The'Journal of Gear Ma,nufacturing

M'ARCH/APRIL 1990

•

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OIRCLE A-3 ON READER REPLYCARD

CAPITAL GAINS, SOCI'ETAL GAI'NS,,OR NO GAINS AT ALL.

Taxes may be one of the only twosure things in Ilife, but that doesn't makethem popular. Nobody is happy to paythem, and the bigger the amount due,the unhappier the taxpayer. Con-versely, politicians know that comingout in favor of a tax cut is the equivalentof voting for apple pie and mother-hood, It's a sure-fire success at the ball'otbox.

Therefore, it should come as no sur-prise that Congress - in spite of themassive federal deficit - is again con-sidenng culting taxes, this time oncapital gains. 0 matter that thebenefits of such a cut are debatable atbest; thai. cutting taxes in the face oft helargest deficit in our history is based onlogic right out of Alice in Wonderland,or that a capital gains tax cut will prob-ably benefit comparatively few Amer-icans. Tax cuts of any kind play well inPeoria and everywhereelse. Accordingto Speaker of the House, Tom Foley, ifyou tell an ordinary taxpayer that hewill reap $1'0 from a measure that willsave the likes of Donald Trump anaverage of $25,000 in one year, he willsay, "Fine, give me my $10.00."

Perhaps that's true. Certainly it makesa convenient excuse for our elected of-flcals when they are explaining whythey are going to take the easy way out- again.

Before we go farther, I think irsimportant to remind ourselves-of a fun-damental fact about tax Jaw. Unlike therules in engineering, mathematics, or

other kinds of accounting, tax laws donot follow any clearly apparent logic.There is no particular inevitability aboutthem because, contrary to what mayappear on the surface, the purpose oftaxes is far more than just raising moneyto pay for running the government. Thepower of taxation is the power toengineer society, to redistribute wealth,to encourage certain economic prac-tices and make others prohibitivelyexpensive.

Given this enormous implicit power,we should think long and hard aboutexactly what we are doing when wesupport tax law changes. Before wejump on the capital gains tax cut bandwagon, we should ask the all-importantquestion, Cui bono?Who benefits? Willanyone? Will a cut in the capital gainslax get the money to where we want itto go to do the things we as a country

want and need to have done?My own feeling is that the answer lis

UNo./~Some of the economic issues which

w need to focus on in the nextdecade(s) include lowering the deficit,increasing the personal savings rate ofAmericans, lowering the cost of capital,and encouraging more long-term in-vestment, all of which will improve oureconomic health and make IUS morecompetitive. I don't think the capitalgains tax cut is the most effecti,ve way toachieve any of these goals.

This tax cut will not, in the long run,do anything 10 lower the deficit -probably our number oneeconomiproblem. All the voodoo economics tothe contrary, somewhere down theroad, the piper will have to be paid. Theonly way the deficit is going to comedown is by raising taxes or cuttingspending or - in vitably, I think -both. The logic that a cuI in the capitalgains tax will spur productive invest-ment is debatable to begin with; andwhile it may provide aninitial surg inrevenues - approximately $3 to $4billion annually for the-first couple ofyears - after that, it would begincosting the Treasury an estimated $5billion a year in lost receipts. That's $4or $5 bi lilian a year we can't afford if weare interested in cutting the deficit.

Treasury Secretary, Nicholas Brady,told the American Business Conferencelast year, 'We have been willing tomortgage our future, to cut com rs in

Morch/Aprill1990 5

pursuit of immediate payoff." Voting fora politically palatable tax cut now forany segment of the economy and wor-rying about the consequences later isanother case of that currently all-too-popular American pastime - takingthemoney and running.

Even if any kind of a tax cut madesense, of equal importance is the factthat the capital gains tax cut does notaddress those other crucial concerns inour economy - encouraging savingsby individuals, lowering the cost ofcapital, and encouraging long-term in-vestment in improving national produc-tivity. Whatever its benefits, thecapitalgains tax will impact a relatively few,relatively prosperous Americans. It willdo little to encourage the averageAmerican, whose income is below$50,000, to save. Consequently, rein-trod uc ing sam e form of tax -deferred ortax-free savings, like the 1M, might bea better move toward the goal of en-couraglng Americans to save more andspend less. It would affect far more peo-ple than the capital gains tax and ad-dress a crucial societal need.

In a recent article in the Wall StreetJournal, Peter Drucker points out thathistorically, tax exemptions ordeferments on savings are most oftentaken advantage of by lower and mid-dle income people. Even when the rateof return on such plans is small, they arepopular with the majority of people -and they work. According to Drucker,the taxexempt savings account wasone of the chief instruments in turningIapan from a country with one of thelowest savings rates (prior to World WarII) to a country with one of the highestin the world.

Increasing savings would go a longway toward decreasing the cost ofcapital in the U.S. The advantage thatother countries have over us in this areais one of the factors that strip us of ourcompetitiveness in global markets. AU.s ..firm might have to pay as much as10-15% to borrow money for a projectwith a lO-year payoff, while a Japanesefirm, only 5%.

It is true that a cut in the capital gainstax would encourage some forms of in-vestment and some industries, butgiven that the taxing power has implica-tions far beyond the mere raising ofmoney, I wonder if these are the in-dustries and investments we wish to en-6 Gear Technology

courage right now. Some of the chiefbeneficiaries of a. capital gains tax cutwould be real estate and the securitiesmarkets, and we must ask if strengthen-ing them benefits the economy as awhole. People in these businesses donot make things; they make deals; andthe question is, are more deals or morecompetitive products what the countryneeds now? Real estate markets arealready over-inflated in many majormetropo'litan areas. Do the buyers andsellers of real estate need more en-couragement to keep up the pace ofturnover just to benefit from a capitalgains tax cut? In securities trading, thequestion is even more pertinent. Manyeconomists and securities traders them-selves are beginning to question thebenefits of leveraged buyout fever.Many viable companies have beengutted and trashed by high-powered in-vestors seeking short-term profits. A cutin the capital gains tax, unless it appliedonly to assets held over some long-termperiod, liike five years, would only en-courage this kind of short-term, short-sighted profit taking.

Instead of subsidizing this kind ofdeal-making mania, a lookat tax incen-tives that would encourage more savingand long term investment in the pro-duction and manufacturing portion ofthe economy would seem to set theright priorities. Those of us in capitalin-tensive businesses like gear manufac-turing are all too aware of the burdenthat the high cost of money places onus. And we are acutely aware of theneed for more money to be spent onresearch, development, and purchasesof resources to get and keep America'smanufacturing base viable and com-petitive. A tax break that would lowerthe cost of borrowing and long term in-vestment would be a direct benefit forus, but, more important, it would be astep in the right direction for oureconomy as a whole.

Alteri ng the taxi ng practices andspending habits of a generation ad-dicted to consumerism and instantgratification will not be easy, but not im-possible. There are plenty of sugges-tions outthere from rei nstituti ng the I!Mto giving tax breaks to companiesengaged in long-term research, butbecause they are not quick, simple, orpolitically popular, they die on the vinefor lack of interest. If the members of

Congress would devote as much crea-tivity to structuring the tax code to en-courage savings and long-term invest-ment as they do to avoidingthe politicalconsequences for voting their own payraises, we might all be better off ..

But in the final analysis, the crucialquestion is not really one oftaxes at all.Which taxes sound like the best oneswill depend to a good extent on whosepockets get emptied. As George Ber-nard Shaw says, 'The governmentwhich robs Peter to pay Pa1uI canalways depend on the support of Paul."

The issue here, it seems to me, iswhether or not we have a Congresswith the courage to use the taxingpower as it was meant to be used. Doesit have the courage to teU the Americanpeople what it can afford and what itcan't? Can it risk the short-term dis-pleasu re of the few - or maybe even ofthe many - for the long-term gain ofthe whole society? Does it have thenerve to do the right thing rather thanthe politically expedient thing?

I don't know. I do know that gettingAmerica back on track economica.lly isnot something that will come cheap oreasy or overnight; that the power to taxis a crucial tool for making theeconomic and social policy required toget the job done; and that it will takeleaders, not politicians, in governmentto use that tool properly.

EDITOR'S NOTE: As we go to press,there is talk that the President mightreintmduce the capital gains tax pro-posal, along with a plan yet to be for-mulated, to encourage savings. We'll allhave to watch how the President andCongress handle vet another oppor-tunity to be statesmen imtead ofpoliticians.

TECHNICAL CALENDAAAGMA Technical Education Sem-mars, Series n..These one- or twCKlayseminars present 'the latest techniquesand information on specific topics ina small group context. tor more in-{ormation regarding fees and registra-tion contact AGMA. 1500 King 51..Suite 201. Alexandria. VA 22314.(703) 684~211.

MARCH 6, 1990. Cincinnati, DHRaUonal Gear Design and Lubrica-ltion.. Design procedures f·or gearsetswith maximum pitting and wear re-sistance, bending strength, and scuff-ing resistance. Discussion ofgeometry. materials. processing, andlubricants.

MAY 8, 1990. Los Angeles. CAControlling the Carb~rizing Process

JUNE 5-6, 1990. AJeJtandria, VAGear System Design .for Noise Control

MAY 1-3, 1990. Houston TooW &Manufacturing Expo itlen, George R.Drown Convention Cent~, Hous'ton"TIC Both national and local exhibitorso.f CAD/CAM equipment, rob tics,communication and informationsystems. precision machine parts, andquality control systems, For more in-formation, contact: Houston Ch'ap~er,NTMA. 515 Post Oak Blvd" Suite320, Houston, TX 77027. (713)439-5890..

