This paper waspresented at ~he8thInternational Power

Transmission and!Gearing Conference,

Baltimore, MItSeptember .2000.11was

later published! in theJaume! of MechanicalDesign~March 2001, bV

the American Society ofMechanicall Engineers.

David G. !Lewickiis a senior aerospace engi-

neer witt: tile U.S_ArmyResearch Laboratory atNASI!. :s Gleim Research

Center ill Cleveland, OH_He has been involved ingear crack propagmion

research. as well as trans-mission life and reliability

predictions ami geardynamics predictions. Healso has worked orl low-

noise, high-strength spiralbevel gears; face gears for

helicopter drive systems:lubricants; and diagnostics,

He lias written or eo-writ-tell.more than 70 technical

article in the field ofdrive systems,

l'isa IE.Sp,ieV8'kis a' structural engineer ill.

the design and analysisgroup of ATA EngineeringInc .. located ill San Diego.

CA. Sire is 011 expert inopplying computer tech-

niques to design, analyzeand test highly stressed

structures and In interpret-ingfracture and fatigue

requirements for thosesl/,llcrure~·. As a graduate

student. she performedresearch with the CornellFracture Group ill which

she simulated three-dimensionalfatigue crack

growth in spiralbevel gears.

Consideration of Moving ToothLoad in Gear Crack

Propagation PredictionsD'avid 'G.lewicki', llsa E. S'pievak, 'PaullA. Wawrzynek,

An1hony R...Iingraffea and Robert F..Handschuh

IntroductionEffective gear designs balance strength, dura-

bility, reliability, size. weight, and cost. Even

effective designs. however can have the possibil-ity of gear cracks due to fatigue. In addition, truly

robust de: igns consider not only crack initiation.but also crack propagation trajectories. As anexample, crack trajectories that propagate

through the gear tooth are the preferred mode of

failure compared to propagation through the gearrim. Rim failures will lead to catastrophic eventsand should be avoided. Analysis tools that predict

Figure I-Location of load.eases for finite elemen;mesh.

crack propagation paths can be a valuable aid 10

the designer to prevent such catastrophic failures.

Pertaining to crack analy is, linear elastic frac-ture mechanics applied to gear teeth has become

increasingly popular. The stress intensity factorsare the key parameters to estimate the characteris-tics of a crack. Analytical method using weight-function techniques to estimate gear tooth stress

intensity factors have been developed (Refs. I and17). Numerical techniques, such as the boundaryelement method and finite element method. have

also been studied (Refs. 12 and 21 ) .. Based onstre s intensity factors, fatigue crack growth andgear life predictions have been investigated (Refs.

2, 3, 5 and 9). In addition, gear crack trajectorypredictions have been addressed in a few studies(Refs. 6, 7, .13, 14 and 19).

From publications on gear crack trajectory pre-dictions, the analytical methods have been numer-

ical (finite element method or boundary elementmethod) while solving a static stress problem. In

actual gear applications, however, the load movesalong the moth, changing in both magnitude and

position. No work has been done investigating theeffect of this moving load on crack trajectories.

The objective of the current work is to studythe effect of moving gear tooth load on crackpropagation predictions. 'Iwo-dimensional analy-sis of an involute spur gear using the finite ele-ment method is discussed, Also, three-dimen ion-

al analysis of a spiral-bevel pinion gear using theboundary element method is discu ed, A quasi-static numerical simulation method is presented inwhich the gear tooth engagement is broken downinto multiple load steps, with each step analyzed

separately. Methods to analyze the steps are dis-cussed, and predicted crack shapes are comparedto experimental results.

Two-Dimensional AnalysisGear ModeUng ..The two-dimensional analysis

was performed using the FRANC (FRactureANalysis Code) computer program developed by

a) load on tooth 12

K"~6 5 4 3 2 1

~K,

35

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-1

-2 L-_-'--_-'-_.....L_-L_~_--'L...----'

2520

f~ 15

:§ ~ 1~

j 0-<n -5 '------'----'---'----'---"

0) Load on looIJi 3 01\D---O:------o-- K,

,"i E 1 18 17 16 15 14 13! is .;

11

.~!l; 0 'll I,? 1§ 1§ H 1.3~ ~~

I ~ _, !.... ..!.. ~~..J.. -!.-_.....L._---L_.......J_--J

o 5 10

j 2

15 20 25

Figure 2-Mode I and mode 11stress intensity fac-lors./ol' ,a'wlit wad alld an, initial crack ,oj 0.26 mm.

