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C. ChangenetMechanical Engineering Department,

ECAM Lyon,France

X. Oviedo-MarlotPSA Peugeot Citroën,

La Garenne-Colombes,France

P. VelexLaMCoS,

UMR CNRS 5514,INSA de Lyon,

Francee-mail: [email protected]

Power Loss Predictions in GearedTransmissions Using ThermalNetworks-Applications to aSix-Speed Manual GearboxA model is presented which makes it possible to predict power losses in a six-speedmanual gearbox. The following sources of dissipation, i.e., power inputs in the model, areconsidered: (i) tooth friction; (ii) rolling element bearings; (iii) oil shearing in the syn-chronizers and at the shaft-free pinion interfaces; and (iv) oil churning. Based upon thefirst principle of Thermodynamics for transient conditions, the entire gearbox is dividedinto lumped elements with a uniform temperature connected by thermal resistances whichaccount for conduction, convection, and radiation. The numerical predictions comparefavorably with the efficiency measurements from the actual gearbox at different speedsand torques. The results also reveal that, at lower temperatures (about 40°C), power lossestimations cannot be disassociated from the accurate prediction of temperaturedistributions. �DOI: 10.1115/1.2181601�

1 Introduction

In the general context of the reduction of fuel consumption andemissions in automotive industry, there is an increased demand formore efficient gearboxes. It is therefore becoming important to beable to predict and control power losses in geared systems, andanticipate the consequences of design modifications prior to test-ing on actual drives. Many factors can affect the efficient perfor-mance of a transmission which may have different operationalcondition ranges, i.e., high/low speed and/or load losses. For thegears, it takes several forms including sliding and rolling frictionlosses �1,2�, windage �3,4� and churning effects �5–7�. Other sig-nificant sources of power dissipation have to be considered,namely, the bearing, synchronizer, and free-pinion losses �8,9�.Some specific studies about automotive transmissions have beencarried out, comprising reduction gears �6�, truck gearboxes �10�,and automotive gearboxes �11,12�, either for a given or a calcu-lated oil temperature. The latter relies on the application of theenergy balance between the losses in the gear unit and the heattransferred to the environment, which makes it possible to derivebulk temperatures in the geared system �6,10,12�.

In this paper, a model is presented which can predict the tem-perature distributions and the efficiency of a six-speed manualgearbox. The system is composed of three shafts �primary, sec-ondary, and reverse� and a differential drive with all shaftsmounted on rolling element bearings �tapered roller, cylindricalroller, and ball bearings�. The following sources of dissipation,i.e., power inputs in the model, are considered: �i� tooth friction;�ii� rolling element bearings; �iii� oil shearing in the synchronizersand at the shaft-free pinion interfaces; and �iv� oil churning, forwhich a new model based on dimensional analysis is presented.Comparisons with experimental measurements on the actual gear-box reveal some discrepancies mainly at the lowest temperatures.A second model based upon a thermal network is set up whichallows temperatures and power losses to be simultaneously calcu-lated. The corresponding efficiency evaluations are in very good

Contributed by the Power Transmission and Gearing Committe of ASME forpublication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received December23, 2004; final manuscript received August 18, 2005. Review conducted by Teik C.
Lim.
618 / Vol. 128, MAY 2006 Copyright © 20

rom: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 09/10/201

agreement with the experimental data; this highlights the potentialimportance of the interactions between temperatures and powerlosses.

2 System Under ConsiderationThe system under consideration is the ML6C six-speed manual

gearbox developed by PSA Peugeot Citroën, France. The wholeset is enclosed in a housing made from an aluminum alloy and itis composed of �Fig. 1�:

• A primary shaft �S1� with four fixed pinions used to drivethe forward gears and one fixed pinion for the reverse gear,the pinion used for the fifth and sixth gears are free andassociated with a sleeve synchronizer set.

• A secondary shaft �S2� with four free pinions and two syn-chronizers; the driven pinions for the fifth, sixth and reversegears are fixed on the shaft as well as the pinion of thedifferential.

• A reverse shaft �SRev� with two free pinions mounted onneedle bearings.

• A differential drive �Diff� transmitting rotations to the axles.