JUNE 7-10, 1990. AGMA AnnualMeeting. The Homestead. HotSprings, VA. For more information,contact AGMA headquarters,

SEPTEMDER 5-13, 1990.. mMTS-90,McCormick Place, Chicago, It.Largest trade show in the Westernhemisphere with exhibitors fromaround the world. A new feature thisyear will be the UO,ooo sq. ft,Fanning and Fabricating Pavilion. f~rmoreinfermation, contact: ]MTS-90,7901 Westpark Drive, Mclean, VA.22l024269. Ph:(703) 893-2900. Fax:(703) 898-1151.

.oCTOBER .29'-31,1990. AGMA FallTechnical Meeting. Hilton Interna-tional Hotel, Toronto, Canada.Papers and seminars on gear design,analysis, manufacturing, applications,gear drives, and related products.Contact AGMA Headquarters for fur-therinJormation. CIRCLE A-S.oN READ'lR IR1iPlV CARD

Marchi Aptll 11990 ,

Gear Noise and the Making of Si'ent GearsJ. Liu, H.I. Wu, M.H. (Jian, c.P ..Chen,

DaGan University of Technology, DaHan, China

Summary:QUI research group has been engaged in the study of gear

noise for some nine years and has succeeded in cutting thenoise from an average level of some 81-83 dBto 76-78 dB byboth experimental and theoretical research. Experimentalresearch centered on the investigation into the relation be-tween the gear error and noise. Theoretical research centeredon the geometry and kinematics of the meshing process ofgears with geometric error. A phenomenon caned "out-of-bound meshing of gears" was discovered and mathemat-ically proven, and an in-depth analysis of the change-overprocess from the meshing of one pair of teeth to the next isfollowed, which leads to the conclusion we are using to solvethe gear noise problem. The authors also suggest some op-timized profiles to ensure silent transmission, and a newdefinition of profile error is suggested.

IntroductionFor some nine years, OUI research group has been engag-

ing in the study of gear noise and the making of silent gears,six papers in English have been published on the internationalconferences and periodicals (see Reference 1-6). Experimentswere done on machine tool headstocks' power gears, as theyrepresent the sort of light-load, medium-speed gears for whichsilent transmission is such a problem. A new single flank totalcomposite error tester (Initiated by Huang Tonglian of China.See Reference 4,) was used to find the relation between geargeometry and the noise. The tester is able to .indicate the pitcherror, the profile error, the final transmission error and thechange-over character (from the meshing of one pair of teethto the next). Conclusions drawn from these experiments havealso been varified by machine testing. When we reached therequired accuracy, the gears offer the expected silent transmis-sion. (Under 78 dB measured at 300mm away from pitchpoint, minimum 73 dB.)

for theoretical analysis, a deep investigation has been madeon the meshing process of involute spur gears. A phenome-non called "out-of-bound meshing" of gears was discovered;that is, with the presence of error, the actual contact line ofa pair of gears is much discrepant from the theoretical one.It usually goes along a broken line linked by a sector of theaddendum circle of one gear and a part of the theoretical con-tact line, (4-5) With the disclosure of this phenomenon aprecise and thorough statement on the change-over processof gear meshing (a geometry and kinematics of the change-over process of gear meshing) is made.8 Gea(Technology

Summarizing from the above mentioned experiments andanalysis we fonned the following hypothesis concerning gearnoise.

Dynamic Behavior-and Change-OverProcess of Gear Transmission

Out-of-bound meshing and change-over impact of gears.Since gear noise is largely induced from change-over impact,thorough analysis of the change-over process must be made.But before entering the study of it, two things must bementioned.

a. Out-at-bound meshing (ORM) of gears. - When a gearis meshing with its pinion, theoretically speaking, the contactpoint will move along the common tangent 1-4 (See Table Ia),starting from 1(the intersecting point of addendum. circle ofdriven gear with glg2), and ending at 4 (the intersecting pointof addendum circle of driving gear with glg2l. On most oc-casions, there are two adjacent pairs of teeth in mesh, but,ac-tually, there is always a difference in normal pitches of driv-ing and driven gears, Assume t'o>to as in the case of TableIa, the contact point, after passing 4, wi1l continue themeshing and the gap at l' will be narrowed until at point 4",the next pair of teeth come into meshing at e' and take overthe transmission. So the actual contact line is a broken linee'44". There is a shock at the instant of take-over, called "leav-ing impact," of OBM. Table 1b shows the other case, t'o

Z - Number 01' leeth

b)

fig. 1

research work, It is an all-round gear tester developed fromthe ordinary single Aank gear transmission error t·ester.(4)The result coming out is the transmission error of every in-dividual pair of teeth (from tip to root), but overlapped intoa polar coordinate, where e of the polar coordinate representsthe transmission error, while (J of the polar coordinaterepresents the angular position of gear, which detereunes theposition of contact point on the profile, expressed by itsgenerating angle tPc. (See Fig. 1a) The greatest merit of thistester is its thorough exposure of the change-overcharacteristic. Fig. 2 shows the expression of different errorson the tester's record. Fig ..1 indicates the characteristic pointson the profile. T and R are the tip and root points of the wholeprofile. The contact line is g1g2. The pitch point is pt. Thegenerating angle of point c is tPc' which also indicates theangular position of the gear when the contact point passes c.B C is the basic section of profile which offers a contact ratio

1. Fig. Ib shows the location of the points on the tester'srecord. where 9b=Oc=360 /(2z), where z = the number ofteeth. Dot limes show the OBM. TB and CR are sectionsgenerally considered for tip/root relief,

Now let us see how the change-over takes place. FromTable Ia, lb. one may think that the way of change-overcould be either as shown in Table 1a or Table lb. Butremember that both Table 1a and Table Ib are based on theassumption that the profile is an involute curve with error inpressure angle (generated from a base circle with error indiameter), but without tip (root) relief. (TN in Fig. 3a.) Herethe change-over point falls on the DaM zone, Now let's seeFig. 3. If the profile has tip relief 1-2IaJ'ge enough to.cover thenormal pitch error. then the change-over point e' will fallwithin the profile 2lR. It iscalled inside change-over as shownin Table IcIt is a change-over without OBM. The convexpro. iles (See Fig. 14, Fig. 15) also belong 10 this group.

Fig. 3b shows the case with profil shape 'error. It is a~ypical saddle form profile. The pressure angle is too small.with a relative reHef, and the outside change-over will betransformed into inside change-over .

Table Ic shows a profile with normal pressure angle, butwith tip relief to obtain inside change-over. Since the tip reliefshould be large enough to cover the normal pitch error, afairly large intersecting angle 6'results, and consequenently,little benefi't could be gained in reducing noise. Table 1d

,I

b) Error in circuldJ" pitch (irUxing error)

dl Pressure angle 100 large (rool hlgher)

e) Concave profile (OU! of involute) + lip too high

a) Perfect in geometry

~~ , I :'T T T R]I .l ]

Fig. Z - Geometric 'errors expressed on tester's record.

TR - original profile12 - tip relievinge - original change over pointe1 - chang ov r point after tip relieving

T 1

y~.. el./\I~",.. Z .' I \I I I

b)

Flg. 3 - Transformation of outside change over to inside chang over bytip relieving

shows another way of transforming outside change-over intoinside change-over. It is done by adapting the pressure angleto obtain. correct normal pitch (normal pitch to = t·,c~sa,varying a, we can get right to). expressing on the testersrecord, the geometric expression of right to is !that th· adja-cent curves willi link up very welL So if the pressur a.nglecanbe adapted to make the adjacent curves meet very we'U,a verysmall "relative relief" can guarantee the inside change-ever,

Mcrch/Ap~1I1990 9

Table 1 Change-over character of gearsOutside change-over Inside ~~ver

driven I ~ driven

~iQf\.,\:'i 0 \. -- g1.

~

Figure of .~• 4 \:.......... !contact \t>- < --~-.- , e ,- \\.gl ~ = \::::

..-- .\ ;:~~ ./driving I\)~

-----driving

Contact line e'4-4" 2H-2-e e'-e

RI R2 R T2 T3 eTester's

~~TI~ ~~ ~record r- I \ I " , e' \: I IIT tTl IT) , - R\ I R2 I R3 " ' I I II II I I f

Type (a) Type (b) Type (c) Type (d)Impact happens at leaving at approaching at change-over point e' eBrief Change-over impact Change-over impact Pressure angle nor- Inside change-overDescription induced by too small induced by too large mal. inside change- obtained by adapting

pressure angle pressure angle over obtained by tip pressure angle to(root) relief which obtain correct norma1 ,should be great pitch (in tester's R!COrdenough to cover the to link up the adjacentindexing error (error curves), with slightestin circular pitch) tip (root) relief to

avoidOBMCharacteristic Normal pitch difference Intersecting angle 9 at change-over pointparameter for .:1to=to (driving)-to(driven)

£gear noise to= normal pitch of gearI

\I Ir~

while intersecting angle (),and hence the impact can be keptvery small. It is apparent that this way of obtaining insidechange-over (Table Id) is much more favorable than theprevious one (Table Ic),

Outside change-over is generally considered as un-favorable, This is because, in the OBM process, the tip of onegear tooth actually does not mesh with the profile of the othergear. There is no common tangent for them. The tip edge ofone gear tooth just scrapes the profile of other gear, and whatis more, theOBM curve in .thetestet'sl'€Cord is steep. (See Fig,2c.) This results in a very large intersecting angle, causing im-pad. So a gear with approaching OBM is a typically noisygear.