Wawrzynek (Ref. 23). The program is a general-purpose finite element code for the static analysi»of two-dimensional cracked structures. The pro-gram uses principles of linear elastic fracture

mechanics and is capable of analyzing planestrain, plane stress. or wej-symmetric problems. Aunique feature of the program i the ability to

model crack and crack propagation in a true-ture, A 1'0 ene of quarter-pciat, ix-aode, triangu-

lar elements is used around the crack tip to model

the inverse square-root stress singularity. Mode Iand mode [J tre inten ity factors, K1and Kifrespectively, can be calculated u ing a variety ofmethods. (Asa refresher, mode m loading refers to

loads applied nonaal to the crack plane and lendsto open the crack. Mode WIrefers to in-planeshear loading.) The stres intensity factors quan-

tify the tate of tre in the region near the cracktip. In the program. the tre inten ily faCIOcan be used to predict the crack propagation tra-jectory angle . again using II variety of methods.

In addition. the program has a uniqaere-meshingcherne to allow automated proces ing of the

crack imulation,A spur gear from a fatigue test apparatus was

modeled to demon trate the two-dimensionalanaly i .The modeled gear had 28 teeth. II 200

pre sure angle. II module of 3. 175 mm {diametral

pitch of 8tin.). and a face width of6.35 mm (0.25in ..)..The gear had a backup ratio (defined a the

rim thickness divided by the I.ooth height) of 3.3.The complete gear was modeled using mostly 8-node, plane sires .. quadrilateral 'finite dements.

For improved accuracy" 'the mesh was refined. onone of the teeth in which a crack was inserted.

The total mode] had 2,353 elements and 7.295nodes. Four hub nodes at the gear inner diameterwere fixed to ground for boundarycoedirions,The material used was steel.

'lootll Loading Scheme. To determine. theeffect of gear tooth movlng load on crack propa-gation, the anaJy is was broken downinto 18 sep-

arate load case (fig. 0. An initial crack of 0.26.mm (0.010 in.) ill length was placed ~n the filletohooth 2. oormal [0 the surface. at the location ofthe maximum tensile tres (uncracked condi-tion). Six load cases were analyzed eparatelywith the load on the tooth ahead of the crackedtoojh, six 0.11 lhe cracked tooth, and six on thetooth after the cracked tooth. The calculatedstressintellsity factors for unit load at each of theload position are hewn in Figure 2. These'Stress. intensity factors were calculated u ing theJ-integraJ technique (Ref. 20). Load. on tooth 2.(crackedtooth) produced ten ion at the crack tip.

2,0 _load on looln 2

\ - lOl!d on tOOl!> J~,/I \I \

\\\

Load on toolh 1'-,I \I \

fII

40Gear rotation, degreea

50

PaulA.Wawrzynekis a senior research associ-(Iff! in /1.1(' CompulaliOlwlMlIIl'rials tnstitute til

Cornell U'lil'l'rsil)'"s TheoryCenter and ill Ih CornellFracture ,Group. A civitengineer; he has focusedhis research primarily ondew:/oping ~'oftwareforsimula/ing crack gro ....th in11 range oj!"llgint'l'ringstructure sand marerials.Abo. Ire manage« FractureA"aIJ'~'isConsultunrs Inc..which provides crackgrowth analysis and soft-waTl"developmen: servicesfor differeru industries.

Anthony R.lngraHeslis u pro[r: Jar 01 1M Schoolof Cil·jf and £rwirollmenlaiEligilleering tu CornellUniversity. Ioctued inIthaca. NY. A civil engineer;Irlfl.raffl!a focuses hisTI!SI!(lrc/I' 011computer simu-

lution (wd physical testing(Jf complex fracturingprocl'SSl!s. He and his stu-

dents have performedresearch in usi/II interac-tive computer graphics in'campillaliona/ fra('/uremechanics. Wilh Iris 51u-

dents, he has written mort(han l80 papers 011 com-plex jracwrinp, proc-essesand compuuutonat fracturemechanics.

IRobert F. Handschuhis llll (I rospac« engineerl1:ilh the U.s. Arm)'Reuarch Laboratory atNASA:r Glenn Res arr:hCemer Hi research hasconcentrated on powertransmission; his researcbin gt'oring has focused .onI:_tperimental and anaiyticalstudies of spiral bevel; fact •.and lIigh'lpt'l'd gearing: Hthas lO'riCltl1 or co-writtenmort than 65 reports in tMfields ,,[ seal; gro" lindrrUlllEricaJ methods.

