L OoZa

NASA

AVSCOMTechnical Memorandum 103281

Technical Report TR- 90- C- 023

Dynamic Measurements of GearTooth Friction and Load

Brian Rebbechi, Fred B Oswald, and Dennis P. TownsendLewis Research CenterCleveland, Ohio

Prepared for theFall Technical Meeting of the American GearManufactures AssociationDetroit, Michigan, October 21-23, 1991

US ARMY

NASA AVIATION

SYSTEMS COMMAND

https://ntrs.nasa.gov/search.jsp?R=19910018256 2020-03-26T08:05:28+00:00Z

Dynamic Measurements of Gear Tooth Frictionand Load

Brian Rebbechi R , Fred B. Oswald, andDennis P. Townsend

National Aeronautics and SpaceAdministration

Lewis Research CenterCleveland, Ohio 44135

1. INTRODUCTION

The dynamic forces at the point oftooth contact are of considerable interestto the designers of high-speed, light-weight gearing. Accurate prediction of thedynamic loads can assist in minimizing thesize and weight of a transmission. In ahelicopter application, where the transmis-sion is a significant fraction of vehicleweight, such a reduction would be an impor-tant factor in overall vehicle performance.

A program to experimentally and theo-retically study fundamental mechanisms ofgear dynamic behavior is being undertakenat the NASA Lewis Research Center in sup-port of a joint research program betweenNASA and the U.S. Army. This paper pre-sents the results of dynamic tooth-filletstrain gage measurements from the NASAgear-noise rig, and it introduces a tech-nique for using these measurements toseparate the normal and tangential(friction) components of the load at thetooth contact. Resolution of the contactforce is desirable for several reasons.Two of these reasons are the following:

(1) A primary output of analyticalmodels of gear dynamic behavior is typi-cally the normal force at the point ofcontact (e.g., [1] and [2]).

(2) The measurement of dynamicfriction of meshing gears does not appearto have yet been carried out successfully.

An interesting trial was carried outby Benedict and Kelly [3], but it was dis-continued because of dynamic response

Visiting scientist from AustralianAeronautical Research Laboratory.

problems. Anderson and Lowenthal [4] com-puted overall losses due to friction andfound good agreement between theoreticalpredictions and experimental data. Krantzand Handschuh (5] applied a similar tech-nique to an epicyclic gear rig, obtaininggood correlation at low oil temperatures,but poorer correlation at higher oil tem-peratures. However, this technique cannotdetect the variation in friction during thetooth engagement cycle. There is also theproblem of separating the power loss due togear tooth friction from power losses dueto other sources such as bearings, windage,and so forth.

Extensive measurements of lubricationconditions at a sliding-rolling contacthave been carried out on disk machines [6].These experiments are of considerable valuein confirming the existence of elastohydro-dynamic lubrication and in identifying theseparate regimes of lubrication that pre-vail under the various slide-to-rollratios. However, the usefulness of themodes of behavior and friction coefficientsin predicting lubrication conditions at anactual tooth contact, where the degree ofsliding changes throughout the toothengagement cycle (typical duration,250 µsec), needs to be verified. In thisshort period of time, large changes occurin the lubricant temperature, shear, andviscosity atpressures up to 1.4 GPa(200 000 lbf-in. ). Dyson (7] reportedtemperatures up to 400 °C and oscillatoryshear rates up to 10 7 sec -1 . These con-ditions cannot readily be produced outsideof an actual tooth mesh.

Friction at the tooth contact isimportant for determining not only powerloss and efficiency, but also for under-standing gear-tooth scoring and wear. Animportant parameter in scoring is the fric-tion coefficient [3]. Friction greatlyaffects the heat input to the lubricantwhen sliding velocities are high.

This report presents dynamic, gear-tooth strain measurements from low-contact-ratio spur gears tested in the NASA gear-noise rig. The technique used to convertthese strain measurements into normal andtangential (friction) tooth loads isdescribed. Plots of normal and tangentialforces, for both static and dynamic condi-tions, are presented for a representativerange of loads and speeds. The normalforce and dynamic strain data have beenused to verify a gear dynamics code inanother related report [8].

2. APPARATUS

2.1 Test Facility3. TEST PROCEDURE

tooth-root fillets on both the loaded(tensile) and unloaded (compression) sideof two adjacent teeth on the output(driven) gear (Fig. 4). To measure maximumtooth bending stress, the gages were placedat the 30 0 tangency location [9].

Strain gage signals were conditionedby two methods: for static calibration andmeasurement, a strain gage (Wheatstone)bridge was used; for dynamic measurements,the strain gages were connected via a slip-ring assembly to a set of constant-currentstrain gage amplifiers.

A 4-channel, 14-bit digital dataacquisition system was used to record thedynamic strain data. Sample rates of 20 to50 kHz per channel were used, depending ontest gear speed.

An optical encoder was mounted on theinput shaft to measure roll angle and hencedetermine load location; the position ofthe encoder was adjusted so it would pro-duce 1 pulse/revolution at a known rollangle.