But leaving OBM acts quite differently. This is due to thephenomenon of losing contact in gear meshing, As shown inFig. S, it goes along the dot line. (For detail see Fig..16c.)

Experiments show that a gear system with nonnal pitch ofthe driving gear is a little bit larger than the normal pitch ofthe driven. gear offers fairly silent transmission,

So there are four typical forms of change -over as listed inTable 1. for the outside change-over, the characteristic10 GearTechnology

parameter is the normal pitch ditference .:1to; and for insidechange-over, it is the intersecting angle (J which determines theimpact and hence the noise. (See Fig. 4.)

Some discussion on low noise gear profile and the defini-tion of profile error ..In the preceding section, we have dis-cussed the gear noise at the instant of take-over. Now let ussee what happens during the whole process of meshing of apair of tooth profiles (from root to tip), They work just likea pair of cams. According to the theory of conjugation, theyshou1d be made .into accurate involute curves (not concavenor con vex on tester's record) to ensure smooth transmissionduring this period. The characteristic parameter for noise inthis case is the reciprocal of the radius of curvature of thecurve on the tester's record, (See Reference 4.) But as men-tioned above, in order to 'ensure smooth change-over,pressure angle should be adapted to link up the adjacentcurves. So we suggest redefining the profile error, splitting itinto two items:

a) Discrepancy from involute curve, that is, draw the twoclosest involute curves of the same base circle (not necessarilythe nominated base circle), The normal distance of them, M

Table 2 Gear noise and its geometric error

Il.O.... .... c::l! l! '1:;

I I, 1100 E E sI ::;j c ::;j ~c:; c:; t: ..... ~ ....·s CII " 0 ~.... > .. v 0~

ci f§ '1:!II '1: nj ...... - 0 >Z c CII c CII

J041..3ifo - -1..----..3if - -0.5

2251 At. = +3

--

Table - 3, An index to Hnd tester's record with known number of gear

Number of gear 0151 0152 0311 0331 0941 702L 703L 706L 709L 710LNumber of Fig. 10 10 10 10 09 13 11 12 13 11

, 1432 1641 2031 2032 2161 2251 2252 3041 711L 1331I 10 09 10 10 09 09 10 09 12 10

703L type: (b)

..--_ ~4 =-3M~ -1

7061. type: (d)

_'::11,,= =/M-O

70Zl type: (d)

.....---- !lt~- II1f - 0

710L type: (b)

,.....- .... ~to =. -2.:11- -2

711L type: (dial

!lt~= +2.....--... M =-1

709L.o.t" - I»: - ......I1f - -0.5

Fig. 11- Ground on Chinese wormwheel gear grinder 7232A

Fig. 12 - Ground on Swiss wormwheel gear grinder NZA

gear grinder 7232A(703L 710t).Fig. U shows gears groundon Swiss gear grinder NZA (706L 711L).Fig. 13 shows gearsground on Chinese gear grinder 7232Awith final free grind-ing. (702L,709l).Table 3 is an index to find the tester's recordof known gear number.

Comparison of noise level of gears ground on different geargrinders. Experiments were done mainly on worm abrasivewheel gear grinders because they are productive, and becausethey show promise for manufacturing silent gears. Table 4shows the noise level measured at 300mm from the pitchpoint of the meshing gears. From these test results, one seesthat with the addition of afinal worm wheel free grinding, thenoise level was dropped froman average of 8O.7dB to 76.7dB.Fig. 14 shows the tester's records from Swiss grinder AZA.The machine tool isaccurate, but because the wrong diamonddressing wheel was used the profile thus obtained is a convexform.

Some gear makers think that a convex profile may offersilent transmission, perhaps because most gear makingmethods produce a concave profile andtypically noisy gears.Therefore, it is understandable to suppose that a.convex pro-file will offer silent transmission. However, we have shownexperimentally that this is not the case. A batch of some 20gears with profiles as shown in Pig. 15 were made. Theyturned out to be very noisy (over 85dB).

Experiment for investigating the phenomenon of losingcontact in gear meshing. Gear noise researchers have longnoticed the phenomenon of losing contact. We did some ex-periments on this phenomenon as well. The gear noise testerwas isolated between driving and driven gears. A signal

Fig. 13 - Ground on Chinese worm wheel geargrinder 7232A plus Rna] free grinding

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Table 4 - Comparison of noise of gears ground on different gear grinders

NO. Type of gear grinder Way of profile generationand indexing No. of Average No. of Fig.

noise noise of tester'stest record

1 Chinese horizontal gear steel belt and rolling disc 83dB (data offeredgrinder 7132A with virtual rolling radius by factory)

adjustable, indexing bychange gear and wormwheel

2 Chinese worm wheel gear Combine profile generation 28 BO.7dB Fig. 15grinder 7232A and indexing just like

hobbing3 Swiss worm wheel gear 32 78.OdB Fig. 16

grinderNZA4 Swiss worm wheel gear 4 81.SdB Fig. 18

grinder AZA5 Chinese worm wheel free 34 76.7dB Fig. 17

grinder

Swiss worm wheel geargrinder AZA

'Fig, 14 - Tester's record of gear ground on AZA

2.S!,

Fig. 15 - Tesler's record of gear with convex profile

voltage with some resistors in series was connected to thegears. If the gears were in contact, the signal voltage was shortcircuited to low voltage. If they lost contact, high voltage ap-peared. Our experiment not only confirmed the existance ofthe phenomenon, but also defined some of its characteristics.Three sorts of losing contact were observed .. (See Fig. 16.)

a) Jump over teeth - The driving gear loses contact withthe driven gear; then, after crossing several teeth, it comesagain into contact.

b) Frequently lose contact for short instant within the

14 'Gear Technology

separate(a)

a 76.7ms

contact

WW~~~'W)\IU1~W,~,U'W~)~\mW,'WW~I(separate (b)

contact

separate (c)

Fig. 16 - Ti me series curves of losing contact

period of one tooth, lose contact and come into contact againseveral times.

c) Jump over teeth and come to frequently losing contactalternatively.

Experiment for investigating the influence of contactratioon noise level of gears, A pair of non-standard high precisiongears were made with very high thin teeth, which GOuld pro-vide a contact ratio of more than two. By varying the centerdistanoe on the tester, a relation between the contact ratio andthe noise level was found as shown inFig. 17. Minimum noise

seems to fall on a contact ratio a little bit less than 2 or 1.Thisindicates that even under the slight load, the influence of tor-sional stiffness is still of some importance.

A 'tentative suggestion on the optimized geometry for lownoise gears. For the time being, we can suggest only the type(d) in Table 1.That is, doing best to reach better indexing (cir-cular pitch), for the existing error, adapt pressure angle ex toensure correct normal pitch, so that on the tester's record theadjacent curves will meet very welt and a smooth change-over can be realized with slightest tip/root relief. Table 5shows an example of silent gear pair,

Research on the Making of Silent GearsFrom the previous sections, it is dear that gear noise is

produced:a) at the instant of change-over,b} during the process of meshing of a pair of gear

teeth.The demand for accuracy is also directed at reducing noise,

that is;a) The normal pitch diHerence.1.to as small as possi-

ble. If it is not possible to obtain this figure, apositive at" is preferable.

b) The profile form error cf should preferably be asdose to the theoretical invelute as possible withminimum relief needed.

From Table 5, it can be seen that to guarantee ideal silentgear transmission, the demand on key items of accuracy isremarkably high.

1) To ensure final accuracy, the key technology is in thefinishing method after hardening.

2) To reach high accuracy in normal pitch, free finishingmethods aile very favorable. They can correct the normalpitch error of the gears with the accurate normal pitch of thetool (gear hone, worm-shaped, abrasive wheel, internal gearhone, etc. In Table 4, it has been shown that an additional freegrinding to the worm wheel gear grinder cut down theaverage noise level by 4dB. Another factory in Dalian suc-ceeded in cutting down the gear noise by some S-6dB by in-temalhoning. (The gear hone is with a shape of an internalgear.) Further theoretical analysis can prove mathematicallythat fre·efinishing can guarantee the type of gear shown inTable Id, while grinding with constrained motion of wheeland job will alwayslead to the geometry shown in Table Ic,

3) To obtain the ideal profile accuracy, a finishing methodwith controllable metal removing rate along the full profile(from tip to root) is keenly wished. The field controlled ECHof gears is one of the new technologies we are developing withthis intention. (See Reference 3.)

ConclusionAs the introduction of this article has stated, all we are

writing here is just an idea formed in our minds by the results

Table - 5 Example of silent gear pair

Driving gear Driven gearGear No. Fig. No. Gear No. Fig. No. Noise level

(average)7S.6dB74.2dB

Fig. 17Fig. 17

709L702L

Fig. 17Fig. 17

702L709L

11 - 1000 rpm110

1.4 2.2

Conlolct ratio (a)

Fig. 17 - Relation between contact ratio and noise level

of experiments. It can basically explain the phenomenon wecan see now, but any disclosure by further experiments maymake it necessary to revise the explanation ..Besides, there isstal a.distance between experiment and application in.produc-tion to cover But so far as we can see now, a compa:rativelysilent gear (say below 78dB) looks feasible. Far further reduc-ing noise, other measure such as damping and absorbing, andsound isolation, should be considered too.

