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Experimental investigations and analysis on churninglosses of splash lubricated spiral bevel gearsS. Laruelle, C. Fossier, C. Changenet, F. Ville, S. Koechlin

To cite this version:S. Laruelle, C. Fossier, C. Changenet, F. Ville, S. Koechlin. Experimental investigations and analysison churning losses of splash lubricated spiral bevel gears. Mechanics & Industry, EDP Sciences, 2017,18 (4), pp.412. �10.1051/meca/2017007�. �hal-02924625�

Mechanics & Industry 18, 412 (2017)© AFM, EDP Sciences 2017DOI: 10.1051/meca/2017007

Mechanics&IndustryAvailable online at:

www.mechanics-industry.org

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Experimental investigations and analysis on churning lossesof splash lubricated spiral bevel gearsS. Laruelle1,2,3, C. Fossier1,2, C. Changenet2,*, F. Ville1, and S. Koechlin3

1 Univ. Lyon, INSA-Lyon, LaMCoS, UMR 5259, Villeurbanne, France2 Univ. Lyon, ECAM Lyon, INSA-Lyon, LabECAM, Lyon, France3 Leroy Somer, Emerson Corp., Angoulême, France

* e-mail: c

Received: 19 October 2016 / Accepted: 24 January 2017

Abstract. Churning losses are a complex phenomenon which generates significant power losses whenconsidering splash lubrication of gear units. However, only few works deal with bevel gears dipped lubricationlosses. The objective of this study is to provide a wide variety of experimental tests on churning losses, especiallygetting interested in geometry of spiral bevel gears influence. A specific test rig was used in order to study a singlespiral bevel gear partially immersed in an oil bath. Experiments have been conducted for several operatingconditions in terms of speeds, lubricants, temperatures and gear geometries to study their impact on splashlubrication power losses. These experimental results are compared with the predictions from various literaturesources. As the results did not agree well with the predictions for all operating conditions, an extended equationderived from previous works is introduced to estimate churning losses of bevel gears.

Keywords: spiral bevel gears / churning / lubrication / power losses

1 Introduction

In a development stage, it is important to predict thebehaviour of industrial gear units. System efficiency isestimated through the evaluation of power losses. Industrialgear units are often dipped lubricated in the case of low tomedium speed geared transmission. In these cases, churninglosses are an important part of no load losses, as shown instudies on both cylindrical [1] or hypoid gears [2]. Anaccurate evaluation of these dissipation sources is needed.Theoretical prediction of churning losses is complex to solvesince the fluid mechanics problem is related to a free surfaceand a two-phase flow. Extensive experimental measure-ments of churning losses for different operating conditionscan lead to empirical or semi-empirical formulas. Analyticalformulas allow users to point out the influencing parametersleading to design guidelines.

The drag torque due to the rotation of discs submergedin a fluid has been theoretically analysed by Soo andPrinceton [3], Daily and Nece [4], or Mann andMarston [5].In the case of gears, different experimental studies havebeen conducted with the aim of developing empiricalrelationships to quantify churning losses. As an example,some tests have been conducted with high viscosity oils andlow rotational speeds by Terekhov [6] for spur cylindrical

hristophe.changenet@ecam.fr

gears or Kolekar et al. [2] for hypoid gears. Furtherextension of Terekhov work was done by Lauster and Boos[7], who applied this approach on a real truck transmission.Boness [8] conducted experiments on discs and gears in lowviscosity fluids and developed an associated relationship.Based on previous works performed by Walter [9] andMauz [10] on no-load gear losses for splash lubrication,Höhn et al. [11,12] also proposed a model of spur gearchurning based on a single flow regime. From this widevariety of formulas, Luke and Olver [1] have shown thatdeviations on power loss calculations might be significant.