These tests were conducted in the NASALewis gear-noise rig (Fig. 1). This rigcomprises a simple gearbox powered by a150-kW (200-hp) variable speed electricmotor, with an eddy-current dynamometerthat loads the output shaft. The gearboxcan be operated at speeds up to 6000 rpm.The rig was built to carry out fundamentalstudies of gear noise and of dynamicbehavior of gear systems. It was designedto allow testing of various configurationsof gears, bearings, dampers, and supports.

A poly-V belt drive served as a speedincreaser between the motor and inputshaft. A soft coupling was installed onthe input shaft to reduce input torquefluctuations caused by a nonuniformity ofthe belt at the splice.

Test gear parameters are shown inTable 1, test rig parameters in Table 2,and gear tooth profile traces in Fig. 2.The tooth surface roughness was measured byusing an involute-gear-checking machinewith a diamond stylus of approximately10-µm (0.0003-in.) radius. The surfaceroughness varied along the length of thetooth, with the region near the rootappearing to be lightly polished. Themaximum surface roughness was estimated tobe 1.34 µm, (34 Ain.) peak-to-peak, or anaverage of 0.43 µm (11 Ain.) (Fig. 3). Thegear rig was operated at an oil fling-offtemperature of 54±2 °C (130±5 °F). At themean temperature of 54 °C, the viscosity ofthe synthetic oil (Table 2) used in thetests was 14 cSt (11.6 cP). Natural fre-quencies from a four degrees-of-freedomeigensolution [8] are also shown inTable 2.

2.2 Instrumentation

General-purpose, constantan foil,resistance strain gages (gage length,0.38 mm (0.015 in.)) were installed in the

3.1 Calibration

Calibration of the strain gages on theinstrumented (driven) gear was conducted toprovide a matrix of strain output versusapplied load. Before commencing the straingage calibration, the gears were demagne-tized. This demagnetization reduced theapparent strain resulting from the gagesmoving through the magnetic field of theadjacent gear. At normal gear operatingspeeds, magnetic effects can induce anerror signal in the gage.

For calibration, the instrumented gearwas meshed with a special gear whose adja-cent teeth had been ground away; this per-mitted loading of a single tooth only. Thecalibration was carried out for each of thetwo instrumented teeth for roll anglesranging from 12° to 30°. At each test po-sition (roll angle) the torque was appliedat three levels - 45 percent, 88.5 percent,and 132 percent of the nominal value of71.8 N-m (635 in.-lb). At each of theseload levels the sliding direction wasreversed (by reversing roll direction), anda linear curve was fit to the data for eachsliding direction. By reversing the rolldirection, the instrumented gear was effec-tively tested as both the driven gear (out-put) and driving gear (input). In eachinstance the gear was rotated a small angle(approximately 1 0 ) in the intended direc-tion of roll until the desired roll anglewas reached, so as to definitely establisha sliding direction.

The strain gage calibration apparatusis shown in Fig 5. The results of the cal-ibration for gages 1 to 4 are given inFigs. 6 and 7, for loading on tooth 1. Thearrows indicate roll direction. Theresults for loading on tooth 2 were verysimilar.

Cn

Dynamometer -^

OptionalTest output flywheel ^.

(a) Layout.

(b) Detail of gearbox.

Figuret.—NASA gear-noise rig.

TABLE 1. - TEST GEAR PARAMETERS

Gear type . . . . . . . . standard involute, full-depth toothNumber of teeth . . . . . . 28Module, mm (diametrial pitch in.^l ) . . . . . . . . 3.175(8)Face width, mm (in.) . . . . . . . . . . . . . . 6.35 (0.25)Pressure angle, deg . . . . . . . . . . . . . . . 20Nominal (100-percent) torque, N-m (in.-lb) 71.77 (635.25)Theoretical contact ratio . . . 1.64Driver modification amount, mm (in.) 0.023 (0.0009)Driven modification amount, mm (in.) . . . . . 0.025 (0.0010)Driver modification start, deg . . . . . . . . . . . . . . 24Driven modification start, deg . . . . . . . . . . . . . . 24Tooth root radius, mm (in.) . . . . . . . . . . . . 1.35(0.053)Average surface roughness, µm (,dn.) 0.43(11)

3

(a) Gage instanation.

RootZero reset

iTip

Tooth 1 Tooth 230° tangency r i

to root fillet\

(location of1 to 4)gages 7 2^

,30, \^Q3

i LO LO \Gage

Tooth 0 J

Rotation

(b) Gage location.

Figure 4. Strain gage installation and location on test gear.