ReJerences;1. CHEN, C.P. et al. "Electrochemical Honingof Gears." OUT.

Al1nals of ClRP. VoL 30/10/1981.2. WEI, G.Q. et al., "An Investigation Into. The Ability of Cor-

recting Error in ECH." OUT. Al1nals of ClRP, Vol.351111985.

3. WEI, G.Q. et al., "Field Controlled ECH of Gears." DUT,Preprint of 4-IPES, Precision Engil1eeril1g, No. 4, 1987.

4. LIN. J. et al.. "Single Flank Total Composite Error Analysisand Noise Prediction of Gears." OUT, Preprinl of 4-IPES,Precision Engineeril1g. No.2, 1987.

S. LIANG, Y.R. "Out-of-bound Meshing of Gears in Shavingand Its Influence on the Accuracy of Shaved Gears." Mastersthesis, 1982.

6. LlU, J. et 31.our, "Gea:rNoise &: the Making of Silent Gears"Preprint from 1988lntemational Gearing Conference, China.

7. SISIMU, KOAYIZIMY. "Gear Noise and its Pr vention." Ap-plicational Machine Engineering. Vol. 15, No.9. (1974)

8. HUANG, L.l. AND ZHONG. X.Q. "Gear's Dynamic Com-posite Error Curve and Its Testing Method." Scientia Sinica,No. 4,1973.

9. Machil1e Tool Noise - Principie and Control. Zhang Co.,Tianjin University, 1984..

10. LlU, J. Advanced Theory of Gear Meshing - A Textbook ForPostgraduate Students. 1980. (In Chinese.)

11. WU, H.J. "Spectrum Analysis of Gearing Noise." Master'sthesis, DUT, 1983.

12. QIAN. L]. "Gear Geometric Errors and Gear Noise." Master'sTheses. OUT, 1988.

13. WEI, L. "Experiment of Gear Noise." Master's thesis. OUT,1985.

Acknowledgement: The achievement of this research is virtu !ly the sum ofthe achievements of OUJ research group over the past 9 years. II should becredited to all members and postgraduates of our I"E5eMChgroup. ,especially.V.f. Iia, x.s.uo. D.L. Wang, W. U, Y.W. Llu, M.H. Wu, and V.P, Guo.

March,/April 1990 15

Infl'uenc,e~of tubrication onPitting and Micropitting

IResistance of !GearsProf. Dr.-Ing. H. Winter &; Dr-Ing. P. Oster

Technica1 University of Munich, Munich, West Germany

Abstract:Pitting and micropitt lng resistance of

case-carburized gears depends on lubri-cants and lubrication conditions. Pit-ting is a form of fatigue damage. Onthis account a short time test wasdeveloped. The test procedure is de-scribed. The "pitting test" was devel-oped as a short ti me test to examine theinfluence of lubricants on rnicropitting,Test results showing the influence ofcase-carburized gears on pitting andmicropitting are presented.

IntroductionPitting and micropitting are essential

failure limits of the surface strength ofgears, which are not only affected bymaterial, but also by lubricants andlubricating conditions.

Fig. 1 shows the increasing torquelimit or pitting and rnicropitting as afunction of increasing tangential speed.The tooth fracture limit may be raisedby increasing the module, so that theruling factor is the pitting limit, even forcase-carburized gears. (See Fig. lb.)Therefore, the surface strength has to bedetermined by a life time test with a geartest rig. Because life time tests are veryexpensive, other test procedures for pit-ting and micropitting load capacity areworked out. (4-8)

Problems of Surface StrengthFig. 2 shows typical pitting damage of

two gear pairs .. The damage appearsbelow the pitch circle at pinion andwheel.

Pitting causes the failure of the gearpair within the testing procedure. Ingeneral, micropitting occurs during sur-

....5iS ~

I- -.I I-

.... '- ...E ~:; ...JfU OJ I::0 ".0- cr'- ...0' 0I- .....

bL-~ __+-~ __~~~~~.Speed vt

fig. la-Main limits of the load capacity of through- Fig. Ib-Main limits of the load capacity ofhardened steel gears. case-carburized steel gears.

Werkstoff: 42CrMo4VLas,twechselzllill:

LW;S8"I06HV2" 270

Pc = 603 N/m,m2

•CIc".CI~~OJ.cIII--~.5

_._ ......-.;.;._~w

Werkstoff: l'6Mn.cr5 ELastwechselzQhl:

LW= 24-106

HV2 '" 780; e'HT=1,3mm

PC'" 1750 N/mm2

Fig. 2- Pitting damaged tooth flanks. Application conditions: a = 200 mm; m = 9 mm; Zl!Z2 =21122; injection lubrication using mineral oil; v, = 11.5 m/s.

16 Gear Technology

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

face strength tests of hardened gears.Operative conditions (especially pres-sure, tangential speed, and tempera-ture), tooth geometry (especially curva-ture and sliding conditions), quality ofthe surface, material, and lubricantfeatures determine stress and deteriora-tion of the gear pair. For the most part,ease-carburized gears show serious pit-ting damage only ata ~ewflanks, whichlimit the life of the gear pair. The pittingdamage varies over the gear periphery.

Because failure shows fatigue charac-teristics, repeated tests under similarconditions show gear lives scattered ina wide range.

The method of calculation in ISO6336/2(31 covers the influence of thelubricant only by the lubricant factorZ,-, which takes :the nominal viscosityof the lubricant and the strength ofmaterial into account. (See F,ig. 3.)

The range of uncertainty is very largebecause varyingadditives of the lubri-cant cannot be taken into account.

Fundamental rolling tests usingthrough-hardened steel disks havesystematically examined the influenceof friction coefficient andlubricant film(referred to the surface roughness) onpining resistance, (See Fig. 4YOl ) The

AUTHORS:

PROFESSOR DR. -INC. HANSWINTER is the head of the Laboratory ofGear Research and Gear Design at theTechnical University of Munich. He nas heldpositions in the Gennan gear industry since1956 working for Zafmuifabrik Fried-.rid15hafenand DEMAG Duisburg in man-ufacturing, droelopment, design, and re-search. He received ftis doctorate /Tom theTechnical University of Munich and st~ldiedas associate of Prof. Dr-Ing h.c. GU5tauNiemann.

DR. - INC. PETER OSITR. .is the chiefengineer at the Laboratory for Gear Researchand Design (FZG) in Munich. whe~e he hasbeen 011 the staff since 1965. He studiedmachine engineering at the Technical Univer-sity of Munich and earned his doctoraldegree there.

transfer of absolute strength valuesfrom these tests to the gear pair is uncer-tain, because conditions differ.

used, and Wohtercurves are deter-mined by running tests. fig. 5shows thetest rig and testing conditions. By v.ary-ing gears and conditions, many applica-tions can be closely simulated.

Failure limit of pitting depends on 3.P""plication conditions and characteristic~eatures of the gear. Fig. 6 shows thedeterioration development du_ring 3.fatigue test as a function of flankpressure and load cycles.

Failure criteria of case-carburizedgears are defined as 4. % pitted area

The Determination ofRank Load Capacity Using FZG

Back- "J:o-BackTest RigsSystematic, statistically proven tests

simulating real-life conditions shouldonly be performed on a cost-effectivebasis, depending on the reliability of thegears required.

Back-to-bark test rigs are normally

iltilrl cant Fae tor ZL wi. tlVIII·thoutEP-Add:1 tlves

Ncml na,1 VI sees !tn'110at IIO·C In rrflS -

soo

B ',,10'u~ 1,00-1-1--t-;Hf:::::l;:""",~r:-.-to..!...!:~-I--I' ~1I200

I ~ I - _+-...., for All. r:: 0,90 ~-Jt>9'-l-"""~... t'atenals.g 'D eO'H-4--~-+--+---+--+-~.J , 0 20 60

synthetIC 011.5 ha,V!1II9 I!;ow

CgcffJZlent or Fmuon:

case CarburlZed :zi-1.II·ZtThroogh Hardened : Zi-1•1• ZL.

Ncmlnal ViSCOSity "soat 50·C In rrf/s

Fig. J -Influence of lubricants on pitting load capacity - calcul tion method ref r nee ISO6336/2.(l.l1

Surface Strengtll,of Through--Hard'ened;IDisks (Material: 42C'rMo4)I Following Factors Inflluencing the Life Time' are Tested SeparatelY:

- PH Hertzian F'ressure; - p.. Coeffilcient of Friction; -r:lBI Flash Temperature;- V2 Tangential Speed; - La Lubricant-Film-Thickness /IRoughness.The following equation is the result of these invesUgat:ions:

LW. = 1(.PH1)-7. (P..')-3. ,(-.!?Bll)· -2. 1(~)1:.2. (ILa1)2.2LW2 PH2 /-!2 !?B12 V22 La2

_ L __

If La'.2 > 1,0 - La'.2 = 1.0,

Surlac·eStrength of Through Hardened Gears (Materlall: 42CrM04)The dominant influence of the coefficient of friction, apart from the influence ofthe pressure. is also proven at Gears.

CoeHiciient. of Friction at EHL ··CondIUonsMineral Oil: /-I - Pl-t°.6 P. - v-O.25p. _ AaO.2;Synthetic Oil: The coefficient nas to be evaluated by tests.·EHL: IElasto-Hydrodynamic Lubrication

Fig. -4 - Effects of EHL-parameters on pitting load capacity - surnm ry.

