Model for Prediction of Churning Losses

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Journal ofMechanical

Design Technical Brief

Model for the Prediction ofhurning Losses in Gearedransmissions—Preliminary Results

. Changenetechanical Engineering Department,

CAM,yon, France

. Velex1

aMCoS,MR CNRS 5514,

NSA de Lyon, France-mail: [email protected]

series of formulas are presented which enable accurate predic-ions of churning losses for one pinion characteristic of automo-ive transmission geometry. The results are based on dimensionalnalysis and have been experimentally validated over a wideange of speeds, gear geometries, lubricants, and immersionepths. The case of a pinion-gear pair in mesh has been consid-red, and it has been proved that, depending on the sense ofotation, the superposition of the individual losses of the pinionnd of the gear leads to erroneous figures. A new formula devotedo a pinion and gear rotating anticlockwise has been derived andalidated by comparison with experimental evidence.DOI: 10.1115/1.2403727�

IntroductionIn automotive applications, continuously increasing power den-

ities require the development of accurate methods of estimatinghe efficiency of candidate transmissions in order to prevent over-eating and subsequent failures. Gearbox efficiency and relatedhermal consequences have therefore become significant issueshich have to be predicted and taken into account at the design

tage. It is accepted that, for low to medium speed gears, powerosses mainly stem from tooth friction and lubricant churninghereas, for high speed applications, windage can become promi-ent. Focusing on churning, the drag torque due to the rotation ofiscs submerged in a fluid has been analyzed by Daily and Nece1�, Mann and Marston �2�, Soo and Princeton �3�, etc. However,n the case of gears, there are fewer models and, because of ex-erimental difficulties, measurements of thermal performance and

1Corresponding author.Contributed by the Power Transmission and Gearing Committee of ASME for

ublication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received January 27,006; final manuscript received August 21, 2006. Review conducted by Professor
avid Donner.
28 / Vol. 129, JANUARY 2007 Copyright ©

ded 15 Mar 2011 to 138.250.82.157. Redistribution subject to ASM

power losses have been limited. The first in situ temperature mea-surements date back to the classic works by Blok �4�, Niemannand Lechner �5�, while the specific studies on churning lossescomprise those of Terekhov �6�, Lauster and Boos �7� and, morerecently, Boness �8�. Terekhov �6� conducted numerous experi-ments with high viscosity lubricants �from 200 to 2000 Cst�, lowrotational speeds, and tested gears with modules ranging from 2 to8 mm. The resulting churning torque Cch is expressed in terms ofa dimensionless torque Cm as

Cch = ��2bRp4Cm �1�

where � is the lubricant density, � is the rotational speed, b is thegear face width, and Rp is the gear pitch radius.

The analytical expression of Cm is deduced from dimensionalanalysis and, depending on the flow regime, it takes the followingforms.

For laminar flows �10�Re�2250�if Re−0.6Fr−0.25�8.7�10−3

Cm = 4.57 Re−0.6 Fr−0.25� h

Rp�1.5� b

Rp�−0.4�Vp

V0�−0.5

�2.1�

otherwise

Cm = 2.63 Re−0.6 Fr−0.25� h

Rp�1.5� b

Rp�−0.17�Vp

V0�−0.73

�2.2�

For turbulent flows �2250�Re�36,000�

Cm = 0.373 Re−0.3 Fr−0.25� h

Rp�1.5� b

Rp�−0.124�Vp

V0�−0.576

�2.3�

In the above formulas, h is the immersion depth and Vp /V0represents the ratio of the submerged volume to that of the lubri-cant.

Using a similar approach but in the specific case of truck trans-missions, Lauster and Boos �7� proposed a unique expression forCm as

Cm = 2.95 Re−0.15 Fr−0.7� h

Rp�1.5� b

Rp�−0.4�Vp

V0�−0.5

�3�

Terekhov’s and Lauster’s models rely on numerous tests but itcan be noticed that the expressions �2� and �3� are independent ofgear tooth geometry which seems unrealistic. This finding prob-ably comes from the particular experimental conditions, i.e., highviscosity lubricants combined with low speeds, which limit thequantity of lubricant between the teeth being expelled by centrifu-gal effects and makes the gear behavior close to that of a disk.