The engine power is transmitted from the primary to the sec-ondary shaft by the engaged gear, to the differential wheel by thedrive gear and finally, to the wheels of the vehicle. The primaryand secondary shafts as well as the differential are supported bytwo bearings, i.e., �a� tapered roller bearings on the primary shaftand the differential and �b� the association of a ball and a rollerbearing on the secondary shaft. The gears are splash lubricated bya lubricant whose properties are given in Table 1.

3 Power LossesPower losses in geared transmissions are traditionally decom-

posed into no-load and load-dependent contributions. The lattercomprises friction at the mating teeth and in the rolling elementbearings while no-load losses mainly stem from oil churning andoil shearing in journal bearings and synchronizer cones. Windagelosses are neglected as the rotational speeds are limited, and alsobecause only some of the pinions on the primary shaft are notsubmerged in oil. For each individual dissipation source, the fol-
lowing formulas have been implemented in the simulation code.
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3.1 Tooth Friction Losses. During the meshing of a loadedpinion �subscript 1� and a gear �subscript 2�, the heat generatedper unit of time by friction between the teeth can be evaluated as�13�

Pgear = PinH�f �1�with

H� =�

cos��b�� 1

Z1+

1

Z2��1 − �� + �1

2 + �22�

�� =gf + ga

pb= �1 + �2, profile contact ratio

The prediction of the friction coefficient is still a challengingtask and its value depends on the lubrication regime, on the lubri-cant, on the surface texture of the mating parts, and on kinematics,etc. A review of the friction coefficients suited for geared systemscan be found in �14� and a contribution to tooth friction modelinghas been recently proposed by �15�. In what follows, the Benedictand Kelley formula �16� has been employed in order to be able todeal with multicomponent systems. It reads

f = 0.0127 log10�291205.8*10−6

��VgU2

Fnu� �2�

Formula �2� is recognized as an interesting compromise be-tween accuracy and simplicity; one major drawback is that thefriction coefficient becomes infinite at the pitch point but, in thecontext of power loss calculations, this is of very limited influence

Fig. 1 Kinematic mod

Table 1 ML6C lubricant properties

Kinematicviscosityat 40°C

Kinematicviscosityat 100°C

Fluid densityat 15°C

47.6 cSt 8.3 cSt 887.3 kg/m3

Journal of Mechanical Design


since the sliding velocity is nil at the same time. Because of thechange in geometry, kinematics and load vary as contacts moveon the tooth flanks. The friction coefficient given by Eq. �2� altersslightly along the path of contact but, for the sake of simplicity, anapproximated approach based on averaging f over a mesh periodhas been retained.

3.2 Bearing Losses. The friction torque for rolling elementbearings �Crol� can be estimated by the following equation:

Crol = Cl + C� �3�

where Cl is the friction torque due to the applied load and Cv, theviscous friction torque.

Harris �8� has empirically evaluated the bearing torque gener-ated by all mechanical friction phenomena and, for a bearing ofmean diameter dm under a load F and at a rotational speed N, thefollowing expression has been proposed:

Cl = f lFdm �4�

where f l depends on the bearing design and load.On the other hand, the viscous friction torque can be expressed

as

C� = 103f0��N�2/3dm3 �N � 2*10−3 m2/s min

C� = 16f0dm3 �N � 2*10−3 m2/s min �5�

where f0 is a factor which depends on the type of bearing andlubrication.

3.3 Oil Shearing. Oil shearing takes place at the free pinion-shaft interfaces since the pinions rotate at a different speed com-pared to that of the shaft on which they are located �N is thedifferential speed�. This effect exists also between the synchro-nizer rings and the unengaged gears. By assuming a Couette lami-nar flow �9�, the drag torque due to oil shearing can be calculatedas follows:

Cshear =4�2LR3N

�6�

of the ML6C gearbox

30j

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3.4 Oil Churning. Some models �5,7,10� can be found in theliterature but they are mostly related to one particular test rig andcan hardly be extrapolated to other gear arrangements. A moreversatile approach is proposed in this paper which is still based onexperimental data but can be generalized by using dimensionalanalysis. The specific test rig used for creating the experimentaldata base is shown in Fig. 2; it consists of a 1.5 kW electric motorwhich drives a shaft via a belt. A pinion mounted at one end ofthis shaft is enclosed in a housing and is partly submerged by thelubricant when the system is at rest. One housing face is made ofPlexiglas in order to visualize the flow of the lubricant-air mixturearound the pinion. The test bench is started with given operatingconditions, and once oil temperature is stabilized, the electricpower provided by the motor is measured. In a second phase, thetest bench is operated without oil at the same rotational speed, andthe corresponding electric power is also determined. The powerlost due to oil churning is finally derived by subtraction. Measure-ments have been carried out for various gear geometries �seeTable 2� and operating conditions, i.e., a pinion rotational speedfrom 1000 to 7000 rpm, several oil levels �with a maximum cor-responding to the height of the shaft axis�, and different oil prop-erties �see Table 3�.