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I(~ler Cyr\feJ Basln9 uPon1... 2li TIlt' Polnu.

Fig. 5 - Determination of pitting load capacity us-,ingFZG test rigs.'

t..8 2.5 ...., -,-,-,"'--.----.--.-.-..----.-,I Z.O:g'.5 t-I-t--t-I-H-I-'H-Ul"'+~-ItI

~ 1.0 r+-+-I--IH--1't--i-;~II:f--i'~.;, '0,'5

il I -_~~:=;t~~hlL--+-Ii! Of"105 "5 ,a lOT ~

LOId CYcles"

re!. '!O' FlG/Ret! Ig

EYalUltlOll of • II!IIller Curve

IInerl.1 : lo;cr~16 •!IodUle : 5.5 .. I

,. Pc • L.20·PeD '

b. Pc • I.02·rCD '

C.Pc"DeD'

d. Pc • o.!I2·Pw '

Pc : Flank Pressure II' !~PilCh 1'II:ln!,

OeD ' Pc Endurance LI.UOCClJr5.

Fig ..6 - Deterioration development as a functionof flank pressure and n umber of load cycles.

6 In !, 100--- DYflalllC Factor Ky

, 'I Kv-.- Total PI ttlng Area G

,

Pc' 600 HidIIIOJ

IlJl: I u eon .~ rPIII ""I LJ

'" t...2 MateriaL Througn 0 '50l'S -'Hardene.... ;..,

1'€CI ..r Pc· 600 NI~ .2!

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20 Gear Technology

dlcated. One gear pair is necessary toachieve two test points, and one testpoint requires about 500 test rig hours toreach theendurance limit. The execu-tion of a S-N curve is very time-con-suming. 170 prove the fatigue stI1e5Sstatistically at the range of limited lifefatigue, shorter funning times arepossible.

Fig. 9 shows the differe.nce of loadcycles between the area of limited lifefatigue strength and endurance limit atpitting tests. It is eviden't that even at thelow range of limited life fatigue(Pc/Pco"'" 1.1). a distinct contractionof the running time differences is to beexpected.

fig ..10 shows S-N curves which ex-amine the influence of additives on pit-ting resistance of case-carburized gears.Only a limited statistical proof ispossible.

Thel/Pitting Test" asa Short TimeTest for Pifl.ing Resistance

Based on results of the influence ofadditives on pitting resistance, theForschungsvereinigung Antriebstechnike.V., Frankfurt, (FVA) has carried out around robin test.

The short time "pitting test" wasdefined and executed by five labora-tories. The test conditions are shown inFig. 11.

FZG gear test rigs having similargeometry (center distance: .3 =91.Smm) to the standardized FZG scuf-fing test rig (DIN 51354) were used.(ll

Gears (case-carburized 16MnCrS.Maag 0° grinding) and lubricant fromone charge were placed at disposal. Theresults of all 'test rigsconcerning twotested lubricants are shown in Fig. 12.

The FZG has taken part in this roundrobin test. Fig. 13 shows the share of theFZG. Although the load cycles widelyscatter, a distinct differentiation be--tween the two lubricants having dif-ferent additives is possible.

The relative scatter (incline of theWei bull curve) of both test series isalmost corresponding. If the FZG test isevaluated sequentially for the use of thetest gear pairs, one can see that, despitethe increasing number of running tests,the statistical mean total enduranceLW50 hardly differs.

It would need about five or six testruns to achieve a correct value if thetests are executed carefully. (See Fig.14.)

In the course of time the FZG has car-ried out further pitting tests using dif-ferent gears (equal geometry, diffel'entmaterial),

Quality of manufacture and heattreatment of these gears were identicalto those used at the round robin test.Reduced load cycles have been theresults of these tests. This is proven bya few comparison tests shown in Fig ..15.Especially in pitting tests, the constancyof test gear features is very important.Besides the difference of endurance, thetwo lubricants show a cardinal dif-ference in the appearance of the flanksurface. The flanks have a general blanklook using FVA oil doped with A99,while using FVA oil doped with lOP ef-fects strong micropitting and heavierflank wear. (See Figs. 16-17.)

Pittingand micropitting affect eachoeher. Depending on intensity,m:icropitting results in continuously in-creasing crater wear at the root of theflanks and in reduced pitting resistance.

MicropiUing andMicropitting Resistance

Description of damage: To the nakedeye, areas where rnicropitting has 0(;-cured appear gray. (See Fig. 18.)Micropitting usually starts at the root ofthe pinion and spreads over the wholeflank. It rarely begins at the teeth tip ofthe wheel.(1.91

There is continuous loss of tooth sur-face material, which results in craterssimilar to wear craters and deterioratesthe profile. Scanning electron micro-scope investigations prove that thefailu~e shows fatigue symptoms. Thereare a lot of broken out, partly conchateparticles (micropittingl ) which give thegray look lathe flanks.

Operative conditions: Fig..19 shows'the main conditions of the micropittingresistance comprehensively. All factorsaffecting the lubricating film betweenthe flanks affect the tendency tomicropitt.ing.(6)

By use of a "thick" EHL film and"smooth" surfaces, micropiUing can. beeluded. Lubricants doped with differentadditives react to test conditions in-dividually. Unlike ZDPadditives, sul-.furphosphorus (S-P) additives are ad-vantegeous to micropitting resistance.

The MicropiUing Test: A short timetest to determine micropitting resis-tance. The micropitting test was devel-oped by 'the ~ZG implementing a FVA

---- ----

---- --- -

I.

PI ttlng Test USing'Different Doped LUbri-cants:

HtI

I , § J ,I-II, ,.,.. .I, ~~-~ fl¥ I

I K I II I!" " , I' , -Ii I , , ! ! , -I I ,,,

.~ • '''IUtE'rNG or,I • T£:ST, .' PO:l:N1"S, I' -'--

I· 5 L-I\ iIIDFi'A TO III I E 5II 'tI' D 0 o 611I I r, 'I-i -I·! ~ Iii

"

1. FVA-OII 2 IoII.th qlAnglamo,199' [,119911

2. FVII-OII 2 With 2%

Lubrlzol 6nA C= IZlncClI th,loohosohate'ClDP».

6 • 101 101

II 99 ! 41 Test RunsZDP 37 Test Runs

Load eYe.1 es -.

Fig. 12 - Results of the first FVA round robin test.

N!J!!Ilm Qf LOad CYcles untilReach! Og the Ea,lI ure lI!!I1 t 1.0

Millions;

01\99 ZOP

6.3 11•. 06.9 21i.68.1 23.18.3 25.19.0 26.49.11 26.910,3 27.5111,1 30.011.2 31.2D.9 32,1.1'..71!L ,8

liO'" 2 ,,& e'107 "6load cvcles

tubrlcar:lt

10.4 ..106

26,0'106EYA2+,a,g9

FVA2+ZDP

Hg. 13 - Results of the FZG pitting test.

)

2

FVA2 • A99 FVA 2 .. ZOPI

,--

I,

If--, l-II i I

!

I

,I'

I,

I , ,610

J 4 56 8 10 12'Nlill'ber or Test RillS

]1 4 S 6 810HlllDer or Tes'! RIJ'IS

FlG Tests wen Eva,luated :In seeeenee or USageor Gear Pairs IEYII ROhlld lIobln Test!

Fig. ]4 =Influence of the number of test runs.------ - - - -- - - - --

March/ApriII1990 2'11

}

Test Gear Wheels16MnCr5 EDIN ooers-s

FVAJ .1..*/ •. A99

- .1F VA 2 .1,,"10 A 99 III-: --LL&.ILU.....IL _

FVAJ.I."!oA99

2 S 10 20 40

load eye Ies '106

Fig. 15 -Influence of lest gear wheels on pitting tests.

Oil: FVA 2 + 2 % ZOP Olil; FVA 2 + 4% A 99

Load Cyc.les; 0,6" 106 load Cycles; W. 9 ~ 1;06

Majior Micro Pitting Minor Micro PUliingbelcw jhe Pilch Circle

Fig. 16 - Examples of gear teeth used on pi tti ng tests.

New State Tested with FVA2.Ag9 Tested with FVA2+ZDP

(and I,t ron Wear at the Root. Heavy Wear frOOI---_. --the Root to theTIP.

Fig. 17 - Tooth profiles of test gears.

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

22 Gear Teohnology

project. To test the influence of lubri-cants, a modified .FZG test (DIN 51354)and a gear with balanced sliding at thetip of the pinion and of the wheel aveused.(7.8,9)The load is raised by loadstages until the amount of the profileer-ror exceeds twice the amount of error atthe beginning. To get reliable judgmentand information about deteriorationdevelopment (partly degressive) atlonger running times, a permanent testrun (10 .... 50'106 Load cycles) is ex-ecuted afterwards.

Results of micro pitting test runs: Fig.20 shows results of a micropitting testusing three lubricants doped with dif-ferent additives.

Obviously there is a relation betweenless of weight and the failure causing in-creased amount of profileerror. It isremarkable that the lubricants' GFf-low and GFT-high base on the same oil.The lubricant GFT -low is doped withZDP additive, and GFT-high with aS-P additive. Thus, the strong influenceof additives en micropitting is obvious.In certain cases the torque limit ofmicropitting is in contradiction to thepitting limit.