TABLE II. - TEST RIG PARAMETERS

Input inertia, J 1 , kg-m' (lb-sec`-in.) . . . . . . 0.0237 (2.10)Gear inertia, J,, J 3 , kg-m' (lb-sec`-in.) 0.0000364 (0.00322)Load inertia, J , , kg-m' (lb-sec`-in.) . . . . . . . 0.085 (7.5)Input stiffness, K,, N-m/rad (1b-in./rad) . . . . . 341 (3017)Gearbox stiffness,'K,, N-m/rad (lb-in./rad) . . . 6158 (54 500)Load stiffness, K,, N-m/rad (1b-in./rad) 12 700 (112 300)Synthetic turbine oil . . . . . . . . . . . . . . MIL-L-23699BViscosity at 130 O C, cSt, (cP) . . . . . . . . . . 14 (11.6)

Natural frequencies (eigensolution), Hz 6.56, 52.5, 1220

1

8

1s 0 0002,n

22 i

^ (a) Driving gear. TEcLO1- 22

15

8

1

6 9 12 15 18 21 24 27 30 33

Roll angle. deg

(b) Driven gear.

Figure 2.—Test gear profile traces.

Figure 3.--Surface roughness measurements of driven gear.

4

Figure 5—Strain gage calibration apparatus.

2000 Driving gear— — Dnven gear

1500

1000 —mQs soo

o0a (a) Gage 2, tensile strain.0

-500

o -1000

-1500 /

-2000

-250030 28 26 24 22 20 18 16 14 12

Roll angle, deg

(b) Gage 1, compressive strain,

Figure 6 —Static strain gage data, single tooth loadingon tooth 1 (arrows show roll direction).

40 Driving gear

m -- Driven gear

Q

n 01 1

o (a) Gage 4, tensile strainO

0MO

_0 28 26 24 22 20 18 16 14 12

Roll angle, deg

(b) Gage 3, compressive strain.

Figure 7.—Static strain gage data measured at tooth 2for single-tooth loading on tooth 1 (arrows show rolldirection).

3.2 Data Acquisition

3.2.1 Static strain data. - Straindata were recorded under static (nonrotat-ing) conditions for the gear set assembledin its normal (running) configuration withthe standard running gear replacing thecalibration gear. The measurements weremade for two reasons: first, as a check onthe accuracy of the method used to resolvetooth force into normal and tangential com-ponents; and second, to provide informationon load sharing characteristics of the gearassembly. A strain gage bridge circuit wasused to record strains for roll angles from12° to 40° relative to tooth 2. Torquelevels of 37, 88, 100, and 132 percent wereapplied, but unlike the single-tooth case,linear curve-fitting of these data was notappropriate because of the kinematic non-linearities introduced by load sharing whenmore than one pair of teeth are in contact.As for the single-tooth case, these meas-urements were carried out for the instru-mented gear acting as both the driven anddriving gear, thus reversing the slidingdirection.

3.2.2 Dynamic strain data. - Dynamicstrains were recorded for the 4 gages, fora speed-load matrix of 28 points: 4 speeds(800, 2000, 4000, and 6000 rpm) and 7torque levels (16, 31, 47, 63, 79, 94,and 110 percent of the nominal value of71.8 N-m (635 in.-lb)). The data wererecorded by 14-bit data recorders via aslip-ring assembly. Sample rates used were50 000 Hz per channel for the 2000-, 4000-,and 6000-rpm speeds, and 20 000 Hz perchannel for the 800-rpm speed. A continu-ous record, consisting of 10 000 datascans, was made at each speed so as to givea record length of 0.2 sec at 50 000 Hz,and 0.5 sec at 20 000 Hz. Because of theinterest in comparing tensile and com-pressive strains on each tooth, data fromthese two gages were simultaneouslyrecorded along with the encoder signal.This procedure was repeated for the secondinstrumented tooth.

The data were then digitallyresampled, by using linear interpolation,at either 1000 or 2000 samples per revolu-tion (depending on speed) and synchronouslyaveraged. Time domain synchronous averag-ing, a technique now in wide use in geardiagnostics (10], was used here to reducenoise effects (especially from the torquefluctuation caused by the belt drive). Itsimplementation requires two data channels -one for timing signal data and one forstrain data. The timing signal data pro-vided resample intervals for exactly onerevolution.

ANALYSIS

For a single tooth, measurement of thestrain outputs Sr, and . S of gagesmounted on the copressive and tensilesides of the tooth respectively (Fig. 4)will, in principle, enable resolution ofthe tooth forces F . (normal) and Fr

(tangential), provided that the response ofthese two gages to the two forces islinearly independent. The response of thegages can then be expressed as

So = a 11Fn + a12Ff (4.1)

S, = a21Fn + a 22 F f (4.2)

or simply as

(S} _ [a] {F} (4.3)

Swhere {S} = S`

z

lF

{F} = Ff

and all is the strain influence coeffi-cient; that is, the strain at i due to aunit normal force (j = 1) or a unit fric-tion force (j = 2).

