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Helix Angle and Root Stress

By measuring the helix angle, the authorand founder of

the Ohio State University Gearlabstudied its effects on rootstresses in helical gears.

B Dod R. Hosr

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THe ISO anD aGMa GeaR RaTInG COMMITTeeS

Have fOR SeveRal yeaRS Been COMpaR-

InG THe ReSulTS Of DIffeRenT RaTInG

MeTHODS fOR SeveRal SeTS Of GeaR

paIRS THaT Have SIMIlaR nORMal SeC-

TIOnS BuT DIffeRenT HelIx anGleS. THe

analySIS pReSenTeD In THIS papeR uSeS a

veRy SOpHISTICaTeD fInITe eleMenT CODe

THaT waS DevelOpeD SpeCIfICally fOR

GeaR anD BeaRInG COnTaCTS TO analyze

THe exaMple GeaR SeTS. analySeS aRe

alSO peRfORMeD uSInG a MORe COn-

venTIOnal lOaD DISTRIBuTIOn analySIS

pROGRaM. THe ReSulTS fOR THe ORIGInal

GeaR SeTS SHOw THaT THe naRROw faCe

wIDTH GeaR TeeTH TwIST SIGnIfICanTly,

THuS MOvInG THe lOaD TO One eDGe Of

THe faCe wIDTH anD eSSenTIally SHOwInG

THaT THe exaMple GeaR SeTS aRe HIGHly

unRealISTIC. wHen analyzeD By THe ISO

anD aGMa RaTInG MeTHODS, THe ReSulTS

DO nOT RefleCT THIS TwISTInG aCTIOn. In

an effORT TO COMe up wITH a valID CO

paRISOn Of STReSSeS fOR DIffeRenT Hel

anGleS, THRee aDjuSTMenTS uSInG wID

faCe wIDTHS weRe aTTeMpTeD. THe fIR

SCHeMe uSeS a wIDeR faCe wIDTH wIT

peRfeCT InvOluTeS. eDGe effeCTS ReSu

In THe peak STReSSeS aGaIn BeInG nea

THe enDS Of THe faCe wIDTH. THe SeCOn

aDjuSTMenT uSeS a wIDeR faCe wID

BuT wITH a naRROw lOaD paTCH In TH

MIDDle Of THe TOOTH paIR anD ReSulT

In THe STReSSeS InCReaSInG wITH Hel

anGle. THe THIRD GeOMeTRy fORM, wHIC

uSeS THe wIDe faCe wIDTH TeeTH wIT

leaD CROwn anD TIp RelIef, GIveS TH

MOST ReaSOnaBle ReSulTS, wITH TH

ROOT STReSSeS BeInG aT a MaxIMuM

THe CenTeR ReGIOn Of THe TOOTH fa

wIDTHS. THe papeR COMpaReS eaCH Of TH

ReSulTS TO eaRlIeR analySeS peRfORM

By OTHeRS uSInG BOTH THe aGMa anD IS

CalCulaTIOnS.

IntroductIonThis r rorts o th ct o hi g o root strsss, toic o discssio or mbr o rs ith

th ISO d aGMa Gr Rtig Committs. Crrt rtig mthods s ithr th lis orm ctor [1] or t

30-dgr tgt mthod [2] id to th trsrs tooth sctio to oct th ositio o mimm root strs

Corrctios r th roidd to ccot or th digo is o cotct tht occr i hic gr tooth cotc

e thogh this toic hs b discssd tsi or m rs, thr r sti disgrmts o ho

ths ctors shod b cctd. Grs rst sigiict chgs h trig to com ith chg

i gomtr tht isot o ct ithot ctig othrs. a gomtr st tht sosd isots hi g

rom othr ribs b sig costt orm cross-sctio or ch hi g gr s roosd

docmts o th ISO Rtig Committ [3, 4]. I ordr to chi this, th rc sd to grt th orm

cross-sctio s t costt. Hor, i ordr to chg th hi g d sti th grs orti

t th sm stdrd ctr distc, it s cssr to rdc th tooth mbrs s th hi g

icrsd. Sic tooth mbrs mst b itgrs, thr r o ossib hi gs tht r ossib

Tb 1 shos th ribs tht r hd costt i this sis, d Tb 2 shos th rtioshi bt

tooth mbr d hi g or th sts.