Summary and OutlookPitting and mieropitting resistance of

case-carburized gears depends onlubricants and lubricating conditions,Pitting is a form .of fatigue damage.Therefore, pitting endurance has to hedetermined by running tests, and theresults shown as S-Ncurves. Becausethese tests need statistical evaluations,they are expensive, therefore, a shorttime test was developed. The test pro-cedure is described. The pitting testdetermines the influence of lubricantsdoped with different additives on pittingresistance.

Test runs have shown that micropit-ting often occurs during pitting tests ofcase-carburized gears. The damagemicropitting is described. The mi.cropit-ting test was developed as a short timetest to prove the influence of lubricantsdoped with different additives on mi-cropitting resistance of case-carburlzedgears.

Lubricants doped with S-P additivesshew a higher micropitting resistancethan lubricants doped with ZDPadditives.

Because qualitative results of bothtest procedures are possible only to the

- - ---

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

respective damage of this test, the FZCaims at a combination of both pro-cedures considering economic aspects.

Referenoes

1. DIN 3990. "Basic Elements for Calcula-I'ion of Load Capacity of Spur andHelical Gears," Draft Jan. 1986 (inGerman).

2. DIN S1 354. 'Testing LubricantsMechanically with the FZG-Back-I,o-Back Gear Test Rig," Aug. 1977 (inGerman),

3. ISO 6336/2. "Calculation of LoadCapacity of Spur and Helical Gears"Part 2: Calculation of SurfaceDurability (Pitting)," ISOTC 60, Feb.1983.

4. GOLL, S. "Influence of Lubricant onPitting Resistance of Gears," Reportsof VDI No. 626 (1987) (in German),

5, cou, S. "lubricant Influence on Pit-tingat Gears, Results of the FVA-Round-Robin Test." Mineral61techMikNo.4, April 1983 (in German),

6, NIEMANN, G.. WINTER, H.Machine Elements 11. Springer Verlag,Berlin (1985) (in German).

7. RETIIG, H. SCHONNENBECK, G."Micro Pitting, A Damage Limit atGears," Arllriebstechnik 19 (1980) No.7 - 8 (in German),

8. SCHONNENBECK, G. 'Testing Pro-cedure to Determinate the Influence ofLubricants on Micro Pitting," An-triebstechnik 24 (1985) No. 1 (inGerman),

9. SCHONNENBECK, G, "Influence ofLubricant on Tooth Profile Fatigue ata Velocity Range 1 .. , 9," mIs,Thesis, Technical University Munich(1984) (in German).

10, SIMON. M. "MeasurementsofElasto-Hydrodynamic Parameters and TheirInfluence on Pitting Resistance ofThrough-Hardened Gears. Thesis,Technical University Munich (1984) (inGerman),

Aclmowledgement: Presented at the AGMA FallTechnicalMeeting, 1987, Reprinted with penn is·sian of the Amen'can Gear Manufacturers'Association. The opinions, statements, and con-c/tGions presented in this article are those of theauthor» and .111 no way represen: the position oropinion of the Anrerican Gear Manl.lfacturers'Association.

a. At the Rootof the DrivIngPInion)

b. Totally Da-maged Flank;

c. At the TIpof the Dr! ven

Wheel I

d. Partly Da-maged Flank.

19427d. 195510,

Fig. 18 - Examples of gear teeth showing micropitting.

Influencing Variable

Flank Roughness: IReduction from 6/im 103/im

Range.. 1:3

Malerial, Heat Trealment(Advantageous Percentage of Austenite) ... , , , 1:2.8

Additives (Equal Viscosity of Base Oil) , , .... , , . , , .. , , . , , . ' .. , , , , , . 1:2

Doubl.ed Working Viscosity, , , .. , , , . , , .. , , .. , , , . , , .. , , , , . , , , ... , l' :.2

Halved Coefficient of Friction , , , .. , , , .. , , , , , , , , , , , 1:1.i

Tangenhal Speed (Advantageous: High Speed) , . , , , . , , , , , , . ,1:1.3

lowering 011Temperature (M = 201

Dynamic loads in ParallelShaft Transm~ssions

Part 1Hsiang-Hsi (Edward) Un

Memphis State University, Memphis, TN

Ronald L HustonUniversity of Cincirman, Cincinnati, OM

John l. CoyNASA Lewis Research Center, Cleveland, OH

24 Gear Technology

IntreductionRecently, there has been increased interest in the dynamic

effects in gear systems. This in.teres~is stimulated by demandsfor stronger, higher speed, improved performance, andlonger-lived systems. This in tum. has stimulated numerousresearch efforts directed toward understanding gear dynamicphenomena. However, many aspects of gear dynamics arestill not satisfactorily understood.

For example, in industrial settings. a high performance gearsystem is oHen obtained by overdesigningand by sacrificingcosts, materials, and compactness. In aerospace and militaryapplication where weight is a premium, gear systems are oftendesigned under conditions very close to. the failure limits,thereby introducting uncertainties in performance and lifeprediction. They are often prematurely replaced to preventin-service failure. Moreover, gear systems are often designedby using static analyses. However, when gear systems operateat high speed, there are several factors which affect their per-formance. These include shaft torsional stiffness, gear toothloading and deformatior; gear tooth spacing and profile er-rors, rotating speeds, mounting alignment, dynamic balanceof rotating elements, gear and shaft masses and inertia, andthe masses and inertias of the driving (power) and driven(load) elements .

There is no. agreement among researchers on the bestmethods for evaluating dynamic load 'effects. Hence, geardesigners are Dfteneontronted with conflicting theories. TheygeneraUy have to rely on past experience, service safety fac-tors, and experimental data with a limited range ofapplicabili ty .

The objective of this report is to provide more insight intothe factors affecting dynamic loads.

Research efforts on gear system dynamics have been con-ducted fer many years. In 1892 lewis(1) recognized that theinstantaneous tooth load was affected by the velocity of thesystem. In 1925, Earle Buckingham(21headed an experimen-tal.research effort, endorsed by ASME, to. measure dynamiceffects. A report published in 1931 represented the firstauthoritative document on gear dynamics. It presented a pro-cedure for determining the so-called.dynamic Ioad incrementdue to. mesh dynamics and gear tooth errors,

In 1959, AttiaU) performed an experiment to. determine ac-

AlmiOR5:DR. HSIANG H51 UN is on the faculty at Memphis State Univer-

sity. His research includes kinematics, dynamics, optimization,finite element methods. and computer-aided design and analysis ofmechanical sys.tems. He received his undergr,u:iuaJ;edegree fromNational Chung Hsing University and did his graduate work at tIll?University of Cincirrnati,

DR. RONAlD L HUSTON is Professor of Mechanics at theUniversity of Cincinruui and Director of the University's Institutefor Applied InterdisciplinQ.ry Research. He earned his degrees inMechanical Engitleering and Engineen'ng Mechanics from theUniversity of Pennsylvania.

DR. JOHN J. COY is Branch Chi1!f. Rotorcraft Systems Tech-nology in tile Propulsion Systems Division at NASA Lewis Re-search!Center in ClevehUld. OH. He took his degrees in MechanicalEngineering from the University of Cincinnati, He is the author ofnumerous papers andis a member of ASME'. Dr. Coy is a licensedprofessional engineer in the State of Dhio.

tual instantaneous loading. He found that Buckingham'sresults were conservative.

In 1958, Niemann and Rettig(4) found that larger massescaused higher dynamic loads, but as the average lead becamelarger, the effect of larger masses became less important. Theyalso found that very heavily loaded gear systems showed no.appreciable dynamic load increment, whereas in lightly andmoderately leaded gear systems, there were consid rabledynamicload increments. In 1958, Hams(S) suggested thatfor gear systems isolated from external stimuli, there are threeinternal sources of dynamic leads:

(1) Error in the velocity ratio measured under the workingload ..

(2) Parametric excitation due to stiffness variation of thegear teeth,

(3) Nonlinearity of tooth stiffness when contact is lost,In 1970, Houser and Seireg(6) developed a generalized

dynamic factor formula for Spill'and helical gears operatingaway from system resonances, The formula took into. con-sideration the gear geometry and manufacturing parametersas well as the dynamic characteristics of the system.

In 1972, Ichimaruand Hirano(7)analyzed heavy-loadedspur gear systems with manufactu_ri-flgerrors under differentoperating conditions. They found that the change in toothprofile showed a characteristic trend to decrease dynamicload, In 1978, Cornell and Westervelt(S) presented a dosedform solution for a dynamic model of a spur gear system andshewed that tooth profile modificafion, system inertia anddamping, and system critical speeds can have significant e£-fects upon the dynamic loads, In 1981, Kasubaand Evans(9)presented a large scale digitized extended gear modeling pro-cedure to analyze spur gear systems for both staticanddynamic conditions. Their results indicated that gear meshstiffness is probably the key element in the analysis ofgeartrain dynamics. They showed that the gears and the .adi centdrive and load systems can be designed for optimum perform-ance in terms ·0£ minimum allowable dynamic loadsIor awide range of operating speeds.

In 1981, Wang and Cheng(1O)developed another dynamicload response algorithm. They reported that the dynamicload is highly dependent on the operating speed. This modelis later modified by Lewicki(1ll to account for the nonlinearHertzian deformation of meshing gear teeth. The geardynamic load found from the revised mode] showed little dif-ference from the original model, since the Hertzian. deflectionwas relatively small incomparison with the total gear toothdeflection. Nagaya and Uematsu(12) stated that because thecontact point moves along the involute profile, the dynamicresponse should be considered as a function ef both the posi-tion and speed of the moving load. In 1982, Terauehi, etal., (13) studied the effect of tooth profile modificat ions on thedynamic load of spur gear systems. According to their results,the dynamic load decreased with proper profile modifica-tions.