The strain influence coefficients allare evaluated by alternately setting Fnand F. in equations (4.1) and (4.2) tozero. In practice, neither F n nor Fcan actually be zero because a normal forcebetween the teeth is a prerequisite for asliding force to develop. However, becausestrain values were recorded for both direc-tions of sliding (that is, for the instru-mented gear acting as both driving anddriven gear) at each roll angle value, weinferred that the average of these twostrain values is equivalent to the fric-tionless case, and that the effect of fric-tion alone will be one-half the differencebetween the two values. Thus, the coef-ficients a 12 and a22 (which relate tofriction) are evaluated from half thedifference between the driving gear anddriven gear curves of Fig. 6. Likewise,the strain coefficients a ll and a21(which relate to normal force) are eval-uated from the average of these two curves.The solution for F and F is found bypremultiplying both n sides of' equation (4.3)by [ a ] -1 ; hence

{F} _ [a]-'(Si (4.4)

The analysis presented above ignoresthe influence on strains S and S t dueto loading on adjacent teeA . In the caseof thin-rim gears [11], this effect can beon the order of 12 percent. For the thick-rim gears used here, however, the influencefrom adjacent teeth is at most 3 percent(compare Figs. 6 and 7). In the data pre-sented in this paper, the influence ofadjoining teeth has been included. Thecomputational procedure is outlined in theAppendix.

5. RESULTS AND DISCUSSION

5.1 Calibration

Tooth-fillet strains for 100-percenttorque were evaluated by fitting a linearcurve to the calibration data for the threetorque levels. These strains at gages 1 to4 are plotted in Figs. 6 and 7 as a func-tion of roll angle, for loading of tooth 1.Notable from these curves is the signifi-cant influence of static friction on strainoutput; the tensile gage (see Fig. 6(a))shows a difference in strain between thedriving- and driven-gear cases (when slid-ing direction reverses) that is 27 percentof the mean strain reading. The signifi-cance of this is twofold: first, it isdifficult to establish a "no-friction',curve; and second, and possibly more impor-tant, these curves (particularly the ten-sile curve) illustrate the effect thattooth friction has on the results. It isapparent from Fig. 6 that the compressivegage is much less influenced by frictionand, thus, would be expected to give thebest indication of normal force if only onegage were used. This is further confirmedby the tooth strain influence coefficients(see Appendix).

5.2 Static Meshing

Measured strain is plotted in Fig. 8as a function of roll angle for staticmeshing of the gears (i.e., for multiple-tooth contact). This figure shows theaverage strain (mean of driving- anddriven-gear values) for 37-, 88-, 100-, and132-percent torque. Figure 9 shows ingreater detail the tooth-fillet strains for

Gage 2 Gage 42000 tooth 1 tooth 2 Torque level,

percent

1500 132

A."

100

loco , ^•\^^;^ j 88—

500 37

0

I I I I I I I I I

(a) Loaded tensile strain side of tooth.

22

-500

—1000

—1500

—200032 30 28 26 24 22 20 18 16 14 12

Roll angle for tooth 2, deg

IIIIIIIII28 26 24 22 20 18 16 14 12

Roll angle for tooth 1, deg

(b) Unloaded compressive strain side of tooth.

Figure 8.—Averaged static strain data on two successiveteeth.

Soo

gages 1 to 4 at 100-percent torque, withthe instrumented gear acting as both drivenand driving gear. The curves of Fig. 9 arethe averaged result of three trials. Fromthe results of Fig. 9, and the influencecoefficient matrix previously described,plots of normal and friction forces(Fig. 10) have been derived from the staticdata for the 100-percent-torque case.

2000 Driving gear-- Driven gear

1500

1000 \\

Soo IN

0IIIIIIIII;J

bo (a) Tensile strain, gages 2 and 4.

-500

-1000

-1500

-200030 28 26 24 22 20 18 16 14 12 10 8

Roll angle on tooth 2, degIIIIIIIII1I3028 26 24 22 20 18 16 1412 10

Roll angle on tooth 1, deg

(b) Compressive strain, gages 1 and 3.

Figure 9.--Static strain data, multiple tooth loading(arrows show roll direction).

Pitch point Pitch pointtooth i tooth 2

500 1720 N = r Normal -2000

aoo --` 386 lb , force

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

300 Tooth 1 Tooth 2 -

1000 200

100

0 0r Frictional -

/ forcei

Z° --------------'y"------------

o ° I I I' I I I

€ £Z Z (a) Driving gear.

500 `-1720 N2000

4001 386 lb

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

300 Tooth t

1000 200 ____ ------- r_ -_ -100

0 0

---------- -- -;---------- ----

I I I! I I I30 26 22 18 14 10

Roll angle on tooth 1, deg

30 26 22 18 14 1CRoll angle on tooth 2, deg

(b) Driven gear.

Figure 10.—Normal and frictional forces at 100-percent torque under staticconditions.

The total normal force between theone- or two-tooth pairs in mesh should beequal to 1718 N (386 lbf). This value isthe torque divided by the base circleradius. The normal-force component of theplots shows agreement within 1.5 percent ofthe expected value.