Recently churning power loss expressions are extendedto a wider domain. To this end, fluid mechanics models aredeveloped to estimate churning behaviour, as presented bySeetharaman and Kahraman [13], who worked on cylindri-cal gear pairs subject to churning and windage. Arisawaet al. [14] also referred to fluid mechanics to simulatechurning in aero engine transmissions. Further works arealso done concerning analytical formulations to quantifychurning power losses, for example, considering multipleflow regimes [15]. The inertia run down method is used toobtain experimental results. Another methodology basedon experimental measurements was done by Changenetet al. [16,17], who used dimensional analysis to propose amodel for cylindrical gears. Then Marques et al. [18]worked on extending previous formulas to a wider range ofapplication showing the shape influence of the oil sump. Itcan be also noticed that several studies [19–21] used

Gearbox

Electric motor

Frequency converter

Data acquisition

Pinion running partly immersed in oil

Fig. 1. Test rig. Fig. 2. Description scheme of the test rig.

Table 1. Gears dimensions.

Gear 1 Gear 2 Gear 3 Gear 4 Cone 1 Cone 2

External diameter [mm] 157 130 188 154 157 160Number of teeth 41 37 41 37 0 0Face angle [°] 72.4 58.1 72.4 58.1 72.4 58.1Module [mm] 2.6 2.4 3.1 2.9 – –

Face width [mm] 27 24.5 32 27.5 27 24.5Width [mm] 22 27 30 30 22 27Pressure angle [°] 20 20 20 20 – –

Mean spiral angle [°] 35 35 35 35 – –

2 S. Laruelle et al.: Mechanics & Industry 18, 412 (2017)

numerical simulations (computational fluid dynamicscode) to predict load independent power losses and morespecifically the churning ones.

However, even if extensive studies are available oncylindrical gears, it appears from this state of art that verylittle experimental data concerning churning losses gener-ated by bevel gears are available in literature. Moreover,different relationships are proposed to estimate this sourceof dissipation for hypoid or spiral bevel gears: technicalreports ISO/TR 14179-1 [22] and ISO/TR 14179-2 [23], orJeon works [24] (details are given in Appendix A).

To extend previous studies on spiral bevel gearchurning losses, a specific test rig has been used which isdescribed in Section 1. Extensive experiments wereconducted and the obtained results are presented inSection 2. The influence of gear geometry on churninglosses is highlighted. Several experimental results arecompared with the predictions from the above-mentionedliterature sources. As the comparisons are not satisfactoryfor all operating conditions, an extended equation derivedfrom Jeon’s approach is introduced to estimate churninglosses of bevel gears.

2 Test rig

A precise description of the test rig shown in Figure 1 isavailable in [16]; only the main features are exposed in thispaper.

The test rig allows measuring the lubricant tempera-ture and the churning torque of an isolated gear or of acouple of gears as detailed in Figure 2. The gear is driven inrotation by an electric motor. The resisting torque isdirectly measured by a strain-gauged, temperature com-pensated, contactless sensor (FGP-CD1140) of accuracy±0.002Nm, a sensitivity shift of 0.009% °C�1, and a zeroshift of 0.0004Nm °C�1 in the 5–45 °C temperature range.The pinion is supported by two ball bearings and theircontribution has to be isolated from the total resistingtorque. To do that, the bearing drag torque has beenexperimentally determined as a function of speed byremoving the gear from the test rig. For example, it hasbeen found that a pair of bearings generates a drag torqueof 0.045Nm at 7000 rpm, which is a significant valuecompared to the accuracy of the torque sensor. A pulse

#1 and #3 #2 and #4

b

btooth

Ro

btooth

Ro

b

Fig. 3. Typical gear geometries.

0

50

100

150

200

250

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

Chu

rnin

g lo

sses

[W]

Ratio h/R [ ]

Gear 1Gear 2Gear 3Gear 4

Fig. 4. Influence of immersion on churning losses (oil C,2000 rpm, 40 °C).

Table 2. Oil data.