A dimensional analysis was conducted in order to generalizethe results of these experiments with the aim of generating em-pirical equations valid for a wide range of conditions and data. Itwas assumed that the most influential parameters were:

• The geometrical parameters associated with the gear geom-etry: m, Dp, b.

• The geometrical parameters associated with the oil sump: h,V0.

• The fluid parameters: �, .• The dynamic parameters: �, g.

The oil churning drag torque Cchurn was therefore sought as afunction of the key parameters listed above as

Cchurn = f�m,b,Dp,h,V0,�,,�,g� �7�

The three fundamental parameters Dp, �, and � representinglength, mass, and time units, respectively, were used for normal-izing all the other factors leading to the following dimensionlessdrag torque coefficient Cm:

Cm =Cchurn

1/2��2Sm�Dp/2�3 �8�

Table 2 Gear data

Module �mm� 1,5 1,5 3 3 5 5 5

Face width �mm� 14 14 24 24 24 24 24Pitch diameter �mm� 96 153 90 159 100 125 150

Fig. 2 Oil churning test rig

620 / Vol. 128, MAY 2006


where Sm represents the submerged surface area of the pinion.According to the Pi theorem in dimensional analysis, Cm de-

pends on six groups of parameters, namely,

Cm = 1� m

Dp� 2� b

Dp� 3� h

Dp� 4� V0

Dp3� 5

Re 6 Fr 7 �9a�

with 1 , . . . , 7: constant coefficients to be adjusted from experi-mental results.

Considering the results of tests with different oils where allother parameters �geometry, oil level, speed etc.� remain constant,the only variations are those of the Reynolds number Re fromwhich the 6 exponent is derived from any two test results �sub-scripts a ,b� as:

6 =ln�Cma/Cmb�ln�Rea/Reb�

�9b�

The exponent of the Froude number is obtained by changing thespeed only, which modifies both Fr and Re but, as the contributionof Re has been isolated in Eq. �9b� one can deduce

7 =

lnCma

Cmb�Reb

Rea� 6

ln�Fra/Frb��9c�

Then, other tests are conducted in which only the geometricalcharacteristics are varied, i.e., the module in the first phase, andthen, the face width, etc., in order to successively determine theremaining exponents 2, 3, etc.

At lower rotational speeds, it appears that the dimensionlessdrag torque coefficient does not depend on parameters related togear geometry. However, its influence on the drag torque ispresent through the expression of Sm, the submerged surface areaand the following equation is obtained:

Cm = 1.366� h

Dp�0.45� V0

Dp3�0.1

Fr−0.6 Re−0.21 �10�

At higher rotational speeds, power losses are found to belargely independent of oil viscosity and it can be concluded thatthe inertia forces are much more important than the viscous ones.Consequently, the Reynolds number can be discarded in the for-mulation when speed increases. On the other hand, the ratio�b /Dp� was found to be significant on the dimensionless dragtorque coefficient. The corresponding equation for the drag torquecoefficient finally reads

Cm = 3.644� h

Dp�0.1� V0

Dp3�−0.35

Fr−0.88� b

Dp�0.85

�11�

The range of validity of the expressions �10� and �11� dependson a critical Reynolds number defined as

Rec =�Dpb

2��12�

Equation �10� will be used when Rec�6000, �11� holds true ifRec�9000 and either Eq. �10� or �11� can be employed in the

Table 3 Oil properties

Kinematic viscosityat 40°C

�cSt�

Kinematic viscosityat 100°C

�cSt�Density at 15°C

�kg/m3�

Oil 1 48 8.3 873Oil 2 145.5 15.7 865Oil 3 320 24 897.8

transition zone.