An absolute value for the frictionforce cannot be determined during calibra-tion since the coefficient of friction atthe tooth contact point is unknown. If anarbitrary value of unity is assigned to themaximum frictional force developed at100-percent torque, then the friction valueshould be either +l or -1 (depending on thedirection of sliding) in the single-toothcontact region. This ideal is nearlyachieved in the static measurements fortooth 1 in Fig. 10(b). For tooth 2, thefriction force is offset by about -0.4 fromthe +1 values. Outside the single-toothcontact region, the friction forcedecreases in approximately linear fashionwith the normal tooth load. This implies aconstant friction coefficient under thesestatic meshing conditions.

It is interesting to note the locationof the zero-crossing of the friction forcein Fig. 10 when tooth sliding changes di-rection. This zero-crossing differs fromthe pitch point by nearly 1 0 of roll. Someof this difference may be due to deflectionof the gear shaft, which causes a shift inthe operating pitch point.

5.3 Dynamic Case

The dynamic tooth strains for the 28speed-load conditions are shown in Fig. 11.To allow direct comparison, the compressivestrain data are inverted (shown as posi-tive) and overlayed on the tensile curves.Notable features of these curves includethe peak tooth strain corresponding withthe high point of single-tooth contact(which occurs at about 23 1 roll angle), adip or notch in the tensile tooth strain

- curves near the pitch point (where thesliding force reverses), and dynamiceffects becoming apparent at higher speeds.

The dynamic effect is particularlynotable in the curves for 4000 rpm at thelowest torque (16 percent). Here, theforce vanishes, thereby indicating toothseparation occurs. By contrast, at 110-percent torque there is very little dynamiceffect, as evidenced by little differenceamong the curves for the four speeds (800,2000, 4000, and 6000 rpm).

In Fig. 12 the computed normal andfriction forces are shown for four speedsat the highest torque (110 percent). Notethe very good agreement with expectedresults at the low speed of 800 rpm(Fig. 12(a)), where we would expect toapproach a static case. Here, the normalforce is very close to the static nominalvalue (a function of the torque divided bythe base circle radius). The frictionresults show a marked transition in forcefrom negative to positive as the tooth con-tact passes through the pitch point, where

3

2

1

0

-1

J -2I I ( v

0

,- Normal 3 LLforce

Tooth 2

Frictionalforce

2

1

0

-1

-2

^ I I I I I I I I I I

(a)800 rpm.

c

0

2000

1500

1000

Soo

0

I

I

^^ I

there is pure rolling. Also the frictioncoefficient appears to be less than that ofthe static calibration case, as can be seenby comparing Fig. 10(a), where the frictionforce has a value of unity for a normalforce of 386 lbf, and Fig. 12(a), where thefriction force is a maximum of approxi-mately t0.75.

5.4 Accuracy

The results obtained herein for thestatic and dynamic tests indicate thefeasibility of using multiple gages toseparate the tooth friction and normalforces. The results of the static case areparticularly encouraging. The value forthe normal force is generally within1.5 percent of the expected value. Thefriction force, whilst at times much lessaccurate, nonetheless demonstrates thetrends we expected to see - that is, the

Tensile tooth strain-- Compressive tooth strain (inverted)

Pitch Pitchpoint for point for

P000 tooth 1 tooth 2Tooth 1 Tooth 2 Torque levels,

Vpercent

1500Vi\^ / //^^ I\ 110

\/ / y\ \

/

// I \ i 94

--79

1000 j/lj^ \ ^ J/I^

^ ^^' X63

l/ \ 11 1 / J _47

jr.^ \ \ `\ `^^ x/31500 / % .I

_16

30 28 26 24 22 20 18 16 14 12 10Roll angle on tooth 1, deg

I I I I I I I I I 1 1

30 28 26 24 22 20 18 16 14 12 10Roll angle on tooth 2, deg

(b) 2000 rpm.

Figure 11.—Dynamic tooth strains at four speeds and seven torque levels.Compressive strains are shown as positive for comparison with tensiledata.

Tensile tooth strain-- Compressive tooth strain (inverted)

Pitch Pitchpoint for point fortooth 1 tooth 2

2000Tooth 1 Tooth 2

I Ii Torque levels,I t percent

1500 /' \{ _-110

\ y \\ ' \ \r. \—94

/ ,

1000 \\/^\ \y / \ \+ \ ^^ X63

\\\

500

//'

,_31

. / !r N \ I .^.\ \y \\\\ 116

o' \

c

o ^L_L_ I I I I I I I I

(c) 4000 rpm.

2000

/ \ Torque levels,/ \^ /\\^ percent

1500 / // /^\ __ _110

—94

/ // \\i ^ \\\ ^/ // \\ \^-^ --79

1000 / / /^\ \ \ \ / / ^\ \ \

// / /^\ \I \\\\\ ^' _-'47

//j //!

\

} ^^ I x-31500

r/ l l^

\\\ . / \\ \ I

Illi

—16

p'I I-30 28 26 24 22 20 18 16 14 12 10

Roll angle on tooth 1, deg

I I I I I I I I I I I30 28 26 24 22 20 18 16 14 12 10

Roll angle on tooth 2, deg

(d) 6000 rpm.