Oil name Oil A Oil B Oil C Oil D

Viscosity @ 40 °C 220 cSt 35 cSt 45.1 cSt 120 cStViscosity @ 100 °C 19 cSt 7.5 cSt 7.7 cSt 16 cStDensity @ 15 °C 895 kgm�3 870 kgm�3 885 kgm�3 860 kgm�3

Oil type Mineral Mineral Mineral Synthetic

S. Laruelle et al.: Mechanics & Industry 18, 412 (2017) 3

counter in the torque sensor ensures the speed measure-ment. The oil sump is a parallelepiped with a Plexiglas faceto observe the flow around gears. The oil level can bemodified. The oil bath temperature is controlled andmeasured by thermocouples. Several heating covers areinstalled on the external faces of the housing. Themaximum deviation between two repetitions of the sametest is equal to 5%.

Four bevel gear geometries are available for these tests.Geometrical dimensions are given in Table 1. Two differenttypes of macro-geometries were tested (Fig. 3). One shapeconsists in a high face angle and a narrow width, and theother one has a smaller face angle and a larger width. Inorder to study the influence of tooth geometry on churninglosses, it can be noted that two smooth cones have also beenmanufactured (Tab. 1). As far as lubricants are concerned,four different oils were used. Their characteristics areresumed in Table 2.

As shown by Leprince et al. [25], oil aeration, which isdefined as the volumetric fraction of air in the lubricant, isnegligible when it is less than approximately 10%.With thesame methodology as the one used in the above-mentionedwork on cylindrical gears, aeration measurements haveconfirmed this observation for this specific study on spiralbevel gears: smaller than 7% on the tested domain.

3 Results

Experiments were performed for different operatingconditions (oil properties � viscosity and density �,rotational speed, gear immersion level) and different spiralbevel gear geometries (diameter, width, face width, faceangle, tooth number, module).

Measurements for different spiral bevel gears as afunction of their relative immersion depth are shown inFigure 4. It clearly appears that this parameter is a firstorder influent one on churning losses. Consideringrotational speed, it can be noticed that the losses areproportional toN1.7 (see Fig. 5, a similar relative immersiondepth of 0.5 was used for the tested gears). For higherrotational speeds, a drag torque decrease is observed, aspresented in Figure 6 and noticed by Jeon [24]. Twotypes of regime were observed between 1000 and 4000 rpm:(i) for the first regime, the oil is projected and the torqueincreases with speed; (ii) for the second one, whichcorresponds to higher rotational speeds, the torquedecreases. The second regime can be interpreted as follows:centrifugal effects generated by the rotating gears becomesignificant and induced a reduction in immersion depth,which in turn leads to a decrease in churning drag torque.

020406080

100120140160180

0 500 1,000 1,500 2,000 2,500

Chur

ning

loss

es [W

]

Rota�onal speed[RPM]

Gear 1

Gear 2

Gear 3

Gear 4

Fig. 5. Effect of rotational speed on churning losses –measurements (oil C, 40 °C, h/R=0.5).

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 1000 2000 3000 4000

Chu

rnin

g to

rque

[Nm

]

Rotational speed [rpm]

Gear 1Cone 1

Fig. 6. Different regimes (oil B, h/R=0.5, 50 °C).

0

20

40

60

80

100

0.3 0.4 0.5 0.6 0.7

Pow

er lo

sses

[W]

h/R [ ]

gear 1cone geometry n°1cone geometry n°2

Fig. 7. Power losses for gear 1 and cones (oil D, 1500 rpm, 40 °C).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

20 200

Chur

ning

torq

ue [N

.m]

Oil viscosity (cSt)

Fig. 8. Flow regimes (gear 3, h/R=0.6, 1000 rpm).

4 S. Laruelle et al.: Mechanics & Industry 18, 412 (2017)

This phenomenon is also observed for a smooth cone. Thepresent paper is focused on the first regime where thechurning torque increases with speed.

To further investigate the influence of spiral bevel geargeometry more tests were performed. Figures 4 and 5 showthat gears 1 and 4 have the same behaviour. These gearshave a similar external diameter (roughly equals to160mm) but present different tooth geometries (numberof teeth, face width, face angle, etc.). In Figure 6, cone 1 andgear 1 have also a similar behaviour.

