an indicationon the dimensions and the material strength of thetoothed parts.

Referenc es :(1) Tragf Eih igkei t sbe recruJuI!g vonVDI-Z Band III,1969, Nr. 4Stirn- und Kegelradern nachDIN3990(2) Lubrication as a Gear DesignFactorProceedings of the International Conference onGearing, London1958