Research ArticleOn the Mechanical Friction Losses Occurring in AutomotiveDifferential Gearboxes

Greacutegory Antoni

Haute Ecole drsquoIngenierie et de Gestion du Canton de Vaud Institut COMATEC Route de Cheseaux 11401 Yverdon-les-Bains Switzerland

Correspondence should be addressed to Gregory Antoni antonigregoryyahoofr

Received 13 August 2013 Accepted 19 October 2013 Published 19 January 2014

Academic Editors C W Leung F Liu and S J Rothberg

Copyright copy 2014 Gregory Antoni This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In the automobile industry themechanical losses resulting from friction are largely responsible for various kinds of surface damagesuch as the scuffing occurring in some mechanical assemblies These scuffing processes seem to be due to a local loss of lubricationbetween certain mechanical elements of the same assembly leading to a sharp increase in the friction which can lead to a surfaceand volume damage in some of them and even can cause in the worst case the whole destruction of the mechanical system if it hascontinued to operate Predicting and checking the occurrence of this kind of undesirable phenomena especially in some principalsystems of the vehicle represents nowadays a crucial challenge in terms of automobile reliability and safety This study focuses onthe mechanical friction losses liable to occur in differential automobile gearboxes which can lead in the long term to the scuffingof these mechanical systemsThe friction losses involved were modeled using a simple analytical approach which is presented anddiscussed

1 Introduction

Although the automobile industry has contributed signifi-cantly during the last few years to increasing the CO

2levels

polluting the atmosphere the introduction of a Carbon Taxhas been inciting car manufacturers to reduce this pollutionIn line with this more environmentally friendly approachautomobile manufacturers are now attempting to decreasethe mechanical friction losses occurring between some vehi-cle partsmdashcausing both an increase in fuel consumption andlevels of carbon dioxide emissionsmdashwhile maintaining theircompanyrsquos competitiveness on the market In practice thesefriction losses are frequently associated with surface damageof several kinds such as scuffing [1ndash6] which is also knownas galling [7] or seizure [8ndash10] Scuffing is an extreme formof adhesive wear which occurs when two mechanical partsadhere locally to each other and amaterial transfer takes placefrom the one surface to the other during a sliding processwhich in turn can result in thewelding or binding of the entiremechanical system [11ndash14] Although the physical causes ofscuffing have not yet been clearly established it is generallyrecognized that they are initiated when the lubricant film

between two sliding parts is suddenly destroyed as theresult of a sharp temperature increase due to the presenceof heavy mechanical loads [15] The most accurate andefficient methods of predicting scuffing processes are thosebased on estimating the thermal energy responsible for thisdamage in comparison with that produced under normaloperating conditions [16 17] The friction energy dissipatedin the mechanical assemblies in question must thereforethus be assessed to define a criterion in order to predictthe occurrence of the resulting scuffing effects [16 17] Themechanical losses to which the differential gearboxes aresubject can lead to a surface and volume damage of one ormore elements of this system and causing in the worst casethe blocking and the destruction of the system if it remainsin operation In this study on the mechanical friction lossesoccurring in differential automobile gearboxes an analyticalmodel was developed (see Section 2) in order to determinethe contribution of eachmechanical component to the overallfriction losses In Section 3 the relative contribution of eachof the constitutive components of the model and the effectsof various parameters on the mechanical friction lossesoccurring in a differential gearbox are discussed

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 523281 11 pageshttpdxdoiorg1011552014523281

2 The Scientific World Journal

Sump

Left axle shaft

Right axle shaft

Figure 1 Differential gearbox (with its sump and the axle shafts)

2 Analysis of the Mechanical Friction LossesOccurring in Differential AutomobileGearboxes Constitutive Equations

21 General Description Generally speaking a differentialgearbox [18] (Figure 1) is a complex device involving gearswhich are connected to output shafts transmitting bothrotation and torque forces In the framework of motorvehicles differential systems of this kind make it possibleto deliver an equal or different speed to all the wheelsdepending on whether the vehicle is taking a straight orcurvilinear course when the vehicle is starting to turnthe inner wheels must rotate more slowly than those inthe outside because they have to cover a smaller distanceduring the same time span Differential systems consist offour bevel gears (two ldquoplanetaryrdquo pinions and two ldquosatelliterdquopinions) a satellite-carrier axis and a plastic shell (Figure 2)all these components are enclosed in a nonsymmetric sumpentrained by the differential ring gear which is itself drivenin a rotating manner by both the primary and secondarygearbox shafts which are made to rotate by the engine ofthe vehicle The two planetary bevel pinions connected toeach axle shaft placed opposite each other are meshed by thetwo smaller satellite pinions (Figure 3) When the vehicle ismoving straight ahead the two planetary pinions rotate atthe same speed (as does the axle shaft associated with each ofthe driven wheel) and the two satellite pinions are stationaryrelative to their axis that is the differential housing and these

pinions rotate at the same speed as the ring gear Howeverwhen the vehicle turns the rotational speed of the wheel (andthat of the corresponding planetary pinions) differs from thatof the two satellite pinions which start rotating around theiraxis in order to compensate for this difference