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Figur.e 3---DANST computer ,program ,0utpuJ ,0/stalic geal' ,too.lll.load. 68 N-m driller torque.K, increased as the load moved toward the toothtip (load cases 12 to 7, Fig. 2b) due to thincrea eel load lever ann. Lo ds on tooth 3 alsoproduced ten ion aI the era k 'lip. bUI at an orderof magnitude le than those produced from theload on tooth 2 (Fig. 2.c)" Loads on toota I gavecom pres ion to the crack lip as shown by the neg-ative K{ values (Fig. 2a).

Next, the actual load magnitudes on the geartooth were considered as il wenl through thmesh. The computer program DANST (Dynamic

ANaly is of SPIl..l' gear Tram mission. Ref. 15)was used for 'the analysis. 1bi. program is based

on a four-degree-of-freedom. torsional, lumpedrna model of n gear tran mi sian. The modelincludes driving and driven gears, connecting

shafts, a motor. and a load. The equations of

motion for this model were derived from basicgear geometry, elementary vibration principles,and time-varyingtooth stiffaesses. For simplicity,the static gear tooth loads of the solution weredetermined ( ig, 3). TIle. e loads were deter-mined from well-established gear tooth stiffne

principle and static equilibrium. The loads arehewn as a function of gear rotation for adriver

torque of 68 N-m (599 in-lb.), Tooth 2 began

contact at a gear rotation of 100• As the gear rota-

tion increased, the load on tooth 2 graduallyincrea ed. Tooth l and tooth 2 shared the load fora rotation from Hlo to 180

• From l8° to 23°. tooth

2 carried the complete load. At 23°. tooth 2 iscan idered at it bighe Ipoint of single tooth con-'tact (HPSTC).

The stre s iDlen"ity factors as a function ofge!U' rotation were then determiaed by multipjy-

i'ng the stressimensity factors determined fromthe units' loads (Fig. 2) by the acreal looth loads(Fig. 3) and applying superposition since linearelastic fracture mechanics was 'U ed. The rr-eultsare shown in Figure 4. As expected. the mode Estress intensity factor (Flig. 4a) was mostly influ-enced by the load on tooth 2. Note that the largestvalue of Kl eceurredat the HPSTC. AI a note that

HlA...!.I_....._....,..Hj;~£ j:~ , , , , ,

..

i 1.5

1.0.~'" e.s ,c:

~ 0.0Zl -0.5,i!!i,ij

-1.010

Figrjre 4--Slress ,ilJlellsity and ,tangential stressfactors as ,afuncrion of gear rotation, 68 N-mdriver tor,que, 0.26 mm in'ilia' crack size,

2.38 crac •• _......_h. mm10' a) Mode I stress in1ensi1y factors 1.58 ,- "',,~.,

0.790.26

b) Mode II stress Intensity ractors

o 10 20 30 40 50

Figure 5--Str:ess ,inl;ensity factors from gear tooth crack propagatitm simuk»tion" backup ratio'" 3.3'.

30 a) Mode' I stress inli3flslty faclars

~L- __ ~ __ ~~ __ ~ __ ~~ __ ~ __ ~

e bl Mode II stress inIensItyfactorsCrac!I length, mm

1.580.790.26

6

4

2

0

·2·10 0

.F:igure ,6-SlTess intensity factors from gear ,toothCTtl.d: propagaJion simula-tiOTl, backup r:atio = O~2.

10 20 ao 40 50Gear rolaJIOll. clegrees

11 the magnitude of K, (Fig. 43) was much largerthan that of K/l (Fig. 41». This implied thai, K, wasthe driving force in the crackpropagation, KIf'

however, affected the crack propagation angle, aswill be shown in the next section,

Cr,ack Propagatiofl Silllulatio,r&. FromWilliams (Ref. 24), the tangential stress near a.crack tip, G9B' is given by

where rand (}are polar coordinate with the ori-gin at the crack tip. Erdogan and Sih (Ref. 8) po -tulated that crack extension starts at the crack lipandgrows in the direction of the greate t tangen-tial sire s. The direction of the greatest taagendalstress is determined by taking the derivative ofEquation I with respect to B. setting the expres-sion equal to zero, and solving lor (;/,Performingthe math, this predicted crack propagation angle,8m, is given by

,9",= 2 tan "

. rom Equation 2, the predicted crack pmpaga-tion angle is a function of the ratio of K{ to Kit

Erdogan and Sih (Ref. 8) used. brittle ptexiglassplates under static loading to validate their pro-posed theorems (ie.,the ratio of K/ to KII was con-stant). For the gear problem in the currern rudy,however. the ratio of K/ [Q KII was not constant dur-ing gear rotation. This is hewn in Hgure 4c (actu-ally plotted as the ratio Kil to K/ for clarity). lnaddi-tion, Figure 4d gives the calculated B;" fromEquation 2 as a function of gear rotation.