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4 Isothermal Model—Comparisons With Measure-ments From the Actual Gearbox

A number of efficiency measurements have been carried out onthe PSA Peugeot Citroën facility schematically represented in Fig.3. The gearbox is operated by an electric motor connected to theinput shaft and the load is imposed by a brake as shown in Fig. 3.The input speed, the output torque, and the oil temperature in thesump are measured and regulated. The global efficiency of thegearbox can be determined from the input and output torquescombined with the speed ratio of the engaged gear as

efficiency =output torque

input torque* ratio of the engaged gear �13�

All measurements were made in steady-state conditions whenthe oil temperature was stabilized. The gear ratios of the ML6Cgearbox can be found in Table 4. The measurement uncertaintiesare about 0.5 Nm on the input torque and 1.5 Nm on the outputtorque which, in the most critical conditions, lead to uncertaintiesof approximately +/−5% on power loss measurements.

4.1 Tests With an Oil Sump Temperature of 80°C. Thefirst set of comparisons deals with no-load conditions and a lubri-cant temperature of 80°C. Because of the inclination of the gear-box, only the differential wheel and pinions on the secondary shaftare partly submerged in oil and contribute to churning losses. Asfar as the differential wheel is concerned, it is supposed that itbehaves as a gear pump and the oil level in the differential istherefore below the inner ring of the tapered roller bearings.

Figure 4 presents some typical results obtained for differentrotational speeds of the input shaft �3000 and 5000 rpm� and dif-ferent gears. In this figure, the dotted lines represent the numericalresults whereas solid lines account for the experimental findings.The results show that the prediction of power losses at no load issatisfactory with a slightly larger discrepancy for the sixth gear at5000 rpm. Since, for this regime, oil churning is the main dissi-pation source �it represents nearly 50% of the total power lossaccording to the simulation�, its accurate prediction is crucial for asound estimate of the drag torque. In these particular conditions,

Fig. 3 ML6C Test rig

Table 4 Gear ratios of ML6C

Number of teeth

Gear Z1 Z2

First 12 41Second 23 41Third 33 37Fourth 44 35Fifth 51 33Sixth 58 31

Differential 17 67



the secondary shaft speed becomes important and one may sup-pose that oil foam is created, thus modifying the boundaries of thesubmerged parts and consequently increasing oil churning losses.In Fig. 5, the gearbox efficiency is plotted against the engagedgear for two different input torque amplitudes of 30 N m and150 N m, and for an input speed of 3000 rpm. A good agreementis observed between the measured and predicted efficiencies overthe whole range of gears. It can be noted that, for the low torque,the efficiency decreases significantly as the engaged gear movesfrom the first to the sixth speed. The cause is the prominent con-tribution of no-load power losses when speed increases as illus-trated in Fig. 4. On the contrary, the variations in efficiency arereduced for the largest torque because of a better balance betweenthe no-load and load-dependent sources of dissipation. It is foundthat, in the first gear, load-dependent losses are prevalent while thesituation is reversed in the sixth gear.

4.2 Tests With an Oil Sump Temperature of 40°C. Thecomparisons shown in Fig. 6 reveal that the simulations are not assatisfactory, and that the theoretical model underestimates thegearbox efficiency whatever the input torque. The maximum de-viation is obtained when the sixth gear is engaged with a torque of30 N m. In this case, the power losses due to oil shearing in thejournal bearings and the synchronizing cones dominate �61.5% ofthe total power lost� and consequently strongly depend on oilviscosity which, in turn, depends on oil temperature. Comparedwith the tests at 80°C, the thermal equilibrium of the gearbox isquite different and local temperature increases, whose influence

Fig. 4 ML6C drag torque versus engaged gear

Fig. 5 Efficiency at 3000 rpm and 80°C versus engaged gear

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cannot be neglected, certainly appear. At these particular places,the rise in temperature changes the viscosity of the lubricant andthe average viscosity deduced by using the sump temperature can-not be employed any longer for power loss predictions.

5 Thermal ModelIn order to simulate the temperature distribution in the tested

gearbox, the thermal network method has been used, which con-sists of dividing the geared unit into isothermal elements whichaccount for the housing, the oil, the bearings, and the pinions, etc.These elements are connected by thermal resistances which, byanalogy with an electric resistance, are defined by generalizingOhm’s law as

thermal resistance =thermal potential difference

heat flow�14�

with the value of the thermal resistance depending on the kindof heat transfer, i.e., conduction, free or forced convection andradiation.

