- -- -- - -- - - - -- -

- - - - ----------

Into-Mesh lubrication of Spur GearsPart 2

L. S. Akin Gear Consultant, Banning, CA and

D. P. Townsend NASA Lewis Research. Center Cleveland, Ohio

AUTHORS:

DR. LEE5. AKIN has been working 111 mechanical engineering since1947, specializing principally in technologies related to rotatingmacflinery. About half of this time has been spent in the gear illdusfl'Yand the other half in the aerospace industry, concentrating onmechanisms involving gears and bearings as well as friction, wear, andlubrication technologies.

Since 1965, when he recei-ved his Ph.D. j'l mechanical engineering,he has bee« extensively involved in 8,earr:esearchespeci.ally related tothe scoring phenomena of gear tooth failure. 1111971he joined forceswith Mr. Dennis Townsend of NASA Lewis Research Center. andtogether they lJave produced numerous p.apers on technologies relatedto gear scoring. Since 1986, Dr. A kin has been working as a gear con-sultant with his own compemy. Gearsearch Inc.

MR. D.P. TOWNSEND is« gear COrlSuitant for NASA andfiumerous industrial companies. Townsend earned a BSME from tileUniversity of West Vir,g:inia. Duri/lg his career at NASA he has authoredover fifty papers in the gear and' bearing research area. For the pastseveral years, he has seroed in active committee roles for ASME.Presently he is Ii member of the ASME Design Engineering ExecutiveCommittee.

Albstract.An analysis was conducted for into-mesh oil jet lubrication with

an arbitra...ry offset and inclination angle from the pitch point, for thecase where the oil jet velod'ty is equal to or greater than pitch line velo-city. Equations were developed for the minimum and the maximumoil je!impingement depth. The ana lysis alsoi ncl uded the mini mumoil jet velocity required to impinge on the gear or pinion and the op-timum oiljet velocity required to obtain the best lubrication condi-tion of maximum impingement depth and gear cooling. It was shownthat the optimum oil jet velocity for best lubrication and cooling oc-curs when the oil jet velocityequaJ sthe gear pitch Ime velocity. Whenthe oil jet velocity is slightly greater than the pitch line velocity, theloaded side of the driven gear and the unloaded side of the pinionreceive the best lubrication and cooling with slightly less impingementdepth. As the jet velocity becomes much greater than the pitch linevelocity, the impingement depth is considerably reduced and maycompletely miss the pinion .

.26 Gear fechnologV

IntroductionIn the lubrication and cooling of gear teeth a va.riety of oil jet

lubrication schemes is sometimes used. A method commonlyused is a low pressure, low velocity oil jet directed at the ingo-ing mesh of the gears, as was analyzed in Reference 1..Sometimes an oil jet is directed at the outgoing mesh at lowpressures. It was shown in Reference 2 that the out-of-meshlubrication method provides a minimal impingement depth andlow cooling of the gears because of the short fling-off time andfling-off angleY) In References 4 and 5 it was shown that aradially directed oil. jet near the out-of-mesh position with theright oil pressure was the method that provided the best imp-ingement depth. Reference 6 showed this to give the best cool-ing. However, thereare still many cases where into-meshlubrication is used with low oil jet pressure, which does not pro-vide the optimum oil jet penetration and cooling. It shouldaJsobe noted that excessive into-mesh lubrication can cause highlosses in efficiency hom gear churning and trapping in the gearteeth. (7) In Reference 8 the case for into-mesh lubrication withoil jet velocity equal to or less than pitch line velocity wasanalyzed, and equations were developed for impingementdepth for several jet velocities.

The objective of the work reported here was to develop theanalytical methods for gear lubrication with the oil jet directedinto mesh and with the oil jet velocity equal to or greater thanthe pitch line velocity. When the oil.jet velocity is greater thanthe pitch line velocity for into-mesh lubrication, the impinge-ment depth is determined by the trailing end of the jet after ithas been cut off or chopped by the following tooth. Theanalysis is therefor-e somewhat different from Part I of this ar-ticie(8) for the case where the oil jet velocity is less than thepitch line velocity. The oil jet location should be offset from thepitch point with an inclined angle to obtain optimum coolingof both gear and pinion for other than one-to-one gear ratios.