In order to simulate gear crack propagation, amodificat.ion to tile Brdogan and Sjh theory waspostulared in the current study. This modified the-ory slate that the crack extension starts at thecrack tip and grows in the direction of the great-est tangentiol tre s as seen during engagement ofthe gear teeth, The procedure to calculate thecrack direction is as follows:I) K/3.Ild Kllare determined as 3 ,fullction of gearrotation (Figs. 4a and 4b, as described ia jhe pre-viees section),2) the ratio of K, to K/I as a function of gear 1:100a-

tion is determined (Fig. 4c),3) 8m (using Eq. 2) as a function of gear rotauonis determined (Fig. 4(1).

4) (100 (u ing Eq . .I) as a function of gear rotationis determined (Fig. 4e),5) the predicted crack direction is the value of 9",

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for which 0'00 i greate t during gear rotation.For the gear example given, th . tangential

stres. fa tor (defined a. "'00' 21rr) is plotted a afunction of gear rotation in Figure <te. Thi plot100 s very similar to the mode Lstre inten fly

factor plot (Fig. 4a) ince KI was much largerthan Ku (see Eq. I). The tangential tres waslargest al the HPSTC (gear rotation of 23") andthe predicted crack propagation angle at thi gearrotation was ,9"," 4.3".

sing thi propagation angle. the crack wasextended by 0.26 mm (0.010 in.), re-me hed, re-an_aJyzed. and a new prepagation angle wa. calcu-lated using the method de. cribed above. This pro-cedure was repeated a number of times 'to producea total crack lengjh of 2.38 mm (0.094 in.), TheO.26-mm crack: exten ion length wac based onprior experience in rderto produce a moothcrack path. Figure 5 shows the stres: intensityfactors versus gear rotation for a number of cracklength . Note that the mode Lstres , intensity fae-tors looked imilar but with increa ed magnitudeas the crack length itncreased. In all. case , theselected crack propagation angle occurred whenthe tooth load was placed a.t me HPSTC. Figure 6. hows a similar analysis bl.l1 with a model of athin-rimmed gear. Here. the gear was modeledbased 011 the previous design, but wi.lIl lOIS incor-porated in the rim 1.0 imuiatea thin-rimmed gear.The backup ratio for thi model was 0.2. A. een,the magnitude of the mode I. stres s inten ity fac-tors during len. ion (gear rotations 1.8" 10 45°)were larger than that of the 3.3 backup ratio gear.Also, there was a significant increase in 'the com-pre sive 1(1 (gear rotation le than 18°) due to theinerea ed compliance of the thin rim gear.

Comparison ,(6 Bxperiment . Figure 7 howthe results of the .allaty i compared to experimen-tal tests in a gear fatigue apparatus. The originalmodel (backup ratio of 3.3). as de eribed before.was compared along with model of backup, ratioof 1.0 and 0.3. These later two models were creat-ed using lOIs in tile gear blank. as previouslyde cribed. The experiments were first reported byLewicki and Ballarini (Ref.. 13). Here, notcheswere fabricated in the looth fiUet region to initiatetooth cracking of [est gears of various rimthick.nes es, The gears were run a!. 10.000 rpm and at avariety of increasing loads unli] tooth or rim frac-ture occurred. A en. [rom the figure, good cor-relation of the predicted crack 'I.fajectorie toexperimental re ults was achieved. For backupratio 'Of 3.3 and 1.0. 'Looth fractures occurred. Forthe backup ratio of 0.3, rim fmc lure occurred.

As a final n te, the llnal)' i indicated thatthe

F~gureB-Boundary element modd of OR-58 spi-ral-bevel pinion'.

~-g 10.5l

~0

" 5

~OL1--~3~--~5--~7---79--~1~1--~13~~15-

b) Loads on tooth CQl'lIacI elhPles

Figure 9-l.ocation' ,of tootl, caatac""ellipses "endmagnitud« ,of load on OH~S8piral.bevel' pinion'tooth,.maximum tangential tress at the crock tip alwayoccurred when the tooth load was positioned at theHPSTC. Thus, for two-dimensional analysi .cra ksimulation ba ed all calculated stress intensity fac-torsand mixed mode erac angle prediction teea-ruques can use a Lmple static analysi in which thetooth load i located at the HPSl1 . Thi was ba edon a modification 'to the Erdogan and Sih crackextension theory and 'the fact that the mode I tre sintensity factor was mu h larger than lhe mode WIfactor.