Boness �8� investigated the drag torque generated by discs andgears rotating in water, or in oil, and proposed that churningtorque be expressed as

Cch =�

�2SmRp3Cm �4.1�

2

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In contrast to �1�, Cch is given in terms of the submerged sur-ace area Sm but here again, the dimensionless torque depends onhe flow regime and reads as follows.

For laminar flows �Re�2000�

Cm =20

Re�4.2�

For intermediate flow regimes �2000�Re�100,000�

Cm = 8.6 � 10−4 Re1/3 �4.3�

For turbulent flows �100,000�Re�

Cm =5 � 108

Re2 �4.4�

It is observed that, in the transition regime, the drag torquencreases with an increasing Reynolds number, suggesting thatow viscosity lubricants generate higher losses in contradictionith experimental evidence.From the observations/remarks above, it appears that a reliable

eneral churning loss model has still to be developed. In thisaper, some new formulas adapted to modern automotive trans-issions are proposed based on the results of a versatile test rig

nd on dimensional analysis. A variety of lubricants �Table 1� andears �Table 2� with several immersion depths have been consid-red for rotational speeds between 1000 and 7000 rpm. Only spurear teeth have been tested. In addition to the results for a singleear, pairs of pinions and gears have been tested and the influencef the interactions between the two bodies in relation to the sensef rotation has been investigated.

Table 1 Lubricant properties

Kinematic viscosityat 40°C �cSt�

Kinematic viscosityat 100°C �cSt�

Density at 15°C�kg/m3�

il No. 1 48 8.3 cSt 873il No. 2 145.5 15.7 865il No. 3 320 24 897.8il No. 4 6,16 31,6 874

Table 2

Gear 1 Gear 2 G

Module �mm� 1,5 1,5Face width �mm� 14 14Pitch diameter �mm� 96 153Number of teeth 64 102

Fig. 1 Oil chur

ournal of Mechanical Design


2 Churning Loss—Formulas for One PinionThe test rig developed for churning loss measurements is de-

scribed in Fig. 1. It is composed of an electric motor which oper-ates a pinion shaft via a belt multiplying the rotational speed by afactor of 2.5. One of the housing faces is in Plexiglas so that theoil flow around the pinions may be observed �Fig. 2�. The electricpower needed to run the gears is measured with lubricant in thecasing and then again in the absence of any lubricant; thus, churn-ing power losses are deduced by subtraction. Thermocouples wereused for determining the ambient and the lubricant temperaturesand particular care was taken to verify that thermal equilibriumwas reached every time power loss was sensed. The maximumuncertainty on the churning torque was estimated to be in therange 0.06–0.08 N m. For high speed conditions, the windagelosses were measured in the absence of a lubricant �9� and weresubtracted from the total power losses in order to isolate the con-tributions of churning.

Figures 3–5 show a series of comparisons between the mea-sured churning torques and the predictions using Terekhov’s �2.1�,Lauster’s �3�, and Boness’ �2.3� equations. Figure 3 focuses on theinfluence of speed whereas Figs. 4 and 5 illustrate the variations

ar data

3 Gear 4 Gear 5 Gear 6 Gear 7

3 5 5 524 24 24 24

159 100 125 15053 20 25 30

Fig. 2 Oil flow around the pinions

Ge

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ning test rig

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ssociated with the immersion depth and the gear tooth geometry.t can be concluded that none of the models gives satisfactoryorrelations over the parameter range of variation thus justifyinghe tentative development of more reliable formulas.

The parameters which are influential on churning loss haveeen categorized and comprise:

�a� the gear geometry described by its module m, its pitchdiameter Dp, and face width b,

�b� the characteristics of the lubricant, i.e., viscosity �, den-sity �, immersion depth h, and the volume of lubricantV0, and

ig. 3 Churning losses „gear 4…—Comparisons between theodels of Refs. †6–8‡ and the experimental measurements

ig. 4 Churning losses „gear 3…—Comparisons between theodels of Refs. †6–8‡ and the experimental measurements

ig. 5 Churning losses „gear 2…—Comparisons between the
odels of Refs. †6–8‡ and the experimental measurements
30 / Vol. 129, JANUARY 2007


�c� the dynamic parameters characterized by the gravity ac-celeration g and the rotational speed �.