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for ths gr sts, ig. 1 shos th rsts or rit o

mthods tht r sd to cct th root strsss. I this ig-

r th strsss r irst cctd or th sr gr (hi g

= 0) d th s or th othr hi gs r ormid to

th cctd or th sr gr so tht crs strt t

o 1.0 or hi g o ro dgrs. prios or b th

ISO docmt thors shos cctios or to c idths, o

r rro ith th c idth big q to th mod d th

Fig. 1: N367 normalized tooth form factor (root stress)

predictions (all data is from [3]-B: Method B from ISO

6336; N: Proposed Norwegian method).

Fig. 2: Transmission3D finite element mesh

template.

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othr or iiit c idth. Cctios

h b md sig ISO 6336 mthod

B, aGMa 2001, d roosd norgi

mthod [3]. Som mthods rdict tht

icrsig th hi g i coti

dcrs root strsss hi othr mth-

ods sho r-icrsig trd i root

strsss ith hi g.

ech o th mthods sd to cc

th root strsss o ig. 1 s stdr

t orms tht r sd i co

tio ith m othr ctors to com

ith strss . This rs tho

thoght tht cctios o th r

strsss tht th grs ric mig

shd som ight o th dircs

Fig. 3: Transmission3D model

of the narrow face width mesh.

Fig. 4: Gear tooth load distribution

from Transmission3D for a narrow

face width spur gear tooth.

Fig. 5: Transmission3D load

distribution and root stress pat-

tern for the 42.833 helix angle

pinion.

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ch cctio mthod. Thror,

dcd iit mt rogrm tht

is sciic dsigd to gr

d brig cotcts, d mor st-

drd od distribtio rdictio rogrm

r mod to rdict th ct root

strsss or th m gr sts. This

r roids tsi ss o

th m gr sts rstd i th

origi n367 docmt d so ss

to roid mor ristic gr gomtr

tht isots th cts o hi g o

root strsss.

ModelIngMethodologyech o th gr sts rstd i this

r s modd ith to srt

rogrms to t th root strsss,

th irst big th high idit thr-

Fig. 6: Predicted

normalized root

stress for narrow

face width pinion.

Table 1: Variables held con-

stant in study 1.

Parameter Variable Value

Gear ratio u 4

Normal module, mm mn 10.0

Center distance, mm a 750.0

Hob addendum, mm ha0 14.0

Hob tip radius, mm a0 4.0

Protuberance, mm p 0.3

Normal pressure angle, degrees 20.0

Face width, mm 1 mn 10.0

Normal tooth thickness, mm t 15.7


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dimsio iit mt rogrm o

s Trsmissio3D [5, 6]. Th scod

rogrm is mor cotio od distri-

btio rogrm (lDp) tht hs b d-

od b Hosr, t . [7, 8].

Trsmissio3D is ir iit -

mt cotct sis rogrm sciic

dsigd or ig gr d brig

cotcts d is bsd o th C cotct

sis sor. Th rogrm hs th biit

to mod com gr gomtris icdig

tooth micro-gomtris tht icd d

d roi modiictios. Th rogrm

is hbrid gorithm tht combis i

mt sis ith th ictio o

smi-tic src itgr sotio

th cotct rgio to rodc comi

trms. Ths comic trms r th

sd i Sim-t sor to

th od distribtio cross th tooth.

usig Trsmissio3D, ch o th

std gr sts is modd s si

msh ith o-rottio dgrs

rdom id to grod. a tooth msh t

t (ig. 2) ith high rsotio i th r

Fig. 8:

Transmission3

display of load

distribution an

stress contour

of the 42.833

helix angle, 10

mm face width

pinion.

Fig. 9:

Predicted nor-

malized root

stress for 100

mm face width

pinion (perfect

involute).

Fig. 10: Flat lead crown applie

to narrow contact patch gears.