Three-DimenionaJ Analy IGear Modelirlg. The three-dirnen ional analy-

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1ront

1,00

Figure 1000tress intensity factors jr"Om'rl,ree.dimenional OH-58 p.iniorl' tootilcrack propagaJion .simulali:On; step I, ,crack area", 5.96 .mm·z.

20- a) MOde' I stress inlens11y 'lICIors

15

o --3 - b) MOOe ll stress inlenslly I'aeIors

Figur:e 11-5tres inten ily factors from 1I,ree-tiime1JsiolU11 OH~S8 p.iniontooth,f:rackpr:opagation' simulation; step 1, normalized position alo/lC ,r,.arkfro.n~'= 0.83.

-.2 !..- -L -'- .L- ----'

0.00 0.25 0,50 0,75Normalized posbon along crack front

sis was performed using the FRANC3D(FRacture ANalysis Code for 3 Dimensions)computer program developed by Wawrzynek(Ref. 23) .. This program uses boundary elementmodeling and principles of linear clastic fracturemechanics to ana1yze cracked suucurrcs. Thegeometry of three-dimensional structures withnon-planar, arbitrary shaped cracks can be mod-eled, The modeling of a three-dimen ionalcracked tructure i performed 'through a series ofprogram. Structure geometry grid point data areimported to a . olid modeler program. Here,

I r appropriate curves and faces (or patches) are ere-I atcd from the grid data. as well as a dosed-loop

surface geometry model. This surface model ithen imported to the FRANC3D program forboundary element model. preparation. The usercan then mesh the geometry model u ing 3- or 6-node triangular surface elements, or 4- or 8-nod.equadrilateral elements. Boundary conditions(applied traction and pre cribed displacement)are applied on the model geometry over faces,

edges, or point. Initial crack. such as ellipticalor penny shaped. can be inserted in the tructure,After complete formulation, the model is hippedto a boundary element equation elver program .Once the di placement and traction unknowns are

solved, the results are exported back to the.FRANC3D program for post-proce ing,Fracture analysis. such as stre sinten ity factorcalculations. can then be performed.

The spiral-bevel pinion of the OH-58 heli-copter main rotor tran rnis ion wa modeled to

demon trate the three-dimen ional anal)' is. Thepinion had ['9 teeth, a 200 pre ure angle, a 30"mean pi ra IIangle. a module of 3.66 rom (diame-tral pitch of 6.94/in.), and a [ace width of 32.51.mm (1.28 in.). 'For OH-58 operation. the pinionmate with a 7 l -tooth spiral-bevel gear. operatesat 6,(}60 rpm, and h a design torque of 350 N-m(3.099 in-lb .).

The boundary element model of the .oH-58pinion developed by Spievak (Ref. 22) was 1.1 edfor the study. Three teeth. tbe rim cone. and thbearing uppert shafts were modeled (Fig. 8).The tooth surface and fillet coordinates weredetermined from the method developed byHandschuh and Litvin (Ref. ] 1) and Litvin andZhang (Ref. 16). The melt of 'the three teem warefined for improved accuracy. A ha1f-eUip ini-tial crack w.ith major and minor diameter of3.175 mm and. 2.540 mm, respectively (0.125 in.and 0.100 in.), wa placed in the fillet of the mid-dle tooth normal to the urface, The crack wascentered along the face width and centered along

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11 b) Mode II stress in!ensiIy I~

o

-1 ~--~--~----~--~----~--~--~1 3 5 13 157 111

Load case

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the fillet. The complete gear model had a total ofabout 2,600 linear elements (both triangular andquadrilateral) andabout 2,240 node. For bound-ary conditions, the end nodes of thelarger-diarn-eter haft were fixedand 'Ille node on the outer

diameter of the smaller-diameter shaft were con-strained in the radial directions, Again, the mate-

rial was steel.Tooth, Contact Analysis and Loading

Scheme. Due to the geometrical complexities, andthree-dimensional action, numerical methods arerequired to determine the contact loads and posi-tions on spiral-bevel. teeth since no closed-formsolution exists. The method of Litvin and Zhang