It is therefore deduced that the churning torque can be writtenas

Cch = f�m,Dp,b,�,�,h,V0,g,�� 5�Boness’ normalization of the churning torque has been pre-

ferred because it seems logical to consider �i� a pressure of refer-ence of the form �1/2��2Rp

2 and �ii� the immersed surface of thepinion �Sm�, thus leading to the expression

Cch = 12��2Rp

3SmCm �6.1�

Sm is the surface area of contact between the gear and the lubri-cant �Fig. 6� which is decomposed into the lateral surface areaSl=Rp

2�2�−sin 2�� and the surface area of the teeth approximatedas St=Dpb�+2Z� Htoothb /� cos .

Dimensional analysis has been used to determine an expressionof the dimensionless drag torque Cm. The ten parameters in Eq.�5� depend on the three fundamental units �mass, length, and time�and, from the theorem of Vaschy–Buckingham �10�, the normal-ized churning torque can be expressed in terms of seven groups ofdimensionless quantities as

Cm = 1� m

Dp�2� b

Dp�3� h

Dp�4� V0

Dp3�5

Re6 Fr7 �6.2�

where 1 , . . . ,7 are constant coefficients which are adjusted fromexperimental results, Fr is the Froude number, and Re is the Rey-nolds number.

Boness �8� has defined the Reynolds number using lm, the chordat the immersion depth �Fig. 6�, but by so doing, the influence ofimmersion and that of the ratio of the inertia to the viscous forcesare mixed. In order to overcome this limitation, the length ofreference in the definition of Re is taken as the gear pitch radiusRp and the Reynolds number employed reads

Re =�Rp

2

��7�

In the same way, the gear pitch radius is used as the length ofreference in the definition of the Froude number as

Fr =�2Rp

g�8�

The parameter identification has been conducted based on nu-merous tests following the procedure described in Ref. �11�. It hasbeen found that, depending on rotational speed, no single formulawas possible. At low-medium speeds, gear geometry is influentialvia the submerged surface area only while tooth number and facewidth play a negligible role as such. At high speeds, churningpower losses are found to be largely independent of oil viscosity

Fig. 6 Pinion partly immerged in the oil sump—Geometricaldata

and the inertia forces become much more significant than the vis-

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ous ones. The Reynolds number is therefore discarded in theormulation but, as opposed to low-medium speeds, the ratiob /Dp� appears to be influential �Fig. 7�.

The identified coefficients 1 , . . . ,7 are listed in Table 3 with ahreshold between the medium and high-speed equations charac-erized by a critical Reynolds number �Rec� in the range of 6000–000.

Due to the influence of tooth face width at high speeds, theritical Reynolds number is defined as follows:

Rec =�Rpb

��9�

In the transition zone �6000�Rec�9000�, a linear interpola-ion between the two formulae is employed. The validity of theroposed formulation is illustrated in Fig. 8 in which the dottedines represent the numerical results whereas the solid lines ac-ount for the experimental findings. These curves represent aample of the results obtained from more than 100 tests and simu-ations with various lubricants �Table 1�, gears �Table 2�, speedsin the 1000–7000 rpm range�, and relative immersions �between.1 and 0.6�. A satisfactory agreement is observed in all cases;urther comparisons are given in Ref. �12�.

Fig. 7 Influence of tooth face width

Table 3 Coefficients in Eq. „4.1…

1 2 3 4 5 6 7

ow-medium speed 1.366 0 0 0.45 0.1 −0.21 −0.6igh speed 3.644 0 0.85 0.1 −0.35 0 −0.88

ig. 8 Comparisons between the experimental results and the
esults from the dimensional analysis for two gear