5.1 Elements of the Network. The ML6C gearbox has beendivided into 44 elements as detailed in Table 5. The primary andsecondary shafts have been divided into two elements �nodes 13and 14 for the primary shaft, nodes 15 and 16 for the secondaryshaft� in order to simulate the axial temperature gradients possiblygenerated by the local temperature rises due to the oil shearingbetween the free pinions and the shafts. On the contrary, the gear-box housing has not been divided into several elements since theobjective was not to calculate the temperature distribution in thehousing, but determine a bulk temperature aimed at quantifyingthe heat exchanges with the air surrounding the gearbox.

The corresponding thermal network of the ML6C gearbox isdescribed in Fig. 7 with the element labels as defined in Table 5.

5.1.1 Thermal Model for the Housing. The ML6C housing isconsidered as an assembly of flat plates which exchange heat withtheir environment in two different ways; �a� natural convectionand radiation with the external atmosphere and �b� convection bythe lubricant on the inside walls. The following relations havebeen used to quantify the heat-transfer coefficient of convection:

• Heat transfer by air �17�

for vertical surfaces, Nu = 0.28�Gr Pr�0.30 �15�

for horizontal surfaces, Nu = 0.2�Gr Pr�0.32 �16�• Heat transfer by oil �18�

1/2 1/3 5

Fig. 6 Efficiency at 5000 rpm and 40°C versus engaged gear

Nu = 0.664 Re Pr for Re � 510 �17�

622 / Vol. 128, MAY 2006


Nu = Pr1/3�0.037Re0.8 − 850� for 5105 � Re � 107

�18�

In these equations, Nusselt, Reynolds, and Grashof numbers aredefined by using the plate length �or height for vertical ones� asthe characteristic dimension leading to an averaged value of Nualong the plate.

The resistances associated with radiation are derived from thelinearized form of the Stefan-Boltzman law assuming the gearboxsurface at temperature Tw is exposed to a large surrounding area atan absolute temperature Tamb, i.e.,

RTH =1

��S�Tw2 + Tamb

2 ��Tw + Tamb��19�

5.1.2 Thermal Model for the Gears. For a pinion and gear inmesh under load, the gear temperature consists in the addition ofthe bulk temperature and the flash temperature, which as demon-strated by Blok �19� is confined to a thin “thermal skin” on thesurfaces of the mating parts. The rotational period of the gearsbeing much smaller than the time required for any changes in bulktemperature, the temperature profiles are such that a pinion-gearpair can be modeled as two smooth cylindrical bodies connectedby a thermal resistance. On top of the classical resistance by con-duction, the influences of striction and centrifugal flingoff have tobe introduced. Striction comes from the dimensions of the ex-change surfaces, i.e., the Hertzian contact zones, which are verysmall in comparison with the characteristic dimensions of thegears, and lead to a constriction of the thermal current from thesurface to the gear center. Following Blok �19�, the associated

Table 5 Elements of the thermal network

Number Element reference

1 Air2 Gearbox housing3 Steel plate4 Differential housing5 Oil sump in gearbox6 Oil sump in differential

7,8 Bearings on the primary shaft9,10 Bearings on the secondary shaft11,12 Differential bearings13,14 Primary shaft15,16 Secondary shaft

17 Reverse shaft18 Differential shaft19 Driving pinion of first gear20 Driving pinion of reverse gear21 Driving pinion of second gear22 Driving pinion of third gear23 Driving pinion of fourth gear24 Driving pinion of fifth gear25 Driving pinion of sixth gear26 Driven pinion of reverse gear27 Driven pinion of first gear28 Driven pinion of second gear29 Driven pinion of third gear30 Driven pinion of fourth gear31 Driven pinion of fifth gear32 Driven pinion of sixth gear

33,34 Pinions on reverse shaft35 Differential drive gear36 Differential gear wheel37 Meshing of gear teeth in the gearbox38 Meshing of gear teeth in the differential39 Synchronizer of first gear40 Synchronizer of second gear41 Synchronizer of third gear42 Synchronizer of fourth gear43 Synchronizer of fifth gear44 Synchronizer of sixth gear

thermal resistance reads



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RTH =0.767a

�lhbk�V�20�

where V represents a mean value of the rolling velocity, lh is theHertzian contact width, and the given value of thermal resistanceis averaged over one mesh period.