(Ref. 16) was used to determine the mean contact

points on tile piral-bevel pinion tooth, Themethod modeled tooth generation and tooth con-tact simulation of the pinion and gear. With themean contact po,ints taken a the centers contact

ellipses were determined u ing Hertzian theory(Ref. 10). Figure '9 shows the estimated contacteflipses on the spiral-bevel pinion tooth. Fifteenseparate ellipses (load cases) were de lenni ned,tarting from the root of the pinion and moving

toward the tooth tip and toe. Load cases 1-4 and12-15 were double tooth contact regions whiletoad cases 5-11 were single tooth contact regions.Note that load case l l corresponds to the load atthe HPSTC, For each load case using the bound-ary element method, tractions were applied nor-mal to the surface, to the appropriate ellipse withthe magnitude equal to the tooth normal force

divided by the ellipse area.Crack PropagaJion: Sim.ulalion. The proce-

dure fer the three-dimen ional crack propagationsimulation of the OH-58 piral-bevel pinion wasas follows. For each of the load case of Figure9, the mode I and mode IT stres inlensity factorswere determined at 25 points along the crackfront (note that for three-dimen ions, there is acrack front, not just a crack tip as, in two-dimen-sions). The extended crack direction at each ofthese 25 points were detennined uing the modi-fied Erdogan and. Sih crack extension theory asdescribed in th.e two-dimensional ana1ysis. Thatis, as the cracked spiral-bevel pinion tOOUl wasengaged in the me 1'1, the crack extension startedat each point along the crack: front and grew in thedirection of the greatest tangential stress at thosepoints during mesh. The amount of crack exten-sion at each point along the crack front was

determined based on the Paris crack growth rela-

tionship (Ref. 18) where

lJ, .:::(J' (~)nI -ma;r K

1.~

Table I-Results of multiple Iliad case crack simulatiol1 analysis.

Crack area(mm2I' Crack front pointls)

!Woad Icase torla rgest (leeSte,p

3.1201 12-25

2 110.35

3

where (J, was the amount of extension of the illlpoint along the crack front, K,.i wa the mode Istress intensity factor of the i,h point along thecrack frontcerresponding to the lend ca e whichgave the largest tangential stress for that frontpoint. K/,IrUU. was, the value of the largest Kl,i alongthe crack front, lJ',HaX was the maximum definedcrack extension along the crack from, and rr wasthe Paris material exponent. From experience, the

maximum extension size. lIma,r' was et to I..27mm (0.050 in.), The Paris exponent, n, was set to2.954 based on material tests For A.[SI 931.0 steel

by Au and Ke (Ref. 4). A third-order polynomialwas then used to smooth the extended crack frontThe new crack geometry was then re-rneshed.After re-rneshi ng, the model was rem II and sol vedfor stress intensity factors and crack propagation

directions. The above procedure was repeated anumber of times to ' imulate crack: growth in the

gear tooth.Table I gives results from the rIC t four step

during this process. Note that step 0 correspondsto the illitial. half-eIHp e crack. For steps 0 and 2,the largest jangennal stress occurred at theHPSTC (load ca ell) for the majorilY of thepoiats along the crack front For teps I and 3, thelargest tangential stress occurred at load ca es g,9,.10, or 11.

As previously stated, the mode I and mode IIstress intensity factors were determined at 25points along the crack front. This WIJI$ true forsteps 0 through 2. For step 3, however, the modeI and mode Il stressintensity factors were deter-mined at 27 points along the crack front. This wadue to the way 'the FRANC3D program extendedthe crack: surface of the third step. For tep 0through 2, the crack front wa a member of one

continuous geometry face (FRANC3D defines .<1

(3) geometry face as a 3- or 4-sided surface.) For

" Heel

Tooth

fillet

Cracksurface

Toe

Figure 12-0B·58 spiral.bevel pinion tooth crackpropagation simulation after seven steps.

Figure J3-Comparisoll ofOB.58 spiral.bellelpin-ion tooth crack propagation simulation toexperi-ments.step 3,. the crack front was a member of threeadjacent geometry faces, thus producing 27 pointsalong the crack front,

Figure 10 shows the stress intensity factor dis-tribution along the crack front for step 1 (crack areaof 5.96 mm2 (0..009 in.2». Similar to the spur gearanalyses, K[ was larger as the load moved from theroot to the tip due to the larger load lever arm.Other than absolute magnitude, the K, distributionsalong the crack front looked similar for the variousload cases. Figure 11 depicts the stress intensityfactors plotted against load case (at a point alongthe front, biased toward the toe, normalized posi-tion along the crack front of 0.83) This figureshows the simulated distribution as the pinionengages in mesh with the gear. Note again that theratio of KI/ to K, was not constant during engage-ment.

noted that the loading was placed only at theHPSTC for the last three steps. This was due tomodeling difficulties encountered using themulti-load analysis. It was felt that this simplifi-cation did not significa:ntly affect the results dueto the smoothing curve-fit used. In addition, thetangential stress near the crack tip was eitherlargest, or near its largest. value, when the loadwas placed at the HPSTC.