Figure 11 —Concluded,

friction force is proportional to the nor-mal load, and a reversal in sign occurs atabout the pitch point. The good resultsfor the static case are believed to bepartly due to using instrumentation thatwas identical to that used for the staticcalibration (i.e., the Wheatstone bridgecircuit). Assessing the accuracy for thedynamic case is more difficult, since we donot fully know what to expect. However,dynamic operation could introduce the fol-lowing problems:

(1) There could be some change in sen-sitivity due to the change in signal condi-tioners (i.e., constant current amplifiersoperating through slip rings).

(2) Resistance variations of the sliprings and other electrical noise can con-taminate dynamic data. This was minimizedhere by the use of synchronous averaging,as described in the test procedure fordynamic data.

(3) Other dynamic effects such as gearbody vibration can also produce unwanted

8

(b) 2000 rpm.

5o0

400 ---------

300

200

100

^ 0

V -----------o

Z 500400 ----------

300

200 -

Z100

- 1 30 26 22 18 14 10Roll angle on tooth 1, deg

0

the summing of the strain magnitudes, as isthe case for normal force (see Appendix).

3 various techniques can be used to minimizeerrors - synchronous averaging, as carried

2 out here, and possibly, an adjustment (com-pensation) of the friction curve to bring

1 about zero friction at the pitch point.The do offset of the strain signal is tri-

o tical. Figure 13 shows the superimposedcurves of normal and friction forces for

-1 four successive revolutions of the gear,using nonaveraged data. Each curve isbased on the corresponding tensile and

3 compressive strains for that particularrevolution. A significant variation in

2 friction estimation is evident from onerevolution to the next; this cannot be

1 ascribed to the expected small torque fluc-tuations caused by the belt drive.

0

Pitch point Pitch pointtooth 1 tooth 2r Normal

500 force^z000

400 --------,--- r ----'-- ----

300 Tooth 1 Tooth 2

1000 200

100 Frictional0 0 I force

--------

(a) 800 rpm.

2000

1000

0ZU0

E

Z2000

1000

0

(c) 4000 rpm.

5002000

---------- -- -- -------- -- -400

300

1000200

1

100

0 0

0

-130 26 22 18 14 10

Roll angle on tooth 1, deg

30 26 22 18 14 10

Roll angle on tooth 2, deg

(d) 6000 rpm.

Figure 12.—Computed dynamic load and friction at four speeds and110-percent torque.

signals. A strain output from the toothfillet gages was observed when the teethwere out of mesh. This is attributed tovibration of the gear body. The effect wasmost obvious at higher speeds, appearing atthree times tooth mesh frequency. Thisfrequency component can be seen in thefriction force trace at 6000 rpm(Fig. 12(d)).

Using strain outputs to detect fric-tion requires accurate measurement ofstrain. A 1-percent error in strain meas-urement will result in a 10-percent errorin force estimation (see Appendix). Thisextreme sensivity to measurement erroroccurs only with friction force estimation.It effectively results from using the dif-ference between the magnitude of the.ten-sile and compressive strains, rather than

Pitch point Pitch pointtooth 1 tooth

f- 3force

Tooth 1 2

i Tooth 22

tional °e 2

0.2A u_

-1

-2

Differences in profile between thesingle-tooth calibration gear and theoperating gear result in the tooth contactpoint being slightly displaced along thetooth profile, thereby causing an error inthe measured roll angle. This error hasbeen estimated to be of the same order(0.25 0 ) as the error in setting the rollangle for calibration.

The friction force results obtainedherein were necessarily qualitative. Alogical next step would be to calibrate thegages with a known friction force. Adevice similar to that of Benedict andKelly [3] (Fig. 14) could be used for thispurpose. In their application, dynamiceffects prevented Benedict and Kelly fromobtaining useful results from this device.If the device were used only for staticcalibration, this restriction would beremoved. Alternatively, with only slightmodification this setup could be used toapply a known force in the friction forcedirection while the tooth contact positionwas held constant.

2 5002000

o0 Z a 400

3 LL U`o 3000 1000 E 200 ____m

2 0 8 too rFricZ fort

0 01

3 30 26 22 18 14 10

Roll angle relative to tooth 2, deg

\2

Figure 13.—Computed dynamic load and friction superimposed for foursuccessive revolutions at 4000 rpm and 110-percent torque.

Appliedtorque —t

Frictional

force 10At

Pivot

i t

Strain gaged bolt —

Figure 14--Gear friction measurement apparatus, modified fromBenedict and Kelley (1961).