Heat transfers between the oil and the gears must also be takeninto account which, after Blok �20�, lead to the additional resis-tance due to the heat removal from the tooth faces by centrifugalflingoff

RTH =2��a

1.14b2ZHtoothk��for � 0.68

RTH =2��a

�1.55 − 0.6 �b2ZHtoothk��for 0.68 � � 1.5 �21�

where is a dimensionless parameter defined as = �Dpa��2 /2�Htooth�1/4 and � represents the time needed for onegiven tooth to leave the oil sump and start to mesh.

5.1.3 Thermal Model for the Other Elements. All the otherelements �bearings, shafts, synchronizer sets� are represented ascylindrical bodies. Consider cylinders of inside radius Rin andoutside radius Rout, the resistances of conduction are given by theclassical formula

RTH =

Ln�Rout

Rin�

2�kL�22�

In the particular case of bearings, a possible radial gradient oftemperature is simulated between the inner and outer rings andtwo thermal resistances of conduction are employed: one betweenthe bearing and the shaft via the inner ring and the second one

Fig. 7 Thermal network of ML6C ge

between the bearing and the housing via the outer ring. Consider-



ing heat transfer by convection, the equation of Churchill andBernstein �21� for cylindrical bodies at rest �reverse shaft for ex-ample� is used to estimate the coefficient of convection, i.e.,

Nu = 0.3 +0.62Re1/2Pr1/3

�1 + �0.4/Pr�2/3�1/41 + � Re

282000�5/84/5

�23�

which is valid for the conditions 100�Re�107 and Re Pr�0.2For cylindrical parts in rotation, the formula given by Becker

�22� for water flows, and validated by Lebeck �23� for oil flows,has been employed, it reads

Nu = 0.133Re2/3Pr1/3 �24�

for 1000�Re�105

In the equations above, the dimensionless coefficients Re andNu have been calculated by choosing the diameter as the charac-teristic length and D /2� as the characteristic speed.

5.2 Numerical Solution. For each node i of the network, thefirst principle of Thermodynamics leads to

Qi = �j and j�i

Ti − Tj

RTH�i, j�+ mici

dTi

dt�25�

Equation �25� can be transformed into

Qi = Ti �j and j�i

1

RTH�i, j�− �

j and j�i

Tj

RTH�i, j�+ mici

dTi

dt�26�

finally rewritten as

Qi = �j

STH�i, j�Tj + micidTi

dt

ox when the sixth gear is engaged
arb
with

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if i = j STH�i,i� = �k

1

RTH�i,k�

if i � j STH�i, j� = −1

RTH�i, j��27�

and using the conventions

1

RTH�i,i�= 0

1

RTH�i, j�= 0 when the thermal resistance RTH�i, j� is not defined.

When considering all the elements of the network, Eq. �27�leads to a nonlinear differential system with temperature and time-dependent time derivatives of the temperature. This Cauchy prob-lem is solved by a numerical integration using a predictor-corrector scheme based on the Adams methods �24�. Adjustabletime steps are employed depending on the convergence rate since,depending on the operating conditions, STH�i , j� can be illconditioned.

6 Combined Temperature-Efficiency CalculationsFigure 8 shows the new set of efficiency curves obtained by

combining temperature and efficiency calculations along with theexperimental results in the case of the oil sump at 40°C. This

Table 6 Calculated temperatures at the nodes of the network„sixth speed, torque of 30 Nm, 5000 rpm, and 40°C…

NumberTemperature

�°C� NumberTemperature

�°C� NumberTemperature

�°C�

1 20 16 65 31 46.32 45.2 17 54.4 32 45.73 43 18 42.7 33 46.44 44 19 41.7 34 445 40 20 41.7 35 43.56 40 21 41 36 42.87 44.9 22 41.1 37 46.38 46.1 23 41.1 38 43.99 53.1 24 41.4 39 68.610 56.9 25 43.5 40 64.211 44.6 26 45.7 41 54.312 44.5 27 52.3 42 47.413 42.2 28 52 43 42.314 44.1 29 51.2 44 43.115 48.2 30 49.7