Comparison to Experiments. Figure 13showsthe results of the analysis compared to experimen-tal tests. The experimental tests were performed inan actual helicopter transmission test facility. Aswas done with the gear fatigue tests describedbefore, notches were fabricated in the fillet of theOH-58 pinion teeth to promote fatigue cracking.The pinion was run at full speed and with a varietyof increasing loads until failure occurred. Shownill the figure are three teeth that fractured from thepinion during the tests (Fig. 13b). Although thenotches were slightly dLfferent in size, the frac-tured teeth had basically the same shape.

A side view of the crack propagation simula-tion is shown in Figure 13a for comparison tothe photograph of the tested pinion in Figure13b. From the simulation, the crack immediatelytapered up toward the tooth tip at the heel end.This trend matched that seen from the tests ..Atthe toe end, the simulation showed the crack pro-gressing in a relatively straight path. This alsomatched the trend from the tests. Toward the lat-ter stages of the simulation, however, the cracktended to taper toward the tooth tip at the toeend. This did not match the tests. One problemencountered in the simulation during the latersteps was that the crack at the heel end of thetooth became close to the actual contact ellipses.It was felt that the crack-contact interaction mayhave influenced the trajectory predictions tocause the discrepancy.

Spievak (Ref. 22) reported on another methodto account for the non-uniform Ku to K[ ratioduring pinion tooth engagement This methodconsidered contributions from. all load cases inthe crack angle prediction scheme and presenteda method to accumulate the load effects. Fromthese studies, reported crack propagation sirnula-tion of an OH-58 pinion also predicted the erro-neous taper toward the tooth tip at the toe end.Again, the crack-contact interaction may haveinfluenced the trajectory predictions to cause thediscrepancy ..Spievak also reported on a simula-tion using only the load at the HPSTC. The crack

Figure 12 shows exploded views of the pinion trajectories from that simulation were similar tocrack: simulation after seven steps. It should be the trajectories in the current study. It should be

20 JANUARY/FEBRUARY 2002 • GEAR TECHNOLOGY·' www.geartechnofogy.com • www.powertrensmisslon.com

ReferencesI. Abersek, B. and J. Flasker, "Stress Intensity Factorfor era ked Gear Tooth," Theorerical and AppliedFracture Mechanics, 1994, VoL 20, No.2, pp, 99-104.2. Abersck, B. and J. Flasker, "Experimental Analysisof Propagation of Fatigue Crack on Gears,"Experimental Mechanics. 1998, Vol. 38, No.3, pp.226-230. .3. ~n, M.A., AJ. Tarhan, and O.S. Yahoj. "LifeEstimate of a Spur Gear with a Tooth Cracked at FilletRegion." Proceedings of lhe ASME DesignEngineering Technical Conference. 1998. Athnta, GA.4. Au, JJ. and I.S. Ke, "Correlation Between FatigueCrack Growth Rale and Fatigue Slrialioil Spacing inAIS] 9310 (AMS 6265) Steel," f"rac/ography andMaterial Science, ASTM STP 733, 1981, pp. 202-221.5. Blarasin, A., M. Guagllano and L. Verganj. "FatigueCrack Growth Predictions in Specimens Similar toSpur Gear Teeth," Fatigue & Fracture of EnRineeringMalerials & Structures, 1997, Vol. 20, No.8, pp.1171-1182.6, Ciavarella. M ..and G. Demello. "Numerical Methodsfor the Optimization of Specific Sliding, StressConcentration, and Fatjgue Life of Gears,"International Journal oj Fatigue. 1,999,Vol. 21, No.5,pp, 465-474.7. Curtin, TJ., R.A. Adey, l.M.W . Baynham and P:

........ pOIill!Ulransm/sslon.com ' .. ww.ge,u'.ecl!nology.com ' GEAR TECHNOLOGY" JANUARYIF:EBRUARY 2002 21

noted that the proposed method in the currentl:udy to account for moving tooth load . for the

three-dimen ionaI analysis was extremely cum-bersome. It is therefore felt that the analysisusing only the load a! the HPSTC appeared accu-rate as long a the crack: did not. approach thecontact region on the tooth.

'ConclusionsA study to determine the effect of moving

gear tooth load on crack propagation predictionswas performed. Two-dimensional analysis of aninvolute spur gear using Ule finite elementmethod wa investigated, AI. o, three-dimen ion-al analysis of a piral-bevel piniongear using the'boundary element method was discu sed. Thefollowing conclusions were derived:

I.) A modified theory for predicting gear crackpropagation paths based on the.criteria ofErdogan and Sih was validated. This theory slat-ed that as a cracked gear tooth was 'engaged inme .h, the crack extension started at the crack tipand grew ill tile direction of the greatesttangen-tial stress during mesh.