In Figure 7, cone 2 has a similar diameter than cone 1,but different face angle and width. As its behaviour is closeto cone 1 and gear 1 it can be deduced that the externaldiameter appears as the main geometrical parameter,whereas the tooth influence can be neglected for the testedconditions.

Looking at the oil properties, Figure 8 presentschurning losses generated by gear 3 versus oil viscosityfor constant rotational speed and immersion depth. Inorder to analyse the influence of viscosity, the oil sump isheated using heating covers. Two flow regimes clearlyemerge: for high values of viscosity, the churning torquedecreases as viscosity is reduced whereas it remainsconstant for lower viscosity values. This behaviour hasalready been pointed out by Changenet et al. [17] forcylindrical gears.

As the published literature contains few calculationmethods to estimate churning losses of bevel gears, somecomparisons of the measured power loss with thosepredicted from the existing relationships have been done.As an example, Figure 9 presents churning losses generatedby gear 3 as a function of speed. This figure highlights thatpower losses calculated with ISO/TR 14179-1 overestimatethe measured ones. Indeed, according to this calculationmethod, the losses are proportional to N3 which is not inaccordance with experiments (proportional to N1.7).Because of this discrepancy, ISO/TR 14179-1 is disre-garded in the following analyses. In contrast, the churningloss levels given by ISO/TR 14179-2 and Jeon formula arecloser from experimental results, even if some discrepancyis observed.

Another comparison, associated with the influence of oilproperties, is given in Figure 10. As it has been underlined,two flow regimes clearly emerge from measurementswhereas calculation methods are based on a single regime:the churning torque always decreases with viscosityaccording to Jeon’s approach and it remains constant byusing the relationships of ISO/TR 14179-2.

4 Results analyses

As spiral bevel gears macro-geometry is close to hypoidones, the methodology used by Jeon [24] on hypoid gearswas considered to further analyse experimental data.

0

100

200

300

400

500

600

900 1,400 1,900 2,400

Chur

ning

loss

es [W

]

Rota�onal speed [RPM]

Measurements

TR 14179-1

TR 14179-2

Jeon's formula

Fig. 9. Influence of speed on measured and calculated churninglosses (gear 3, oil C, 40 °C, h/R=0.5).

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

20 200

Chur

ning

torq

ue [N

.m]

Oil viscosity [cSt]

Measurements

TR 14179-2

Jeon's formula

Fig. 10. Influence of oil properties on measured and calculatedchurning losses (gear 3, oil D, h/R=0.6, 1000 rpm).

0

0.005

0.01

0.015

0.02

0.025

0.03

0 500 1000 1500 2000 2500

Cm

[]

Rotational speed [rpm]

Gear 1Gear 4

Fig. 11. Cm for gears 1 and 4 according to equation (1) (oil B,70 °C, h/R=0.5).

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0 500 1000 1500 2000 2500

Cm

[]

Rotational speed [RPM]

Gear 1Gear 4

Fig. 12. Cm for gears 1 and 4 according to equation (3) (oil B,70 °C, h/R=0.5).

S. Laruelle et al.: Mechanics & Industry 18, 412 (2017) 5

Churning power losses are given by:

P ¼ 1

2rV3R2

obSmCm ð1Þ

where r is the lubricant density, V is the rotational speed,Ro is the outer radius on the large end of the bevel gear(see Fig. 3), b is the width, Sm is the submerged surfacearea and Cm is the dimensionless drag torque expressed asfollows:

Cm ¼ 2:186h

Ro

� �0:147 V o

R3o

" #�0:198

Re�0:25Fr�0:53 ð2Þ

where h is gear immersion depth, Vo the oil volume, Re theReynolds number and Fr the Froude number.