Fig. 8 Efficiency at 5000 rpm and 40°C versus engagedgear—combined temperature/efficiency calculations

624 / Vol. 128, MAY 2006


time, very good agreements are observed which prove the interestof this approach to power loss prediction at the lowest tempera-tures and demonstrate that, in these conditions, local temperaturerises cannot be neglected when evaluating lubricant viscosity. Asan illustration, the temperatures at all the nodes of the network forthe sixth gear with an input torque on the primary shaft of 30 Nmare given in Table 6. Clearly, the temperatures above are verydifferent from the average value of 40°C and strongly modify thelocal oil viscosity at certain points in the transmission. A compari-son with the previous isothermal simulation is given in Table 7which highlights the role of temperature related power losssources.

Similar calculations were performed in the condition of an oiltemperature of 80°C and a much lower influence of localtemperature/viscosity variations has been found. Consideringagain the results when the sixth gear is engaged with the inputtorque of 30 N m, the bulk temperature of the first gear synchro-nizer is found to be 88.5°C, i.e., very close to the oil sump tem-perature. Moreover, for such temperatures, viscosity is not verysensitive to temperature variations, explaining why local heat dis-tributions have a limited influence on power losses as shown inTable 8.

7 ConclusionAccurate power loss predictions in manual gearboxes are cru-

cial especially at the design stage when no tests on actual drivescan be conducted. A numerical simulation is proposed which re-lies on some existing models for the estimation of tooth frictionlosses and rolling bearing contributions. An original approach tooil churning is proposed which is based on dimensional analysisand leads to analytical formulas. Comparisons between calcula-tions and experimental findings show that these models are soundwhen the oil sump temperature is high �near 100°C�. At the low-est temperatures, 40°C in the proposed examples, it is demon-strated that isothermal approaches based on the oil sump tempera-ture underestimate gearbox efficiency because local temperature

Table 7 Power losses when the sixth gear is engaged„5000 rpm, 30 N m, and 40°C…

Total power lost

Without thermalcalculations

6330 W �100%��%�

With thermalcalculations

4082 W �100%��%�

Oil churning losses 22.5 34.8Journal bearings losses 33.3 25.1Rolling bearing losses 14.6 17.6Teeth friction losses 1.3 2.1Synchronizers losses 28.3 20.4

Table 8 Power losses when the sixth gear is engaged„5000 rpm, 30 N m, and 80°C…

Total power lost

Without thermalcalculations

3078 W �100%��%�

With thermalcalculations

2861 W �100%��%�

Oil churning losses 46.3 49.8Journal bearings losses 19.6 17Rolling bearing losses 13.6 13.7Teeth friction losses 4.5 4.8Synchronizers losses 16 14.7



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rises are not taken into account. An integrated model of the gear-box combining bulk temperature predictions by a thermal networkmethod and power loss calculations is presented. The results are invery good agreement with the experimental evidence thus validat-ing the methodology which can be readily applied to industrialapplications.

Nomenclaturea � thermal diffusivity �m2/s�b � tooth face width �m�

dm � bearing mean diameter �m�Dp � pitch diameter �m�

f � friction coefficientFnu � normal tooth load per unit length �daN/m�Fr � Froude numberg � acceleration of gravity �m/s2�

ga � length of recess �m�gf � length of approach �m�Gr � Grashof number

h � submerged depth of a pinion �m�Htooth � tooth depth �m�

j � diametral clearance �m�k � thermal conductivity �W/m °C�L � axial length �m�m � module �m�

mc � thermal inertia �J/K�N � rotational speed �rpm�

Nu � Nusselt numberpb � base pitch �m�

Pgear � power loss by tooth friction �W�Pin � input power �W�Pr � Prandtl numberQ � heat generated �W�R � radius �m�

Re � Reynolds numberRTH � thermal resistance �K/W�

S � exchange surface area �m2�T � temperature �K�T � time �s�U � sum of rolling velocities �m/s�

Vg � sliding velocity �m/s�V0 � oil volume �m3�Z � number of teeth

�b � base helix angle� � emissivity of the housing material � dynamic viscosity �Pa s�� kinematic viscosity �m2/s�� fluid density �kg/m3�� constant of Stefan-



Boltzmann=5.67 10−8 W/m2 K4

� � rotational speed �rad/s�

References�1� Anderson, N. E., and Loewenthal, S. H., 1981, “Effect of Geometry and Op-

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�4� Diab, Y., Ville, F., Velex, P., and Changenet, C., 2004, “Windage Losses inHigh-Speed Gears. Preliminary Experimental and Theoretical Results,” ASMEJ. Mech. Des., 126�5�, pp. 903–908.