The analysis presented here assumes an arbitrary offset and in-clination angle to obtain an optimum oil jet velocity for variousgear ratios. Further analysis is needed to determine the op-timum offset and inclination angle for various gear ratios.

AnalysisThe high-speed cooling jet conditions discussed in this

analysis are used only when a range of duty cycle conditionsdictate a wide operating speed range with a constant oil jetvelocity t.hat must be suitable over the whole range of speeds.Start ing with Fig. 1 the sequence of events forthe pinion in thecase where Vj>Wpf sec {3p is shown in Pigs. 1 through 4. Here,instead of tracking the head of the jet stream as in Part Iof thisarticle, (81 the trailing end Of "tail" of the stream will.be trackedafter it is chopped by thegear toothY' This is shown at "A" inFig. 3, to the final impingement at a depth "dp" on the piniontooth 2; as shown in Fig. 4. Initial impingement on the pinionstarts as the pinion top land leading edge crosses the jet streamline with inclination angle set at {3p and offset Sp as shown inFig. 1.

The position of the pinion at this time is OpJ' defined (fromFig. 1) as:

(1)

where:

inv !Pop = tan 'P3f - !Pop and!Pop = cos (rbfrlll

inv !P = tan !P - !P

Generally the arbitrarily set offset "5" for the gear establishes thevalue of {3p from:

(2)

Given (3p' then Sp. can be calculated from

so that:

The tai] of this jet stream is fmally chopped at "A" in Fig. 3 bythe gear top [and leading edge. The position of the gear at thistime is9g1 calculated:

Ogl = cos -l(Rs/R,J - inv!pog + inv!p (4)

where:

inv !Pog = tan !Pog - !Pog and

'J

. --)

PHI

F .....

July/August 1989 21

• liP\,! 01 (1:l: 4 NIl)/Pd - addendum t time

t! i*Ibt end sear backlath. reepediveIy lE, t.. time of fliaht, rotationiefaJ.. pIalon, gear back1uh at Pd - 1 V,-V, linear velocity of pinion and gear at pitch lineI'4IIiW impingement depth Vl,Vjp oU jet velocity, general, pinion controlled

~L, pJNaa, JIll' final bllpiDaerrent distance x distance &om offset perpendicular to jet lineb RlMCIiate impitt&ement distance intersection

rna l\/N, - Rtr - oJp/&J - .... ratio Vtmax)p maximum velocity at which ~ - 0

~Ns number of teeth in pinion, par V,(min)" minimum velocity at which dp - 0dfittallial number of teth {j arbitrary oil jet inclination angle

p. ....... pitch {jp c:onstraiDed inclination anglefl Jbdort at..-pitch racIil fJpp indinat!o:n angle for pitch poblt intel section

~ distance frqm pinion, gear center tp pre!ISUN angle at pitch circlestoiItUne tppl,tp. pinion and sear pressure angle at points

rtl dIataace along line of c:eId1IIrI to jet line orisin spedfJed at ifill CIiitanoe alOfIIIine of centers to jet line ~/Wi pinion par and angular velocities

iDtInection at x inVf,D tan tp - " - involute function at pitch point orttv pInicm and par outside drde diameter operatins pressure anglefJar!t. pinion and gear base radii VJ(Opt, U)p upper Umit jet velocity to impin&ement at pitch~So,~ adIIrary jet nozzle offset to intenIect O.D.'s line (at upper end of plateau)

o8Iet for pinion only

Then the position of the pinion at time equal to zero (t = 0) iscalculated from:

8p4 = mjJgl + inv I{J (5)

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28 Gear Technology

which locates the pinion at the time of the flight of the tQil of thejet stream when it is initiated. (See Fig. 3.) The jet tail continuesto approach the trailing side of the pinion tooth profile until itreaches the position shown in Fig. 4, when it terminates at timet = tf. The position of the pinion at this time is calculated from:

where:

inv 1{Jp5 = tan ¥?p5 - I{Jp:5

(6)

I{Jp5 = cos"? ([(r _ Lpsin~p/: (~COS~p)2J1f2) (7)

(See Fig ..4.)