6. SUMMARY OF RESULTS

Tooth-fillet strains were recorded for28 operating conditions on the NASA gear-noise rig. A method was introduced thatused the tensile and compressive tooth-fillet strains to transform these strainmeasurements into the normal and frictionalloads on the tooth. This technique wasapplied to both the static and dynamicstrain data. The results demonstrated thatthis technique was viable, and in parti-cular they showed the following:

1. For the static case, the normalforce closely agreeed (within 1.5 percent)with expected results. The frictionalresults were much more variable, but theyexhibited expected trends.

2. In the dynamic case, the estim-ation of normal force was good, the fric-tion results, less so. However, the fric-tion force results showed expected trends;that is, the dynamic friction coefficientwas less than the static coefficient, andthe friction reversed direction near thepitch point. Further refinement of meas-urement techniques will be required toproduce more accurate results.

3. The influence of sliding frictionwas particularly marked on the tensiontooth-fillet gage. The compression gagewas affected by friction to a much lesserdegree.

APPENDIX

TOOTH FORCE INFLUENCE COEFFICIENT MATRIX

The meshing cycle of a tooth on a low-contact-ratio spur gear may be divided into3 cases: (1) the tooth load is shared withthe preceeding tooth; (2) the entire loadis carried by the tooth; and (3) the loadis shared with the following tooth. Wehave shown earlier that loading on theadjacent (preceeding or following) toothwill produce a small influence on a tooth-fillet strain gage.

The normal and frictional componentsof force on a gear tooth may be computedfrom equation (4.4), expanded to a six-by-six matrix. To compute the forces on tooth1, we must know the output of strain gageson the adjacent teeth (designated 0 and 2)due to loading on tooth 1. Although we did

not measure the strain on tooth 0, we knowthat the strain due to loading on adjacentteeth is a small effect. And we assumethat the unmeasured strain reading on tooth0 due to loading on tooth 1 is identical tothe strains measured under similar condi-tions on tooth 1 due to a load applied ontooth 2. Likewise, when we calculateforces on tooth 2, we make a similarassumption that unmeasured readings fromtooth 3 due to loading on tooth 2 are iden-tical to measured readings on tooth 2 dueto loading on tooth 1. Equation (4.4) thusbecomes

a ••• ••• ... ... aOnF

11 16 $,Oc

F ... ... ... ... ... ...S o"of

F in ... ... ... ... ... ... ^S lcF ir... ... ... ... ... ... Sit

Fen ... ... Sec

F 2 a61 a 66

Set

wherea 1 is a function of roll angle.Since there are only one or two teeth incontact at any one time, there are only twoor four nonzero rows and columns in equa-tion (Al), so the matrix is effectivelyonly of order two or four. Any

ai, cor-responding to a tooth outside of the con-tact region is zero.

To illustrate the significance of thedominant terms in this matrix, for thepurposes of this example only, cross-coupling terms' are disregarded. Thenormal force Finsimplifies to

Fin — a33 S IC + a34Sit (A2)

where F = normal force on tooth 1; S =compressive strain on tooth 1; and S1t'c=tensile strain on tooth 1.

The coefficients a and a areplotted in Fig. Al. It is notable4 that thetension gage has less influence on the com-putation of normal force than the compres-sive gage. Indeed, at a roll angle of 281the coefficient a 4 becomes zero; thetensile gage then 'has no effect on Fn.

Similarly, the friction force (seeFig. A2) is described (again disregardingcross-coupling terms) by

Fir = a43SIC + a44Slt (A3)

where F1f = normal force on tooth 1.

'Cross-coupling terms will assume amuch greater significance in the case ofthin-rim gears where the strain at a toothfillet is significantly affected by theloading on an adjacent tooth.

(Al)

10

To aid in the interpretation of thesecoefficients, it is useful to plot thetooth strain versus roll angle, for thetension and compressive gages. This isgiven in Fig. A3; the data shown here wereobtained from taking the average of thecalibration curves in Fig. 6.

The influence coefficients for thefriction force show why accuracy is impor-tant. Recall that tensile and compressiveoutputs are similar in magnitude but oppo-site in sign; therefore the resultant valueof friction is a small (approximately10 percent) difference obtained from theproducts a d3 S 1c and a44sit in equa-tion (A3).

Pitch point —^ r ao

`I 34

/

ra33.2

U I `\'^^•I

30 28 26 24 22 20 18 16 14 12Roll angle, deg

Figure At.—Strain coefficients a 33 and a34 for calculationof normal force (eq. (A2)).

.02II _.

ra 44 I /^^^_

pdo ^ ^ I

V a 43 II

030 28 26 24 22 20 18 16 14 12

Roll angle, deg

Figure A2.--Strain coefficients a43 and a44 for calculationof friction force (eq. (A3)).

2500 Compressive gageI

^ ^ I

o Tension gage T_- ---.I

030 28 26 24 22 20 18 16 14 12

Roll angle, deg

Figure A3.--Single tooth strain data obtained from Figure 6.(Tensile curve is average of two curves of Fig 6(a); com-pressive curve is average of two curves of Fig 6(b) and isshown as positive for comparison with tensile data.)