In this first part of the article, we present a model of aparallel shaft transmission. We consider the effects o.f shaf,tstiffness and inertia, lead and power source inertias, toothstiffness, local compliance due to. contactstresses, load shar-ing, and friction. A parameter study is prov.idedin 'the secondpart of the article.

MarCh/April 1990 25

ModelingFig..1depicts a model of the transmission. It consists of a

motor or power source connected by a flexible shaft to thegear system. The gear system consists of a pair of involutespur gears. They are connected to the load by a second .flexi-ble shaft as shown. Symbolically, the model may berepresented by a collection of masses, springs, and dampersas in Fig. 2 ..

Let 61M!611, 62, and Ol represent the rotations of the motor,the gears, and the load. Then by using standard proceduresof analysis, the governing differential equations for the rota-tions may be written as

J191 + Cs1(91 - OM)+ Ksl(Ol - £1M) + Cg(t)[Rb161 - Rb26z1+ Kg(it)[Rbl(Rbl01 - ~202)1 = TIl{t) (2)

Jz8z + Csz(02- (1) + ·~«(12 - (;11) + Cit)lRb~2 - Rt.1911+ Kg(tHRbZ(~2612- ~lOl)l = Tf2(t) (3)

hOt + Cs2(OL - 02) + ~((h - (h) = - TL (4)

where JM, h, h, and h represent the mass moments of iner-tia of the motor, the gears, and the load; Cs], c..,.,and Cg(t)are damping coeffidenrs of the shafts and the gears; Ksl' ~,and Kg(t) are stiffnesses of the shafts and the gears; TM, T t-Tn(t), and T1z(t) are motor and load torques and frictionaltorques on the gears; Rbt and Rb2. are base circle radii ofthe gears; t is time; and the dots over (1 indicate timedifferentiation.

In developing Equations 1 to 4 several simplifying assump-tions are employed.

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(1) The dynamic process is studied in the rotating plane ofthe gears. Out-of-plane twisting and misalignment areneglected.

(2) Damping due to lubrication of the gears and shafts .isexpressed in terms of constant damping factors.

(3) The differential equations of motion are developed byusing the theoretical line of action ..

(4) Low contact ratio gears are used in the analysis,Specifically, the contact ratio is taken between 1 and 2.

AnalysisA major task in the analysis is to determine the values of

the stiffness, damping, and friction coefficients appearing inEquations 1 to 4. Another task is to determine the ratio of loadsharing between the teeth during a mesh cycle. These factorsdepend on the roll angle of the gears. Thus, Equations 1 to 4are made nonlinear by these terms.

STIFFNESSGear stiffness. - Consider first the stiffness coefficients

Kg, Ksl' and Ksz. Let the tooth surface have the form of an in-volute curve. Let Wi be the transmitted load at a typicalpoint j of the tooth profile. Let qj be the deformation of thetooth at pointj in the direction of Wj. Then the gear stiffness~ for the gear teeth in contact at j is

Kg; = Wi (5)qj

In general qj wiIl depend on the following: (1) the bend-ing of the tooth, the shear deformation of the tooth, and theaxial compression of the tooth; (2) the deflection due.to theflexibility of the tooth foundation and fmet; and (3) the localcompliance due to the contact stresses.

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26 'Gear Technol'ogy CIRCLE A~10 'ON REA'IOER REPLYCARD

III .. , ml~ .-&EAR 1

IIilllTOR SlW'T 1 Ir-. ,

-.! -.!:

~, .,J1= ."r-. SHAfT 2 I LOADI

GEAR 2~.il= ... '''I II

Fig,] -A Simple Spur Gear System.

y

~---- Xi+1

1--------- Xj

Fig, 3- Element Modelling of a Gear Tooth.

as the sum of the deformation in the elements ibeneath pointj. That is,

nqbj =E qbij

i=l

where n is the number of elements beneath iand qbij is thedeformation of element idue to the load Wj'

(continued on page 30)

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Fig. 2 - A Mathemat leal ModeJ of the Transmission System.

To determineq, let the tooth be divided into elements asshown in Fig. 3. Let i be a typical element with thickness T,cross section area Aj, and second moment of inertia Ii' Let~j be the distance between element iand point ialong thex-axis. Let f'Jj be the angle between WJ and the y-axis.(See Fig. 3.)

Consider the tooth to be a nonuniform cantilever beam.Let qbi be the contribution to qj by the bending, shear,andaxial deformation of the tooth. The qbj may be represented

mailto:extemall.straightorlnvolutesplineconfigur.ations.

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r------------------ _.

DYNAMIC LOADS ...(continued from page 27)

~ o~----~----~----~----~--~tb (A) PI.TCHRADII. Rpl =Rp2 = 57 .. 1 M."",'

~,,..22j,~

GEAR TOOTH PAIRDRIVING GEAR TOOTHI:(RTZI ANDEFORI'IA Tl ONFOi.L'OWING GEAR TOOTH

20

15

10

5

ROlL ANGLE. DEG

(BJ PITCH R'ADII. Rp,11 = 57.1 M.Rp2 = 171. 3 Mo>\.

Fig. 4-No.rmalized Deformation of a Pair of Teeth. Module, 3,18 m:rn;Pressure Angle, 20"; Pace Width, P = 25.4 mm; Modulus of Elasticity, E= 2'07 GPa; Applied Load, W = 105 KN/m.

With standard analysis qbij is found to have thevalue:(8. 14. 15)

qb" = (W.(J::)J cos2{3·f-,(.T3+ 3T.2L .. + 3TL2")/3l]II l'-'e l I I - -1 1) III· I

- cos {3jsin {3i[ (T;2Yj + 2TiYj~j) 121;]

+ cos2{3i[12(1 + lilT/SA; I+ sin2{3j(T/ Ai) } (7)

30 Gear Technology

where Y, is the half-tooth thickness at element i(See Fig3.),II is Poisson's ratio, and Ee is the "effective elastic modulus"depending upon whether the tooth is wide (plane strain) or

narrow (plane stress). Specifically, for a "wide" tooth, wher-ethe ratio of the width to thickness at the piltch point exceeds5,114) Ee is

(8)

where E is Yeung's modulus ef elasticity. Fer a "narrow" tooth(width-to-thickness ratio. less than 5), !Ee is

!Ee =E (9)

Expressions similar to !Equation 7 hold fer'lli' the con-tribution to.qj for the deformation due to the flexibility of thetoothfillet and fou:ndation.l1S)

Let qcj be the contribution to qj from the local compliancedue to contact stresses. With the procedures of Lundberg andPalmgren(16) qcj may be expressed as

(10)

where F is the width of the tooth.Hence, by superposition, the deformation at] in the direc-

tion of Wi is

(11)

The above expressions were used to calculate the defor-mations for two different gear pairs. The resul tsare showngraphically in Figs. 4a and b.

Shaft stiffness. - The shaft stiffness K, is given by thestandardexpression

IGK,=-I

(12)

where G is the shear modulus, I is the shaft length, and J isthe polar moment of area. given by

(13)D'I

J- - 1"----32

where D is the shaft diameter.

DAMPINGShaft damping, - Next consider the damping coefficients

C51, C51• and Cg• Damping in the shafts is due to. the shaltmaterial. In Equations 1 to. 4 the coefficients Cs1 and C51 aretaken to. have the form

C" - 2~{t~'tr (14)

.02..~....~'" 0u,\5....z~u

it15'u

(A) BUCK[NGHM·' S """DEl.

ROlL ANGLE. DEG

(B) BENEDICT AND KEtLEY'S """DEL.

Fig. 5-Friclion Coefficient of Gears in Figure 4a at 1500 RPM.

and

Cs2 = 2ts{... ~ .)1/21 1-+-It h

where t. represents the damping ratio. Experiments haveshown that-~. has values between 0.005 and 0.075. (17)

Mesh damping. - Similarly, the effect of damping of thegear mesh is taken as

C = 2~ i[KgR~lRkJlh]- 1/2 (16)g I R~lJl +Rkh

where, as before, ~is the damping ratio. Measurements haveShOWRE to have values between 0.03 and 0.17.(9-10)

Friction. - Equations 1 to 4 contain terms Tn and Tf2which represent the frictional moments of the driving anddriven gears. These moments occur because of the relativesliding of the gear teeth. Buckingham(l8) has recorded a

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Nomenclature

Ai cross section area of ith element of gear teeth, mrrr' (in.2)Cg damping coefficient, gear tooth mesh, N-sec (Ib-sec)C, damping coefficient of shalt, N-m-sec (in.-Ib-sec)E, effective modulus of elasticity, N/m2 (Ib./in2)F tooth face width, mm (in.)G shear modulus, N Im2 (Ib lin. 2)Ii second moment of inertia of ith element of gear teeth,

mm2 (in.2)k polar moment of inertia of load, m2-KgJM polar moment 'Ofinertia of motor, m2_KgII polar moment 'Ofinertia of gear 1, m2_Kgh polar moment of inertia of gear 2, m2-KgKg stiffness of gear tooth, Nm/rad-mK, stiffness of shaft, N-mlradLij distance between elements iand i. mmqb gear tooth deformation due to beam deflection, mmqc gear tooth deformation due to contact deformation, mm~ gear tooth deformation due to foundation flexibility, mmq =~ +

DRIVING GEAR---_._-_.- FOlLOWING GEAR

I:I

z:(AI BUCKINGHAM'S MODEL.