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TABLE 1. - EQUATIONS FOR OIL JET VELOCITY AND PINION IMPINGEMENT DEPTH FOR PHCH LINE AND HIGHER OIL JET VELO([TV

Relative velocity scale Oil jet ve loc ity Pinion Impingement Depth

Critica.l high velocity *[[R2 _ R'j1/2 2 r~jl/2 apJ dp(min. U) •. 0" (m _ 1)sec a~ - (r 0 - sec wI!to mi5S the pinion vj(max)po s g- B - 8

p6 ·op <"hen m ) m (cr ir) and 0 ~ s e So ~nd9 - 9V j(mu)p .., when S - 50 and Bp - Bpp o S. :ap < app

HIgher than pi ten line..~[(Ro _ R )I/Z

- L~l dp - given ( usual design ~olution andsec al!nlocity up to where Y. s d )2 _ ,.2 cos' • ]lf2 + rJ - 8p4 - BpS Lp - [(ro - sin"pthe jet surts to miss p p

the' pinion V • g.i yen (when 0 .5. dp .5. a, ~) Iter-ate Lp from:

[(R2 _ R2)1/2sec ap - Lp] "'p - (Bp2 - ~pI) Vjo 5

dp

_ r - [(r- Lp sin . )2 + (Lp 'cos a/ j1/20 :'p

SUghtly higher and "s (R~ - R~)1/2 sec,pitch \I ne-upper end Y j( opt. U)p . .~ dp(tai 1) - -, (trail Ing profile only)of velocity plateau

891

v J ."pr sec 'p .OlgA sec Ip dp .~.(1 * 4'Np/2) JPd' (both profiles)

The design solution to the problem of pinion cooling when"d," is specified is to solve explicitly for the jet Vi based on thefact that tf = to) as shown in Part I of this article. Thus, the re-quired jet velocity is calculated from:

[(R2 - R2)1/2 sec R - TIv. = 0 . s . tJp '1> wpJ

8p4 - Bps(8)

where:

The analysis solution to the problem when Vi is specified[Vj(Opt, tn, < Vi < 00 J so that the resulting impingementdepth dp' can be calculated implicitly by solving iteratively for"Lp" from:

wp[(R~ - R;)1I2 sec {Jp - Lpl = (8p4 - 8ps)Vi then (IO)

dp = ro - t (r - L, sin {Jp)2 + (L, cos {Jp)2j1!2 (11)

(See Fig. 1.)

Moving up along the velocity scale of Table 1from wpI" sec (Jp= Vj' it can be shown that the upper limit Vi(Opt, U) for the"constant impingement depth range" where dp = a, can becalculated for the pinion from:

Thus, if the jet velocity VI is between the lower limit Vj(Opt,u, < Vj < Vi(Opt, U)p' then the impingement depth will be dp

= a = 1/Pd on at least one side of the tooth profile. If Vi = wprsec (Jp exactly. then d", = a on both sides of the tooth profile. in-creasing the jet velocity above Vj(Opt, U)p reduces the impinge-ment depth dp until at Vj(max)p the tail of the jet chopped by the

30 Gear Technology

gear tooth is moving 50 fast as to be just missed by the pinion topland leading edge "AU in Fig. 6 when mg > ffig(crit). The upperlimit critical gear ratio, as a function of Np and assumingVi(max)p = 00. may be calculated from;

IIlg(crit)=

cos-1 Np - inv (COS-lNpCOS¥') + invll' + w/Np - 2 B p/NpN p + 2 Np + 2

_l(m,(Crit)Np + 2PdS) , . +Icos 0 - JnV coslIlg(crit) + 2 (

Np cose ) + .__ L-----'-__ , IIlV¥'

Np +2/mg(crit) (13)

PlH

where:

and

and

N~ = INpcos2!Sp - {(Np + 2)2: - N~cos2t3p}lt2sint3p](14)

{3p= tan.-1 {S Pdl[mg(crit)Np(1 - PdS)2 - (PdS)Zj1/2}(15)

PdS = (StSo)[mg(crit) - 1][mg(crit) + 1]

when S= So and fJ = t3pp,Vj(max)p < oe , the mg(crit) ceasesto exist.