8. REFERENCES

1. Lin, H.H.; Huston, R.L.; and Coy,J.J.: On Dynamic Loads in ParallelShaft Transmissions. 1: Modelling andAnalysis. NASA TM-100180, 1987.

2. Lin, H.H.; Huston, R.L.; and Coy,J.J.: On Dynamic Loads in ParallelShaft Transmissions. 2: ParameterStudy. NASA TM-100181, 1987.

Benedict, G.H.; and Kelly, B.W.:Instantaneous Coefficients of GearTooth Friction. ASLE Trans., vol. 4,no. 1, 1961, pp. 59-70.

4. Anderson, N.E.; and Loewenthal, S.H:Spur-Gear-System Efficiency at Partand Full Load. NASA TP-1622, 1980.

Krantz, T.L.; and Handschuh, R.F.:Efficiency Study Comparing TwoHelicopter Planetary Reduction Stages.AIAA Paper 90-2156, July 1990 (Also,NASA TM-103106, 1990).

6.. Dowson, D.; and Higginson, G.R.:Elastohydrodynamic Lubrication,Pergamon, Oxford, 1966.

7. Dyson, A.: Frictional Traction andLubricant Rheology in Elastohydro-dynamic Lubrication. Philos. Trans.,vol. A266, no. 1170, 1970, pp. 1-33.

8. Oswald, F.B., et al.: Comparison ofAnalysis and Experiment for Dynamicsof Low Contact Ratio Spur Gears. NASATM-103232, 1991.

9. Cornell, R.W.: Compliance and StressSensitivity of Spur Gear Teeth. ASMEPAPER 80-C2/DET-24, Apr. 1981.

10. McFadden, P.D.: Examination of a Tech-nique for the Early Detection of Fail-ure in Gears by Signal Processing ofthe Time Domain Average of the MeshingVibration. Mech. Syst. SignalProcess., vol. 1, no. 2, 1987,pp. 173-183.

11. Drago, R.J.; and Pizzigati, G.A.: SomeProgress in the Accurate Evaluation ofTooth Root and Fillet Stresses inLightweight, Thin-Rimmed Gears.American Gear ManufacturersAssociation, Fall Technical Meeting,Washington, D.C. Oct. 1980. AGMA Paper229.21.

NASA National Aeronautics Report Documentation PageandSpace Administration

1. Report No.NASA TM -103281

2. Government Accession No. 3. Recipient's Catalog No.

AVSCOM TR 90 - C - 0234. Title and Subtitle 5. Report Date

Dynamic Measurements of Gear Tooth Friction and Load

6. Performing Organization Code

7. Author(s) 8. Performing Organization Report No.Brian Rebbechi, Fred B. Oswald, and Dennis P. Townsend

E -6020

10. Work Unit No.505-63-369. Performing Organization Name and Address

NASA Lewis Research Center 1L162211A47ACleveland, Ohio 44135 - 3191

11. Contract or Grant No.andPropulsion DirectorateU.S. Army Aviation Systems CommandCleveland, Ohio 44135 — 3191 13. Type of Report and Period Covered

Technical Memorandum12. Sponsoring Agency Name and AddressNational Aeronautics and Space AdministrationWashington, D.C. 20546 - 0001 14. Sponsoring Agency CodeandU.S. Army Aviation Systems CommandSt. Louis, Mo. 63120 - 1798

15. Supplementary Notes

Prepared for the Fall Technical Meeting of the American Gear Manufactures Association, Detroit, Michigan, October 21-23, 1991. Brian Rebbechi, Australian Aeronautical Research Laboratory, Melbourne, Australia. Fred B. Oswald andDennis P. Townsend, NASA Lewis Research Center. Responsible person, Fred B. Oswald (216) 433-3957.

16. Abstract

As part of a program to study fundamental mechanisms of gear noise, static and dynamic gear tooth strain measure-ments were made on the NASA gear-noise rig. Tooth-fillet strains from low-contact-ratio spur gears were recordedfor 28 operating conditions. A method is introduced whereby strain gage measurements taken from both the tensionand compression sides of a gear tooth can be transformed into the normal and frictional loads on the tooth. Thistechnique was applied to both the static and dynamic strain data. The static case results showed close agreement withexpected results. For the dynamic case, the normal-force computation produced very good results, but the frictionresults, although promising, were not as accurate. Tooth sliding friction strongly affected the signal from the straingage on the tension side of the tooth. The compression gage was affected by friction to a much lesser degree. Thepotential of the method to measure friction force has been demonstrated, but further refinement will be requiredbefore this technique can be used to measure friction forces dynamically with an acceptable degree of accuracy.

17. Key Words (Suggested by Author(s)) 18. Distribution Statement

Gears; Gearing; Dynamic load; Dynamic friction; Unclassified-UnlimitedFriction; Dynamics Subject Category 37

19. Security Classif. (of the report) 20. Security Classif. (of this page) 21. No. of pages 22. Prioe"

Unclassified Unclassified 14 A03NASA FORM 1828 OCT 88