�5� Terekhov, A. S., 1975, “Hydraulic Losses in Gearboxes With Oil Immersion,”Vestnik Mashinostroeniya, 55�5�, pp. 13–17.

�6� Winter, H., and Funck, G., 1987, “Wärmehaushalt bei Getrieben—erläutert amBeispiel eines Kegel-Stirnradgetriebes,” Tribol. Schmierungstech., 34�4�, pp.234–241.

�7� Bones, R. J., 1989, “Churning Losses of Discs and Gears Running PartiallySubmerged in Oil,” Proc. ASME Int. Power Trans. Gearing Conf., Vol. 1,Design Engineering Division, ASME, Chicago, pp. 355–359.

�8� Harris, T. A., 2001, Rolling Bearing Analysis, 4th ed., Wiley Inter-sciences,New-York, p. 1086.

�9� Candel, S., 1995, Mécanique des Fluides-Cours, 2nd ed., Dunod Ed., Paris, p.286.

�10� Lauster, E., and Boos, M., 1983, “Zum Wärmehaushalt Mechanischer Schalt-getriebe für Nutzfahrzeuge,” VDI-Ber., 488, pp. 45–55.

�11� Roulet, B., and Briec, A., 1995, “Modelling of Power Losses in MechanicalGearbox,” Proc. 5ème Congrès EAEC, Strasbourg, pp. 1–12.

�12� Iritani, M., Aoki, H., Suzuki, K., and Morita, Y., 1999, “Prediction Techniquefor the Lubricating Oil Temperature in Manual Transaxle,” SAE Transactions,108�4�, pp. 337–345.

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�14� Martin, K. F., 1978, “A Review of Friction Predictions in Gear Teeth,” Wear,49, pp. 201–238.

�15� Diab, Y., Ville, F., and Velex, P., 2004, “Prediction of Power Losses due toTooth Friction in Gears,” Tribol. Trans., pp. �in press�.

�16� Benedict, G. H., and Kelley, B. W., 1961, “Instantaneous Coefficients of GearTooth Friction,” Tribol. Trans., 4, pp. 59–70.

�17� Winter, H., Michaelis, K., and Funck, G., 1986, “Wärmeabführung bei Getrie-ben unter Quasistationären Betriebsbedingungen. Teil I: Systematische Unter-suchungen zur Wärmeabführung bei Natürlicher und Erzwungener Zuft-strömung mit Hilfe eines Modellprüfstandes,” Antriebstechnik, 25�12�, pp.36–42.

�18� Holman, J. P., 1989, Heat Transfer, 7th ed., Mac Graw-Hill, New York, p. 260.�19� Blok, H., 1937, “Theoretical Study of Temperature Rise at Surface of Actual

Contact Under Oiliness Lubricating Conditions,” Proc. Inst. Mech. Eng., Gen-eral Discussion on Lubrication and Lubricants, 2, London, pp. 222–235.

�20� Blok, H., 1970, “Transmission de Chaleur par Projection Centrifuge D’huile,”Société d’Etudes de l’Industrie de l’Engrenage, 59, pp. 14–23.

�21� Churchill, S. W., and Bernstein, H., 1977, “A Correlating Equation for forcedConvection from Gases and Liquids to a circular Cylinder in Cross Flow,” J.Heat Transfer, 99, pp. 300–306.

�22� Becker, K. M., 1963, “Measurement of Convective Heat Transfer From aHorizontal Cylinder Rotating in a Tank of Water,” Int. J. Heat Mass Transfer,6, pp. 1053–1062.

�23� Lebeck, A. O., 1991, Principles and Design of Mechanical Face Seals, Wiley-Interscience, New York, p. 800.

�24� The MathWorks Inc., 2000, Using MATLAB-Version 6, pp. 15–6.

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