As experimental results have underlined that toothgeometry has a negligible influence on churning losses, thesubmerged surface area is quantified by modifying thecylindrical gear formula provided by Changenet et al. [17]and by only taking into account the equivalent envelopegeometry of a spiral bevel gear. The detailed Smcomputation is presented in Appendix B. The analyticalformula of Sm concurs with Jeon’s work on hypoid gears

since the retained geometry is only related to the gearenvelope dimensions. Then equation (1) can be used todetermine experimental dimensionless churning torquesfrom measurements. A typical series of results is presentedin Figure 11. As gears 1 and 4 do not have identical values ofCm but similar diameters, it can be deduced fromdimensional analysis [26] that parameters associated withtooth geometry (module, tooth face width, etc.) have alsoto be taken into account to determine Cm, which is not inaccordance with equation (2).

As the experiments show that the outer radius is a firstorder influent parameter, it is proposed to change equation(1) as follows:

P ¼ 1

2rV3R3

oSmCm ð3Þ

Figure 12 demonstrates that this new relationship leadsto identical dimensionless drag torque for gears 1 and 4. Itimplies that the tooth geometry parameters are not to betaken into account anymore in dimensional analysis.

0.0025

2000 20000

Cm

[ ]

Reynolds number [ ]

Oil D - Gear 3

Oil D - Gear 3

Oil D - Gear 3

Oil A - Gear 1

Oil A - Gear 1

Oil A - Gear 1

Fig. 13. Two flow regimes of churning depending on Reynolds number.

Fig. 14. h/R influence (gear 3, 1000 rpm, oil D).

6 S. Laruelle et al.: Mechanics & Industry 18, 412 (2017)

Nevertheless, influential parameters identified in equation(2) can be kept for a further analysis:

Cm ¼ g1Reg2Frg3h

Ro

� �g4 V o

R3o

!g5

ð4Þ

where g1,… , g5 are constant coefficients which are adjusted

from experimental results.Figure 13 presents the dimensionless torque Cm for two

different gears (1 and 3) and two oils (A and D) accordingto the Reynolds number. The two flow regimes that havebeen pointed out in Section 4 clearly appear on this figure.A transition between the two flow regimes is observed for a

Reynolds number around 20 000. For the regime below thetransition, g2 equals �0.25 (in agreement with Jeon’sformula) whereas for higher Reynolds number, g2 is zero.For this last regime, it means that viscous forces becomenegligible.

Concerning the Froude number, its influence can beisolated through tests where only the rotational speedvaries as the ones given in Figure 5. The constant g3 canthen be deduced. A value of �0.53 is found which isidentical to Jeon’s formula.

As far as influence of gear immersion is concerned,equation (2) is used to determine the following ratioðCm=ðRe�0:25Fr�0:53ðV o=R

3oÞ�0:198ÞÞ according to relative

Fig. 17. Position of the deflector and direction of rotation of thegear.

Fig. 18. Housing influence (gear 1, 2000 rpm, oil D, 40 °C).

Fig. 15. Jeon test rig, oil behaviour during tests [extract of JeonPhD illustration].

Fig. 16. Oil behaviour in ECAM test rig.

S. Laruelle et al.: Mechanics & Industry 18, 412 (2017) 7

immersion depth (h/Ro). In Figure 14, this ratio iscalculated in two different ways: either using experimentaldata for Cm or using Jeon’s relationship. It appears thatcomparing these two calculations, experiments lead to amore sensitiveness to the relative immersion depth.Looking at the general geometry of the casing in theJeon’s experiments (Fig. 15) and in the present study(Fig. 16) it can be noticed that the gears do not generate asimilar oil flow probably due to enclosure effect. The axlehousing presents a circular shape around the crown gear,contrary to this study where a parallelepiped defines the oilsump. In order to investigate this assumption, comple-mentary tests were conducted with deflector around thebevel gear (Fig. 17). These tests (with and without adeflector) are reported in Figure 18 and show the impact ofthe housing on the dimensionless drag torque: adding adeflector modifies considerably the evolution of dimension-less torque according to the relative immersion. This isconsistent with the observation mentioned previously.