2) For two-dimensional analy is, crack sirnu-lation based on calculated stre intensity factorsand mixed mode crack angle prediction tech-ruque.s can use a imple tatie analy is in whichthe tooth load i located at the highest point ofsingle tooth contact.

3) For three-dimensional analysis, crack simu-lation can also use a simple static aaalysi .in whichthe tooth load i located at We highest point of In-gle 'looth contact as long as the crack does notapproach the contact regien on the tooth. 0

Marais, "Patigue Crack Growth Simulation forComple~ Three-Dimer» ional Geometry and Loading,"Proceedings from the 2nd Joint NASAlFAAJDODConference on Aging Aircraft, 1998, WHliamsburg,VA.8. Erdogan, F. and G.C. Sih, "On the' rack Extensionin Plates Under Plane Loading and Tran verse Shear,"Journal of Basic Engineering. 1%3, Vol. 85, pp,519-527.9. Glodez, 5., S. Pehan, and 1. Flasker, "Experimental.Resu.lts of the Fatigue Crock Growth in a Gear TootbRoot," International Ioumal oj Fatigue. 1998, Vol. 20,No.9, pp. 669-675.10. Handschuh, R.F. and T.P. Kicher, "A Method forThermal Analysis of Spiral Bevel Gears," Journal ofMechanical Design, 1996, Vol. 118" No.4, pp,580-585.1.1. Handschuh, R.F.and F.L. Litvin, "A Method forDetermining Spiral-Bevel Gear Tooth Geometry fOJFinite Element Analysi ," NASA TP-3096, AVSCOMTR~91-C·020, 1991.12..Inoue. K. and M. Kala, "Crack 'Growth ResistanceDue to Shot. Peening in Carburized Gear ," Presentedat the 30th A1AAlASMEJSAEJASEE Joint PropulsionConference, 1994,lndia_napoJjs, IN.13. Lewicki, D.G. and R. Ballarini, "Effect of RimThieknest on Gear Crack Propagation Path," Ioumaiof Mechanical Design, 1997, Vol. 119. No.1, pp.8&-:95.14. Lewicki. D.G., A.D. Sane, RJ. Drago, and P:A.Wawrzynek, "Three-Dirnensional Gear CrackPropagation Studies," Proceedings of the 4th W~rldCongress on Gearing and Power Transmission, Paris,France. 1999. Vol. 3, pp..231/-2324.15. Lin, H.H., RL. Huston and JJ. Coy, "On DynamicLoads in Parallel Shaft Transmissions: Part I -Modeling and Analysis," Journal of Mechanisms.Transmissions. and Automation ill Desigll.198B, Vol.110, No.2, pp. 221-225.16. Litvin, F.L. and Y. Zhang, "Local. Synthesis andTooth Contact Analysis of Face-Milled Spiral Beve]Gears," NASA Contractor Report 4342,· AVSCOMTechnical Report 9QwC-028, 1991.17. Nicolette, G. "Approximate Sire s Intensity Factorsfor Cracked Gear Teeth," Ellgineering FractureMechanics, 199'3, Vol. 44. No, 2, pp. 231-242.18. Paris, P.e. and F. Erdogan, "A Critical Analysis ofCrack: Propagation Laws," Journal of BasicEngineering, 1963, Vol. 85, pp, 528-534.19. Pehan, S., T.K. Hellen, J. Flasker and S. Glodez,"Numerical Methods for Determining Stress IntensityFactors Y$, Crack Depth in Clear Tooth Roots."International Journal of Fatigue, 1997, Vol. 19, No. 10,pp. 677-685.20. Rice, l.R. "A Path Independent Integral and theApproximate Analysi of Strain Concentration byNotches and Cracks," Journal of Applied Mechanics,1968,. Vol. 35, pp. 379-386.21. Sfakiotakis, V.G.• D.E. Katsareas and N.K.Anifantis, "Boundary Element Analy i of Gear TeethFracture," Engineering Anal}'sis with BoundaryElements. 1997, Vol. 20. No.2, pp. 169~175.22. Spievak, L.E. "SimulatLng Fatigue Crack Growth inSpiral Bevel Gears," Masters of Science Thesis,Cornell University, 1999,23. Wa.wrzynek, P,A. "Discrete Modeling of CrackPropagation: Theoretical Aspects lind Implementa-tionIssues in Two and Three Dimensions," Ph.D.Dissertation, Cornell U ni versity, 1991.24. Williams, M.L. "On the Stress Distribution at theBase of a Stationary Crack," Journal of AppliedMechanics. 1957, VoL 24, No. I. pp. 109--114. .

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