Fig. 7: Transmission3D model of

the 100 mm face width mesh.

Table 2: Gear pairs used in

study 1.

Helix angle, degrees Number of pinion teeth Number of gear teeth

0.000 30 120

14.83 29 116

21.04 28 112

25.84 27 108

29.93 26 104

33.56 25 100

36.87 24 96

39.95 23 92

42.83 22 88

Fig. 11:

Transmission3D

predicted load

distribution and

root stresses

for the narrow

contact patch

25 helix angle

pinion.

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rgio is sd to ccrt ctr th

strss grdits ithi th tir root it.

ech simtio is th r or o msh

cc (o bs itch o rottio). Th

root strsss r od b srchig th

tir root it or th mimm rici

strsss cross th c idth. whi most

tooth root strss cctio rocdrs

o t strsss or o tooth ir t

tim, this mthod ts is o

cotct simtos d so icds

th dctios o th tir gr b.

Th scod rogrm sd to mod th

gr sts is th lod Distribtio progrm

(lDp). lDp is rogrm tht s

sig msh gr irs sig iit t

comic cctio i coctio ith

th icsio o Hrti dctios d

dctios o th tooth bs [7]. Th root

strsss or ch gr st r comtd

sig to dimsio bodr mt

[9] tht hs b tdd to th third

dimsio sig rocdr dod b

jrmio [10]. lDp so hs th biit to

s iit mt crtd comic c-

tios s s rormig iit mt

cctios o th root strsss.

Study 1: narrow FacewIdth gearS [n367]Th origi std s bsd o th n350

d n367 docmts rom th ISO Rtig

Committ rocdigs [3,4] tht rstd

Fig. 12:

Predicted

normalized

root stress

plot for

the nar-

row contact

patch pin-

ion.

Fig. 13:

Pinion

micro-geom-

etry modifi-

cation.

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i gr sts ith rig hi gs s gi i Tb 2. Th

c idth is q to th orm mod, mn, hich roids

r rro gr ith iio c idth to dimtr (F/d) rtio o

0.031. It s ssmd tht this s do i ordr to ssti

crt hic gr tooth orm tht cts i sr gr, sic

o c itr di highst oit o sig tooth ir cotct,

or th highst hi g gr ir.

Th gr sts o this std h rct iot rois, r-

ro c idth o 10mm, d costt orm tooth thicss.

B hodig th orm tooth thicss costt d ig th

c idth rro, th ct o th digo i o cotct oth momt rm distc to th root ctri is rdcd. This

os th rsts to b ormid d comrd to th sr

gr gomtr hr th is o cotct r r to th sht

is. asis s rormd o ch o th i sts sig both

Trsmissio3D d lDp.

figr 3 shos th iit mt msh sd i Trsmissio3D

hi ig. 4 shos both th rdictd od distribtio t th high-

st oit o sig tooth ir cotct d th root strsss or

sig sr gr tooth. Iiti obsrtios shod tht th orm -

id root strss r rdcd d icrsd mch mor rid

th th 1/cos() th tht is simir to th r cr o ig. 1.

frthr istigtio shod sigiict tistig o th odd

tooth s th hi g icrsd. This tist c b s i ig.

5 or th 42.83 hi g gr ir. This tistig shits th od

to o sid o th tooth d ths csig s strss t-

tr i th root ith th mimm root strss big octd oo-

sit to hr th tooth is odd. This tistig idcs icrsd

strsss i th root d to th shit i od distribtio to o

sid o th tooth csig ht sm to b th high ristic

rsts sho i ig. 6. Th lDp rdictios, hich do ot icd

th tistig ct, sho icrsig root strss ith hi g,

bt to mch ssr dgr th th 1/cos() cr.