5 Conclusion

Only few works deal with churning losses of bevel or hypoidgears. Moreover, the published literature contains differentcalculation methods to quantify this source of dissipationand these methods may give widely different results. Inorder to resolve the discrepancy, the authors havecompared some experimental results with the predictionsfrom various literature sources. It is shown that the resultsdid not agree well with the predictions for all operatingconditions, although correlation was somewhat better with

the predictions derived from Jeon’s work. In order toextend Jeon’s formula, numerous tests were conducted fordifferent gear geometries and lubricants. To this end aspecific test rig was used and the following conclusions canbe addressed:

– two lubricant flow regimes appear according to theReynolds number value (around 20 000). For highReynolds number, the viscosity has a negligible effecton churning losses whereas for lower values the observedbehaviour is similar to the one pointed by Jeon;

–

the main influential geometrical parameter is theexternal diameter of the bevel gear and other parameterssuch as module, tooth number and face width are ofsecond order. To take into account this point, thedimensional analysis proposed by Jeon has been modi-fied;

–

a great discrepancy between the present tests and Jeon’sformula was noticed concerning the influence of oil level.It was assumed that it was linked to the difference interms of enclosure between the two tests rigs. Other testswere performed with movable walls and confirm thishypothesis.

The development of new relationships to estimate thechurning losses generated by bevel gears aims to extend thevalidity range of Jeon’s formula to different bevel geargeometries and several fluid flow regimes while it is clearfrom this study that a modelling methodology valid for allcasing types and configurations is still required.

8 S. Laruelle et al.: Mechanics & Industry 18, 412 (2017)

Nomenclature

A1

area of the back end side of the cone, m2

A2

area of the front side of the cone, m2

A3

area of the peripheral surface of the cone, m2

a, a0

parameter of the equation of the slope of theperipheral surface, a0 for immersed dimen-sions

b, b0

parameter of the equation of the slope of theperipheral surface ; b0 for immersed dimen-sions, m

Cm

dimensionless drag torque eo radius of the front side of the truncated cone

associated to the cone of base radius Ro, m

Fr ¼ V2Ro

g

Froude number depending on gear parame-ters

g

acceleration of gravity, m s�2

H

height of the tooth, m h immersion depth of a pinion, m h*=Ro� h difference between the radius and the immer-

sion depth, m

N rotational speed, rpm P

2

churning power losses, W

Re ¼ VRo

n

Reynolds number

Ro

outer radius, m Sm immersed surface area of the pinion, m2

Vo

oil volume, m3

a

pressure angle, rad gi power law coefficient i u angle associated to the immersed surface of

the circular base of the cone, rad

l test number l F test number F n kinematic viscosity, m2 s�1

r

fluid density, kgm�3

c

spiral angle, rad Pi dimensionless parameter number i V angular velocity, rad s�1

Appendices

Appendix A Churning losses formulas(a) ISO/TR 14179-1 [22]

Churning power losses are decomposed as the sum of theones associated with the faces of the gear (Psides) and theones of the tooth surfaces (Ptooth).

For the faces of the gear:

Psides ¼1:474⋅fg⋅n⋅n3⋅D5:7

Ag⋅1026ð5Þ

For tooth surfaces:

Ptooth ¼7:37⋅fg⋅n⋅n3⋅D4:7⋅F ⋅ Rfffiffiffiffiffiffiffiffiffiffi

tanðbÞp� �

Ag⋅1026ð6Þ

Rf is the roughness factor which is defined as:

Rf ¼ 7:93� 4:648

mtð7Þ

where Pi is the power loss for each individual element [kW],fg is the gear dip factor (0� fg� 1), n is the kinematicviscosity of the oil at operating temperature [cSt],mt is thetransverse tooth module, D is the outside diameter of thegear [mm], Ag is the arrangement constant (=0.2), F is thetotal face width [mm], b is the generated helix angle [°] (ifb< 10° then consider b=10°), n is the rotational speed[rpm].