When the maximum jet stream velocity (Vj = Vi(max)p) isreached, the initial position of the pinion as the gear tooth chopsthe tail of the jet stream may be calculated from:

(16)

(See Fig. 5.lThe final position at the point "AU in Fig. 6 is calculated from:

The maximum jet velocity may then be calculated from:

Vi(max)p

= wpHR! - Rs~1I2 sec!Sp - (r! - ~1t2 sec !Spl(Jp6 - (Jop

(18)

PI

r.... l.

Therefore, when Vj ~ Vj(max}p' dp = 0, if mg > mg(crit).Also if 5= So and !S = !Sp = t3pp' then Vj(max)p - oc.Vj(max)p = Vi(Opt, U) then Vj(max)p is set equal to Vj(Opt,U). Stated differentty, when Vj(Opt, U) is greater 'than 01 equalto the calculated Vj(max)p, then Vj(max)p no longer physical-ly represents the solution, and Vj(Opt, U) is the maximumvalue that can be allowed for Vj'

Again. it should be noted that the selection or specification forVi must be kept within the bounds of w" sec (3pand Eq-uation 18if impingement on the trailing side of the tooth profile is desired.

Equations 8 through 18 have been summarized in Table 1 ona velocity scale to add graphic visibility to their usability range.

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The Gear.- When the Jet Vdocity is Greater than Pitch LineVelocity (Vi >w,R sec 1&).

The sequence of events for the gear in the case where Vi >wgR sec /3p is shown in Figs. 5, 7 and 8. Again, instead of track-Ln~'the head, the trailing end or "tail" of the jet stream will betrackedafter it is chopped at time (t = 0) as shown at "A" in Fig.7, to the final impingement point at a depth "d.g" at time t = tw,

as shown in Fig. 8. The position of the pinion at time (t = 0) maybe calculated from O'pa... defined above. The associated gear posi-tion can be calculated from:

(19)

which locates the gear at the time (t = 0) when the flight of thetail of the jet stream is initialed. (See Fig. 7.)Theposition of thegear when the jet stream tail is terminated on the gear may becalculated from,

e '7 = tan-1 (. Lgeos /3p ). + inv I{J '7g . R L . ·8- + g sin i3p

where

inv I{Jg'7= tan 'Pg7-'Pg7

'Pg7 = C05-1 ([(R + Lg sin .Bp)~: (l"g cos i3

p)lJlI2) (21)'

at time (I = t.,), as shown in Fig. 8.Once again. when O~ e,< i3pp and 0 ~ 5 < So, the analysis

solution to the problem of cooling the gear is constrained by thejet velocity limits for the pinion to maintain impingem.ent 'onsame. And, as explained above, a given "gear mesh" must havea common jet velocity. Accordingly, a given impingementdep th is selected for the pinion. Then, the associated jet velocityVip is solved for this velocity, which can then be used to findthe associated gear impingement depth "dg". Thus, after fin-ding Vjp, solve for Lg iteratively from:

and

dg = Ro - [(R + Lg sin {3p)2 + (lg cos {3p?11/2 (23)

(mntil1ued on page 45)

......,

t.

July/August 1989 33

(20)

(contil1ued from p~ge 33)

TA8U 2 .• - EQUATIONS FOR OIL JU vELOCIfY ANVr.tAR I.MPINGEMENT DEPTH FOR P[TtH LlNf ANlJ HI'-HER OIL JH VELOCITIES

, • =wIer! - t{)lI2MC ~ - ~ - -l- f' COl ."a1"2 - • It II 'pil ..- 't7

..... 'l,ata" j, 'ot j, 'Jl-),'J tt_

I Relub' ... ee-I ttJ' tell.

Ire......,.tch H.weleettJ'lIPao........ onjet ItvtI coIt II tile .....

CrfUCI. IIflltweloctt,so t.~II tile ....

Slttllll1 ......llid "tell 1t1l-..,... ... ",

.. Ioctt)' pllte1u

d,(IIII, U) • '0 - It. • l,I") .tn " 1 • lL,C_) ca .,f 111........., ~ _.(cr") ellMIt 1I,(lI1n, ., • 0 ... 5 • So ItICI .., ....

d, ,'MI'! (I11III1 .... IOl11Uon)

l • ((R. _ ~t ". C012 .,)111 _ • I ...