Study 2: 100 MM Face wIdth gearSwhi th rios gr sts rstd i th n367 docmt

hd o id or isotig th cts o hi g, th rro

c idth rtd th crrt ss rom rodcig ristic

rsts mor commo to most gr ictios tht h mch

idr c idths. To rdc th tooth tist ct, th c idth

s sim icrsd rom 10 mm to 100 mm (F/d = 0.31 ith

th c cotct rtio rig rom 0 to 2.16, ddig o hi

g) ith othr gomtr dt big t th sm s or th

rios gr sts. Th iio torq s icrsd b ctor o

10 i ordr to rmi t th sm od r it c idth s i

th rios rro c idth std. prct iot rois r

ssmd so som ocid tistig is sti ctd t th corrs

hi ti itrrc d to tooth dctio is so ctd.

athogh ot o iiit c idth tht is ottd or th scodst o dt i ig. 1, th rsts or this c idth r ctd

to b comrb to th iiit c idth gr ir.

agi, th i gr sts r modd d d sig th

to rogrms. figr 7 shos th iit mt msh sd i

Trsmissio3D, d ig. 8 shos th odig d strsss or

o msh ositio o th 42.83 hi g iio. athogh th

tooth tist hs b rdcd, th rdictd hic gr

Fig. 15: Root stress distribution

for the 36.87 helix angle pinion

using the three analysis meth-

ods: a) Transmission3D load and

root stresses at one meshing

position; b) LDP finite element

composite root stresses; 3) LDP

boundary element root stresses

at nine difference face width

positions.

Fig. 14: Contact stress distribution for 36.87

helix angle.


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root strsss sho i ig. 9 sti occr t th dg o th tooth

c idth d sm borm high rti to th gr rtig

mods. It is itrstig to ot tht th highst idit mod,

Trsmissio3D, rdictd th highst strsss sic it sti

shos th tistig ct s s hig icrsd strsss

d to th rdcd tooth bcig bcs o th gd tooth

dg. Th lDp fe mod rdictd ss tist d hc ss

strss d i, th sim lDp mod tht dos ot mod th

tist, rdicts strsss tht r sight or.

Study 3: narrow contactPatch gearSI ordr to isot th cts o hi g, hi t th sm tim

imitig th dg cts, it s dcidd to tr ig gr

ir tht hd trs o ch o th to rios stdis. I this

std, th c idth s t t 100 mm, bt cotct tch

o 10mm i idth s id do th ctr o th tooth. This

tch s chid b ig borm high-d ri cross

45 rct o ch d o th c idth, ig o 10 rct

o th c idth i cotct. figr 10 shos th d ri sci-

ictio. Th Trsmissio3D iit mt msh is th sm s

tht sd i Std 2. I ordr to m th gr st ct i sr

gr ir, th otsid dimtrs r rdcd so tht th roi

cotct rtio s t cos to 1.0. figr 11 shos th od dis-

tribtio d root strss ttr or th 25 hi g iio. Th

cts o hi g o root strss tht r sho i ig. 12 r

o mch bttr bhd, ith o th mods giig somht

simir rsts d ch o thm rogh ooig th irs o

th cosi o th hi g ot. Hor, i o stos to thi

bot this bit, th tooth orm cross-sctios r r simir,

bt th orm od icrss b th cosi o th hi g, soooig th irs o th hi g trd is ctd.

Study 4: tyPIcal gear PaIrech o th rios stdid gr irs hs som tr tht

ms th gr st ristic. I ordr to gt mor ristic

grig sittio, th c idth s t t 100 mm d circr

roi d d modiictios r id sch tht d cts

d ti itrrc r rdcd. Th tic micro-gomtr o

th iio is sho i ig. 13. Th rir iit mt mod

s sd or th Trsmissio3D sis.

Th cotct strss distribtio ot o ig. 14 shos comt

cotct cross th tooth ith roig o o strss t

th trms o th tooth c idth d th roi. figr 15

shos th root strss ttrs o th thr mods, m,

Trsmissio3D, lDp-iit mt, d lDp bodr -

mt, rscti. Th lDp bodr mt rsts sho root

strsss ottd i 11 rct icrmts cross th c idth.