(b) ISO-TR 14179-2 [23]

The churning loss is expressed as:

P ¼ TH⋅v ð8Þ

TH ¼ Csp⋅C1⋅eC2ðvt=vt0Þ ð9Þwhere v is the rotational speed [rad s�1],TH is the churningtorque [Nm], vt is the peripheral speed at pitch circle[m s�1] and vt0= 10m s�1.

The other factors are determined by the followingrelationships:

Csp ¼ 4⋅hemax

3⋅hc

� �1:5 2⋅hc

lhð10Þ

C1 ¼ 0:063he1 þ he2

he0

� �þ 0:0128

b

b0

� �3

ð11Þ

C2 ¼ he1 þ he2

80⋅he0þ 0:2 ð12Þ

where he1 and he2 are the tip circle immersion depths withoil level stationary [mm], he0 is the reference value ofimmersion depth (=10mm), hc is the height of point ofcontact above the lowest point of the immersing gear [mm],lh is the hydraulic length [mm], b is the tooth width [mm]and b0= 10mm.

As far as the hydraulic length is concerned, it iscalculated by:

lh ¼ 4⋅AG

UMð13Þ

where AG is the enclosure area, UM is the enclosurecircumference.

(c) Jeon PhD work [24]

In this approach, power losses are expressed as a function ofa dimensionless drag torque which in turn depends onseveral non-dimension numbers. Detailed equations aregiven in Section 5.

Fig. 19. Scheme from the back side of the pinion (tooth notrepresented), with the cone angle. Fig. 20. Function representing the slope of external surface.

S. Laruelle et al.: Mechanics & Industry 18, 412 (2017) 9

Appendix B Submerged surface area Smformula

The immersed areas of a bevel gear are identified inFigure 19.

To simplify the problem, two areas are isolated: thefront and back areas {a}, the peripheral area {b}.

(a) The front and back areas (orange areas in Fig. 19)

The areas are computed as follows for the front and backside of the gear:

– h* is defined as a constant value computed from the backend of the gear:

h� ¼ Ro � h ð14Þ

– Then, the orange area for the back side is:

A1 ¼ 1

2R2

o 2a cosh�

Ro

� �� sin 2a cos

h�

Ro

� �� �� �ð15Þ

For the front side, it is:

A2 ¼ 1

2e2o 2a cos

h�

eo

� �� sin 2a cos

h�

eo

� �� �� �ð16Þ

eo represents the radius of the front side of the truncatedcone associated to the cone of base radius Ro.

(b) The peripheral area

It represents the area of the side face of the truncated conewhich is immersed in the lubricant sump. It is coloured inblue in Figure 19. A function shown in Figure 20 is definedwhich represents the slope of the external surface.

b0 is equal to zero so a0 is defined. umax is set as:

umax ¼ acosh�

Ro

� �ð17Þ

From Figure 20 and these notations, the area of theexternal surface of the cone of radius Ro is expressed as:

A3 ¼ ∫aþba 2uðxÞrðxÞdx ð18Þ

Finally the area is equal to:

A3 ¼ ðaþ bÞ a0ðaþ bÞa cos h�

a0ðaþ bÞ� ���

�H

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� h�2

a02ðaþ bÞ

s ��

� a a0a cosh�

a0a

� ��H

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� h�2

a02a2

r !" #ð19Þ

H is the height of the tooth.

References

[1] P. Luke, A.V. Olver, A study of churning losses in dip-lubricated spur gears, Proc. Inst. Mech. Eng. G J. Aerosp.Eng. 213 (1999) 337–346

[2] A.S. Kolekar, A.V. Olver, A.E. Sworski, F.E. Lockwood, Theefficiency of a hypoid axle – a thermally coupled lubricationmodel, Tribol. Int. 59 (2013) 203–209

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Cite this article as: S. Laruelle, C. Fossier, C. Changenet, F. Ville, S. Koechlin, Experimental investigations and analysis onchurning losses of splash lubricated spiral bevel gears, Mechanics & Industry 18, 412 (2017)