IWitt Lg "..I 2

[(ro - ro) soc "

d, •• 0 - Ul L, st • .,,2+ (l, _ ."

1111 •• • ~ .. t: (tramng prom.)d II

dg ••• (1 "',;12) (IIotll profUes)

The design solution to the problem when Vj(min)g < Vi <Vj(max)g may be calculated from:

v. = [(r~-r~l!2sec.Bp- {(Ro-dgf-R2cos1,Bp}1I2+Rsin.BpJwgJ

(Jg6-8g7 (24)

with the additional restriction that Vj(Opt, U}g< Vi < Vj(max)g= Vj(max)'p- As for the others, Equation 24! is shown placed onthe velocity scale of Table 2.

Also if Vj is specified within the range allowed for Vjp torEquation 8 and 24, thendg can be calculated implicitly by solv-ing iteratively fo·r "Lg" hom:

I(~ - ~)11~sec ~p - LgJWg = (8g6 - I':Ig7)Vip and (25)

dg = Ro - [(R + Lg sin {3p)2 + (Lg 'cos ,Bp)2]112 (26)

When the Jet Velocity is Equal to Pitch Line Velocity(Vj = wiR sec @g)

Continuing up the velocity scale of Table 2 from VI = wgRsec .Bp, it can 00 shown that the upper limit for the constant im-pingement depth range, where dg = a, can be calculated for thegear from:

Thus, if the jet velocity Vj is between V;(Opt, L)g < Vi<Vj(Opt, U)s, the impingement depth will be dig = a = lIPd. onat least one side of the gear tooth profile. If Vi = wgRsec .Bpex-actly, the dp = "a" on both sides of the tooth profile.

Increasing the jet velocity above Vj(Opt, U}greduces the im-pingement depth dg until Vi = Vj(max)g' When S < So' the tailof the jet chopped by the gear tooth is moving so fast as to be justmissed by the pinion top land so that dp= 0 and ffig =mg(lim.). When S = So and Vj(maxJg - 00, then dp - o.

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and as shown inFig. 5 (for S = 0). The limit position "Au in Fig.6 as the jet tail just missesthe pinion top land is calculated from:

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(31)

If Vi(max)g 5 Vj(Opt, U) then set Vj(max)gequal. to Vj(Opt, U).Also, it should beobse:rved that since only one ViPcan be used,we must set Equations 18 and.29 equal: Vj(max)p· Vj(max)gfor the given ms' making the design mg = mg(lim) when mg >mg(crit). .Vj(max)g in gear parameters may be calculated from:

SummaryAn analysis was conducted for into-mesh all. jet lubrication

with an arbitrary offsetand inclination angle from the pitehpoint for the case where the oil jet velocity is equal. to or greaterthan pitch line velocity. Equations were developed forminimum and maximum oil jet impingement depths. The equa-tions were also developed for the maximum oil jet velocityallowed, so as to impinge on the pinion and the optimum oil jetvelocity required to obtain the best lubrication conditlon ofmaximum impingement depth and geas tooth cooling, ThefollOWing results were obtained:1. The optimum operating condition .for best lubricaticn and

cooling isprovided when the jet velocity isequal to pitch linevelocity Vi = Vgsec {3p =wprsec (3p = wgR sec :BPI whereby

Note that as Vg(max)g - co for mg::; mg{crit):and when mg >mg(crit),then Vj(max)g is finite at d, = O.

The impingement distance Lg(max) when Vi = Vj(max)gmay be iterated from:

Then, when S < So

dg = dg(min, U) = Ro - {[R + Lg(ma:x) sin m2 +

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IForestCity Gear featuf,ing the mostmodem gear machinery available toany gear job shop in the worl'd seeksa well q,ualifled set-up individuaJ forhobbing and shaping. Iinspection ex-perience would be desir,aJ:)'le.Goodwages and benefits plus an exoenento,pportunity to 'grow with modemtechnology. Please send yourresume to:

FOllest City 'Gear Co.P.O. Bo)(80

IRoscoe, u 61073(815) 62~2168,

Bring in new customers for your business by advertising inGEAR TECHNOLOGY, The Journal of Gear Manufacturing.