3 Churning Loss—Formulas for a Pinion-Gear PairIn actual transmissions, the most common situation corresponds

to a pinion and a gear in mesh and the current practice consists inadding their individual contributions to derive the total churningloss. In order to assess this calculation method, a series of experi-ments was conducted in which one gear was driven by the pinion�cf. Sec. 2� in the test rig. Two pinion-gear pairs were examined�one with a module of 1.5 mm and the other with that of 3 mm�with �i� two relative immersion depths of 0.6 and 0.45, respec-tively �immersion measured on gear�, and �ii� two senses of rota-tion �clockwise and counter-clockwise as described in Fig. 9�.Power losses were determined following the same methodology asfor the experiments with a single pinion. Depending on the senseof rotation, very different trends have been found showing that theassumed linearity �addition of the two individual contributions�works well for clockwise rotations but leads to erroneous resultsfor bodies rotating counter-clockwise �Table 4�. The latter condi-tion systematically gives rise to higher power losses than thoseestimated by the simple addition of the individual pinion and gearlosses. From a physical point of view, this difference is probablydue to the trapping of lubricant by the meshing teeth �13� and bya swell effect which dissipates energy and increases the immer-sion depth on the pinion as illustrated in Fig. 10. Several experi-ments were conducted in order to quantify the additional loss��P� associated with counter-clockwise rotations. These com-prised the testing of �i� four pinion-gear pairs formed by combin-ing the gears of same module in Table 2, �ii� the four lubricants inTable 1, and �iii� relative immersion depths h /Rp between 0.4 and0.7 �the gear being taken as the reference for the definition ofimmersion parameters�. The rotational speeds on the pinion werein the 500–1500 rpm range. When comparing power losses for agiven pinion-gear set and lubricant, numerical findings have beencorrected in order to account for the viscosity variations associ-ated with the different temperatures of equilibrium in the sump.According to the low-medium speed formula proposed in �6.2�, itappears that viscosity is embodied in the Reynolds number onlyand that churning losses are therefore proportional to � 6, with 6=−0.21. From the latter, a viscosity correction factor for cal-culating power loss variations is defined by

Fig. 9 Definition of the senses of rotation

Table 4 Results for a pinion-gear pair of module 1.5 mm

Rotational speed �rpm� 500 750 1000 1250 1500Power loss measured foranticlockwise rotation �W�

11 25 41 56 76

Power loss calculated byaddition of the twoindividual contributions �W�

5.3 15 23.5 31.5 39.5

Fig. 10 Schematic representation of the swell effect

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�P = Pac − Pcc��ac

�cc�0.21

�10�

here �ac represents the kinematic viscosity at oil sump tempera-ure for counter-clockwise rotation and �cc is the viscosity forlockwise rotation.

As in �5�, the variation in churning power loss �P is expressedn terms of a dimensionless variation of churning torque �Cm andP is sought under the form

�P = 12��3SmRp

3�Cm �11�

n which all the geometrical data are those of the gear.From the experimental results obtained, the following conclu-

ions have been drawn

�a� As shown for isothermal conditions in Fig. 11, the vis-cosity of the lubricant has a weak influence on �P andRe is discarded in the formulation.

�b� �P is sensitive to rotational speed �Fig. 12� and, by anal-ogy with the structure of Eq. �6.2�, a proportionality toFr−0.68 is considered.

�c� For identical pinions and gears, no swell effect can begenerated due to the symmetry. In these conditions, �Pwas found to be close to zero, thus demonstrating that theair-lubricant trapping by the teeth is negligible and con-sequently proving that the swell effect is prominent.

The latter observation is taken into account by considering theifference in relative immersion depths between the gear and theinion defined as

� h

Rp�

GEAR− � h

Rp�

PINION= �u − 1��1 − � h

Rp�

GEAR� �12�

here u is the speed ratio�Cm is supposed to be proportional to �12� but, for very high

peed ratios, it can be inferred that, in practice, the swell effect

Fig. 11 Oil viscosity influence on �P

Fig. 12 Rotational speed influence on �P

32 / Vol. 129, JANUARY 2007


disappears, as is schematically illustrated in Fig. 13. In order toaccount for these extreme situations, a correction factor of theform �1/u�n was tested and has led to good agreement with theexperimental evidence for n=8. Finally, the following expressionof �Cm has been retained

�Cm = 17.7Fr0.68�u − 1�u8 �1 − � h

Rp�

GEAR� �13�

Numerous power loss measurements were conducted and theexperimental �P and the corresponding values derived from �11�and �13� are shown in Fig. 14. A good correspondence is observedover a large range of parameters which proves that the proposedformulation is sound.