12ROLL ANGLE. DG

(B) SHOlCT AND KELLEY'S MODEL.

Fig. 6 - Friction Torque Variation Along the Contact Path for Gears in Figure4a at 1500 RPM.

When the tooth pair reaches B, the recessing tooth pairdisengages at D leaving only one zone. When the tooth pairreaches point C. the next tooth pair begins engagement at Aand starts another cycle.

In the analysis. the position of the contact point of the gearteeth along the line of action is expressed in terms of rollangles of 'the driving gear tooth.

Fig. 8 shows typical stiffness and load sharing character-

,,,,,- ........ \

~~DOUBlECONTACTlOlE

0,

fig. 7 -Illustration of Gear Meshing Action.

7S

so

, lB/IN. - 0.0175 NII'I

25

3000

z: 2000

~o...J

i!O8I-

o 123 ~NORMALIZED CONTACT POSl liON (BASE ,pITCH)

Fig. B- Typical Stiffness and Load Sharing of Low Contact Ratio Gears ofFigure 4a.

istics through a mesh cycle. let a series of mating tooth pairsbe denoted as a, b, c, dand let points A. B, P, C. Dbe thesame as those in Fig. 7. Then AB and CD represent thedou-ble contact regions, Be represents the single contact region,and, as before, P is the pitch point.

The stiffness values at double contact regions are dearlymuch higher than those at single contact regions. Whengears rotate at appreciable speed, this time-varying stiffness

March/April199Q 33

as shown in Fig. 8 is the major excitation source for thedynamic response of the system.

DiscussionThe objective of this anal ysis is to establish the governing

differential equations and to present a procedure for solu-tion. As noted, the equations themselves are nonlinear,However, they may be efficiently solved by using the follow-ing linearized-iterative procedure.

The linearized equations may be obtained by dividing themesh cycle into n equal intervals, Let a constant input tor-que TM be assumed. Let the output torque TL be fluctuatingbecause of damping in the gear mesh, because of friction,and because of time-varying mesh stiffness.

Let initial values of the angular displacements be obtainedby preloading the input shaft with the nominal torque car-riedby the system. Initial values of the-angular speeds maybe taken from the nominal operating speed of the system,

The iterative process is then as follows: the calculatedvalues of the angular displacements and angular speeds afterone period are compared with the assumed initial values.Unless the differences between them are sufficiently small, theprocedure is repeated by using the average of the initial andcalculated values as new initial values.

Finally, observe that the term (Rb181 - Rbzl;l2) in the equa-tion of motion represents the relative dynamic displacementof the gears. Let 0 represent the backlash. Let gear 1 be thedriving gear. The following conditions can occur:

Case 1

The normal operating case is

The dynamic mesh force F is then

Case 2«

Rb101 - Rb20Z = 0 and IRb181 - Rb282 I = 0 (21)

In this case, the gears will separate and the contact betweenthe gears will be lost. Hence,

F=OCase 3

In this case, gear 2 will collide with gear 1 on the backside,Then,

CondusionA low contact ratio spur gear transmisson model is

devleoped. The model includes inertias of load and powersource, stiffness of shaft, time-varying mesh stiffness, anddamping and friction inside gear transmissions.

Govemmgequations of the model are derived and a linear-ized iterative procedure for the solution is presented.34 Gear Technology

Parameter study including rotating speed, diametral pitch,applied load, damping, stiffness, and inertia will be presentedin Part 2 of this article,References

1. LEWIS, W. "Investigations of the Strength of Gear Teeth," Pro-ceedings of the Engineers Club of Philadelphia, 1893, pp.16-23,

2. BUCKINGHAM, E. Dynamic Loads on Gear Teeth, ASMEResearch Publication, New York, 1931.

3. AITIA, A.Y, "Dynamic Loading of Spur Gear Teeth." r Eng,Ind" Vol. 81, No, 1, .Feb, 1959, pp. 1-9,

4, NIEMANN, G. & REITIG, H. "Error-Induced Dynamic: GearTooth Loads," Proceedings of the International Conference onGearing, 1958, pp. 31-42.,

5. HARRIS, S.L "Dynamic Loads on the Teeth of Spur Gears,"Proc. Inst. Mech. Eng" Vol. 172, 1958, pp, 87-112,

6. SEIREG, A, & HOUSER, D.R. "Evaluation of Dynamic Fac-tors for Spur and Helical Gears." J. Eng, Ind. , VoL 92, No, 2,May 1970, pp. 504-515,

7. ICHlMARU, K. & HIRANO, F, 'Dynamic Behavior ofHeavy-Loaded Spur Gears." J. Eng, lnd., Vol, 96, No, 2, May1974, pp. 373-381.

8, CORNEll, R.W. & WESTERVELT, W,W, "Dynamic ToothLoads and Stressing for High Contact Ratio Spur Gears." [.Mech, Des" VoL 100, No, 1, Jan. 1978, pp, 69-76,

9. KASUBA, R, & EVANS, J,W. "An Extended Model for Deter-mining Dynamic Loads in Spur Gearing," r Mech. Des., Vol.103, No, 2, Apr. 1981, pp. 398-409,

10, WANG, K. L. &; CHENG,. H,S. "A Numerical Solution to theDynamic Load, Film Thickness, and Surface Temperatures inSpur Gears, Part 1 - Analysis." J. Mech. Des" Vo1.103, No.1, Jan. 1981, pp. 177-187,

11. LEVVICKl,D ,G, "Predicted Effect of Dynamic Load on PittingFatigue life for Low-Contact-Ratio Spur Gears," NASATP-2610, 1986.

12, NAGAYA, K. & UEMATSU, S, "Effects of Moving Speeds ofDynamic Loads on the Deflections of Gear Teeth." J. Mech.Des" VoL 103, No, 2, Apr. 1981, pp. 357-363,

13. TERAUCHI. Y., NADANO,H. &NOHARA,M. "On the Ef-feet of the Tooth Profile Modification on the Dynamic Loadand the Sound Level of the Spur Gear," Bull, ]SME, Vol. 25,No. 207, Sept. 1982, pp. 1474-1481-

14, CORNElL, R.W. "Compliance and Stress Sensitivity of SPW'Gear Teeth," r Meeh. Des" Vol. 103, No.2, Apr. 1981, pp.447-459.

15. UN, H,H. & HUSTON, R.L 'Dynamic Loading on ParallelShaft Gears " (UC~MIE-051586-19, University of Cincinnati;NASA Grant NSG-3188) NASA CR-179473, 1986.

16. LUNDBERG, G. & PALMGREN, A "Dynamic Capacity ofRolling Bearings," ActRPolytech. Meck Eng, Sci., VoL 1, No.,

(23) 3, 1947.

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17, HAHN, W.F. "Study of Instantaneous Load to Which GearTeeth are Subjected." Ph.D, Thesis, University of Illinois, 1969.

18. BUCKINGHAM, E, Analytical Mechanics of Gears. Dover,1949.

19, BENEDICT, G,H, & KELLEY W.W. "Instantaneous Coeffi-cients of Gear Tooth Friction," ASLE Trans., Vol. 4, No. 1.Apr. 1961, pp. 59-70.

20. ANDERSON, N,E, & LOEWENTHAL S,H. "Spur GearSystem Efficiency at Part and Full Load," NASA TP-1622,1980,

Aclmowledgements: Published inASME Journal of Mechanisms, Tmnsmis-sions, and Automation in Design, VoL 110, No.2, June, 1988, pp, 221·229.and as NASA T~calMemoranduml00280andAVSCOMT,M,87-C-2,Reprinted with permission.

Shaper Cutters-Design &ApplicationPart 1

William L. Janninckrnv - Illinols Tools, lincolnwood, IL

field of ApplicationGear hl\aping is one of the mo t popular production choices

in gear manufacturing. While the gear soaping process isreally the most versatile of all the gear manufacturingmethods and can cut a wide variety of gears. certain types ofgears can only be cut by this process. These are gears doselyadjacent to shoulders; gears adjacent to other gear. such ason countershafts; internal gears, either open or blind ended;crown or face gears; herringbone gears of the solid configura-tion or with a small center groove; racks; parts with filled-inspaces or teeth removed; and gear or splines with thick andthin teeth, such as are used in some Clutches. Fig. 1 graphicallyillustrates the flexibility of the shaping process. and Fig. 2shows some examples of gears which are mainly suitable tothe shaping process, as well as some which may sui't alternatemethods. such as the rack. External spur and helical gears andpinions are the other conventional' gears widely manufacturedby the shaper cutter process.

Fig. 1 A ~in&le cutler can cut vanous ~eilr t~.

Other special involute and non-involute forms that can beshaped are roller chain sprockets, silent chain sprockets. andtiming belt pulleys, as well as ratchets. saws. parallel keysplines, involute splines, and many miscellaneous items.

Anoth r haper cutter type tool is used with a specialmachin to produce worms and similar screw items. In thisconfiguration the tools are called thread generators.

The Shaping MethodThe machining or cutting of gears with gear shaper cutters

is a planing or shaping process involving a reciprocating mo-tion of the