Callf·3;12J ,437-6604

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both sides of the pinion and gear will be wetted. and themaximum impingement depth to the pitch line will beobtained.

2. When the jet velocity is slightly greater than the pitch linevelocity, w" sec fJ,p< Vi < Vi(Opt. U), the loaded. side of thedriven gear is favored and receives the best cooling withslightlyless oil impingement than when Vj = wpr sec wp'

3. As the jet velocity becomes much greater than the pitch linevelocity, Vj(Opt, U) < Vi < V1(max)p, the impingementdepth is ,considerably reduced. As a resuh, the pinion maybe completely missed by the lubricantso that no direct 0001-

ing of the pinion is provided when V1(max),pS Vi'

Refer· nees1. AJGN, t, and TO\IVNSEND. D. '~CoolingofSpurGearswithOil

Jet Di.rectedinto the Engaging Side of Mesh at Pitch Point", 1SME.Proc. of Inter. Symposium 011 Gearing and Power Tr:ansmissions.Vol. 1. M, 1981. pp. 2.61-274. -

2. TOWNSEND, D. and AKIN. L. "Study of lubricant Jet FlowPhenomena in SPill' 'Gears - Ou.I-of~Mesh Condition", Trans.ASME,]. Medl. Design, Vol. ioo, No.1 .• Jan. 1978, pp. 6,1-68.

3. HEIJNlNGEN. G.J.J. VAN and BlOK. H. "Continuous asAgainst In'termittent .Fling-Of£ Cooling of Gear Teeth". Trans.

GEAR GRINDERSIPrecision aircraft gem rnanutac-tursr i;n Mt. Clemens, Mil, hasopenings for 'experienced RedIAing& Heishauer Operators.Must Ibeable to do own set ..up '&work overtime. We are in a newstat&Of·th~al1facility located ina 'fural area. Please send a let-ter wi:th your work history to;

A'Oft I'NDUS:l1R1ES,. INC.15375 Twenty Three lMile' RoadMt Clemens, Mil 48044·9680

Equal Opportumty Employe!

A5ME Journal of.Lub. Tech., Vol. 96. No.4, oe. 1974, pp.529-538. --

4. AKlN. L., MROSS,J., and TOV\INSEND, D. "Theory forth a-feetof Windage on the Lubricant Flow inth Tooth Spare; of SpurGears". Trans. ASME. t. Eng·rg. for Industry, Vol. 79, Ser. B, No.4.1975. pp, 1266-1273.

5. AKIN. L. and TOWNSEND, D. "Study.of Lubricant Jet FlowPhenomena in Spur Gears", Tra,1S ASME. /. Lubrication J:l!ch.,Vol. 97, Ser. F, No.2, 1975, PI' ..283-.258.

6. TOWNSEND. D. and AKIN. L. "Analytical and Experim nt ISpur Gear Tooth Temperature as Affected by OperatLngVariables", 1980 Trans. I. Mech. Design. ASME, Vol. 103. No.4., Jan. 1981, pp. 219-226.

7. TO\l\i'NSEND. DENNIS P. "The Applications of Elaslohydro-dynamic Lubrication in Gear Tooth Contacts". NASA TMX68142, Oct. 1972.

8. AKiN. L.S. and TOWNSEND D.P. "Into Mesh Lubrication ofSpur Gears and Arbitrary Offset. Part 1 - For Oil Jet VelocityLess TI1~n or Equal To Gear Velocity", Gear Tech,IO!'ogy. Vol. 6.No.3, MayIJune. 1989, p, 30.

Adc.nowledgemmt: R.eprintM courtesy of American Sod£ty of MecmmiclllEngineers v from their WinteT.Annual Meetil'lg. PhO('f!iz, AZ. Nowmbe, 15-19,1982. and ASME Journal of MKhanlsms, Transmisions, and Autornillion in[)esjg;n, Vol. 105. D c.. 1983. pp. 713-724.

Chlel Englneers/ManufacturingProcessing Engineers,

S45,QOO.$65.000.Gear ProcessingJOesignlOuotes.

Ann Hunsucker, Excel AaaoclatelP.O. Box 520. Cordova, T1N13801B

(901) 757·9600 ,or Fax: i(90117~2B96

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