4 ConclusionA new model has been presented which enables reliable predic-

tions of churning losses in automotive transmissions. It has beenvalidated by more than 100 experimental measurements from aspecific test rig. Particular attention was paid to the influence oftemperature on lubricant viscosity and that of windage on highspeed measurements in order to avoid bias. One specific charac-teristic of the proposed model is that it accounts for a pinion anda gear in mesh and for their senses of rotation. It has been provedthat, for counter-clockwise rotations, the total loss is not equal tothe sum of the individual losses associated with the pinion and thegear when considered apart. The comparisons with the experimen-tal evidence over a wide range of speeds, gear geometries, andlubricants are good and, moreover, the corresponding churningtorque equations have been indirectly validated by simulating andmeasuring global power losses in gearboxes under various load/speed conditions �11�. In the context of minimizing energy losses,one possible way to improve churning losses consists in changingthe shape of the casing. So far, the casing contribution is intro-

Fig. 13 Schematic representation of the immersion for apinion-gear pair with a large speed ratio

Fig. 14 Additional churning power loss �P

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uced via a global factor related to the lubricant volume which,or instance, cannot account for flanges and deflectors. The devel-pment of a new test rig with movable walls and enclosures isurrently under way in order to systematically analyze the influ-nce of radial and axial clearances and derive a more sophisticatedhurning loss model capable of simulating these different runningonditions.

omenclatureb � tooth face width �m�

Cch � churning torque �N m�Cm � dimensionless drag torqueDp � pitch diameter �m�Fr � Froude numberg � acceleration of gravity �m/s2�h � immersion depth of a pinion �m�

Htooth � tooth height �m�m � module �m�

Pac � power loss for anticlockwise rotation �W�Pcc � power loss for clockwise rotation �W�Rp � pitch radius �m�Re � Reynolds numberSm � immersed surface area of the pinion �m2�

u � speed ratioV0 � oil volume �m3�Z � number of teeth � pressure angle

�Cm � dimensionless variation of churning torque�P � additional churning power loss �W�

� � dynamic viscosity �Pa s�� kinematic viscosity �m2/s�



� � fluid density �kg/m3�� rotational speed �rad/s�

References�1� Daily, J. W., and Nece, R. E., 1960. “Chamber Dimension Effects on Induced

Flow and Frictional Resistance of Enclosed Rotating Disk,” ASME J. BasicEng., 82, pp. 217–232.

�2� Mann, R. W., and Marston, C. H., 1961, “Friction Drag on Bladed Disks inHousings as a Function of Reynolds Number, Axial and Radial Clearance andBlade Aspect Ratio and Solidity,” ASME J. Basic Eng., 83�4�, pp. 719–723.

�3� Soo, S. L., and Princeton, N. J., 1958, “Laminar Flow Over an EnclosedRotating Disk,” Trans. ASME, 80, pp. 287–296.

�4� Blok, H., 1937, “Les Températures de Surface Dans les Conditions de Grais-sage Sous Extrême Pression,” Proc. 2nd Congrès mondial du Pétrole, Paris,pp. 471–486.

�5� Niemann, G., and Lechner, G., 1965, “The Measurement of Surface Tempera-ture on Gear Teeth,” ASME J. Basic Eng., 11, pp. 641–651.

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

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

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

�9� 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.

�10� Candel, S., 1995, Mécanique des Fluides—Cours, 2nd ed., Dunod, Paris, pp.151–183.

�11� Changenet, C., Oviedo-Marlot, X., and Velex, P., 2006, “Power Loss Predic-tions in Geared Transmissions Using Thermal Networks—Applications to aSix-Speed Manual Gearbox,” ASME J. Mech. Des., 128�3�, pp. 618–625.

�12� Changenet, C., 2006, “Modélisation du Comportement Thermique des Trans-missions par Engrenages,” PhD. thesis, Institut National des Sciences Appli-quées de Lyon, pp. 70–75.

�13� Diab, Y., Ville, F., Houjoh, H., Sainsot, P., and Velex, P., 2005, “Experimentaland Numerical Investigations on the Air Pumping Phenomenon in High SpeedSpur and Helical Gears,” Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci.,219�8�, pp. 785–800.

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Model for Prediction of Churning Losses