Rsts r ir simir i od distribtio d so sho simir

root strss trds ith mods shoig th highst strssd

rgio big o o sid o th tooth. as s i ig. 16, th root

strsss r ss th th rrc sr gr or hi g

ss th 25, hich shos tht this mod ot o ct

th ct o hi g, bt so icororts th ct o t

hic ctor tht is rt o th ct rdictd b th rt

stdrds. It is itrstig to ot tht Trsmissio3D d t

lDp iit mt gi most idtic rsts ith th ct

o th highst hi g dt. at o d mdim hi g

th irs o th hi g is highr th o th rdictio

athogh th aGMa rdictio sho i ig. 1 is or dir

c idth st, its s od i bth o th rdict

s o ig. 16.

SuMMaryThis r hs rstd th rsts o high idit iit

mt sis o th n367 gr sts tht r rios sd

th tio o th hi g ct o root strsss o hi

grs [3]. Th iiti sis shod ristic od distri

tio shit or highr hi g gr irs d to sbstti t

o th rro c idth tth.

atrt gr gomtris r th roosd tht might b

tr isot th cts o hi g ritio hi so s

mor ristic tooth odigs. O th or stdis tht r

ormd, th Std 4 rrgmt, hich ss rsob

c idth tooth ith modiictios tht ctr th od o t

tth gis rsts tht od b bst or comrig sohi

ctd mod rsts ith rtig cctios.

O mor cocsio is tht th d cts o rro

idth hic grs c rst i tooth tistig tht might b

m icrs dg odig. pror roi d d modii

tios c miimi ths cts d dcd od distribt

sis is ms o dtctig sch isss.

Fig. 16: Predicted normalized root stress plot fo

the more typical pinions.

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reFerenceS1) lis, w., 1893, Istigtio o th Strgth o Gr Tth,

procdigs o egirs Cb, phidhi.

2) Dd, D., 1984, Hdboo o prctic Gr Dsig. Gr-

Strgth Cctios, Chtr 2, McGr Hi, 1984.

3) Sdbrg, e., 1989, ISO/TC 60/wG 6 n367. lttr. 04 ar.

1989.

4) ISO/TC 60/wG 6 n 385. 11-13 S. 1989.

5) vir, S., 1991, a Combid Src Itgr d fiit

emt Sotio or Thr-Dimsio Cotct probm, It.

j. nmr. Mthods eg. 31, . 525545.

6) vir, S., Hosr, D.R.,1991 Cotct asis o Grs

usig Combid fiit emt d Src Itgr Mthod,

procdigs o th aGMa f Tchic Mtig, pr

91fTM16.

7) Cor, T.f. d Sirg, a., 1973. a Mthmtic progrmmig

Tchiq or th etio o lod Distribtio d Otim

Modiictios or Gr Sstms. j. eg. Id., Trs. aSMe, vo.

95, no. 4, . 1115-1123.

8) Hosr, D., 2009, Thortic Bsis o Th Ohio Stt lod

Distribtio progrm (lDp) Th Ohio Stt uirsit, Combs,

OH.

9) Cr, M., d Hosr, D., 1994, a Bodr emt

procdr or prdictig Hic Gr Root Strsss d lod

Distribtio fctors, procdigs o aGMa Tchic Corc,

St. lois.

10) jrmio, T.j., 1950, Dctios d Momts D to

Coctrtd lod o Ctir pt o Iiit lgth, j.

a. Mch., Trs. aSMe, vo. 72, . 67-72

about the author:

acknowledgements:

Dod R. Hoser is roessor emerits d odero The Ohio Stte uiersitys Ger d powerTrsmissio Reserch lbortory (Gerb). Gooie to www.gerb.org.

The thor wod ike to thk adced nmeric Sotios

lCC or the se d sort o Trsmissio3D. aso,sicere thks go to the sosors o the Ger d powerTrsmissio Reserch lbortory s we s the aGMafodtio, who roided feowshi sort or the secodthor o this stdy. prited with ermissio o the coy-right hoder, the americ Ger Mctrers associtio,1001 n. firx Street, 5th foor, aexdri, virgii 22314.Sttemets reseted i this er re those o the thorsd my ot rereset the ositio or oiio o the americGer Mctrers associtio.


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