22 Power and Yield Equations The total gearbox yield 120578gbis obtained by dividing the sum of the output powers of eachof the driven wheels by the gearboxrsquos input power

120578gb =119875lw + 119875rw119875en

=119862lw120596lw + 119862rw120596rw

119862en120596en (1)

where 119875119894 119862119894 and 120596

119894(with 119894 = (lw rw en)) are powers

torques and angular velocities associated with the left wheelright wheel and engine respectively

Upon introducing both the primary and secondary gear-box shaft yields the ratio between the power of the differentialring gear and that of the engine is equal to the product of theseshaft yields that is

120578ps120578ss =119875rg

119875en=119862rg120596rg

119862en120596en (2)

where 120578ps and 120578ss denote the primary and secondary shaftyields respectively and 119875rg 119862rg and 120596rg are the power torqueand angular velocity associated with the differential ring gear(drive gear) respectively

On the other hand the total gearbox yield can be writtenas the product of the primary and secondary shaft yields andthe differential gearbox

120578gb = 120578ps120578ss120578di (3)

After combining (1)ndash(3) the total differential gearboxyield 120578di can be obtained by dividing the sum of the outputpowers of each of the driven wheels by the input power

120578di =119875lw + 119875rw119875rg

=119862lw120596lw + 119862rw120596rw

119862rg120596rg (4)

Since a differential gearbox can both transmit and dis-tribute the differential ring gear power (the input power) tothe output gears associated with each of the driven wheels itfollows that

119875rg = 119875lw + 119875rw + 119875ab (5)

where 119875ab is the power absorbed in the differential gearboxdue to the friction losses

In the equations governing the angular velocities and thetorques in the differential gearbox (Figure 4)

120596lw119877119901 = 120596rg119877119901 minus 120596119904119877119904 (6a)

120596rw119877119901 = 120596rg119877119901 + 120596119904119877119904 (6b)

119862rg = 119862lw + 119862rw (7)

The Scientific World Journal 3

(a) (b)

Figure 2 (a) Sump of a differential gearbox (b) Various components of a differential mechanism

Satellite

Satellite

Left wheel planetary Right wheel planetary

Satellite-carrier axis

Left axle shaft

Right axle shaft

Plastic shell

Figure 3 Components of a differential gearbox two planetary bevelpinions two satellite bevel pinions with their shaft and the plasticshell

where 120596119904denotes the angular velocity of satellite pinion and

119877119901and119877

119904are the pitch radii of planetary and satellite pinions

respectivelyAdding or subtracting (6a)-(6b) we obtain the classical

relations

120596rg =120596lw + 120596rw

2 (8a)

120596lw120596rw

120596rg

Clw Crw

Crg

120596s

120596s

Figure 4 Angular velocities and torques in the differential gearbox

120596119904=120596rw minus 120596lw

2

119877119901

119877119904

(8b)

Note that from now on Δ120596 = |120596lw minus 120596rw| ge 0 andΔ119862 = |119862rw minus 119862lw| ge 0 will be used to denote the differencein the angular velocity and torque between the driven wheelsrespectively (where | sdot | is the absolute-value function)

Combining (5) and (6a) (6b) and (7) the powerabsorbed in the differential can be written as

119875ab =1

2Δ119862Δ120596 (9)

Based on the above equations the total differential gear-box yield is

120578di = 1 minus119875ab119875rg

= 1 minus (Δ119862Δ120596

2119862rg120596rg) (10)

4 The Scientific World Journal

Comments

(i) Under straight driving conditions the angular veloc-ities of two satellite pinions in relation to the satelliteaxis are zero (Δ120596 = 0) and the speed of thedriven wheels is equal to the angular velocity of thedifferential ring gear that is 120596lw = 120596rw = 120596rg

(ii) In curvilinear driving situations the angular velocityof two satellite pinions in relation to the satellite-carrier axis is no longer zero (Δ120596 gt 0) and the speedsof the driven wheels are no longer equal that is 120596lw gt120596rw (right turn) or 120596lw lt 120596rw (left turn)

(iii) If one of the drivenwheels undergoes slipping [18]mdashifit is the left (resp right) wheel 120596lw = 0 (resp 120596rw =0) that is Δ120596 = 2120596rgmdashthe differential gearbox yieldreduces to 120578di = 1 minus (Δ119862119862rg)

23 Mechanical Losses Friction Torques Thepower absorbedin the differential gearbox is the sum of the power dissipatedat the various contact points existing in the overall mecha-nism which can be decomposed as follows

119875ab = 119875sapl + 119875axsa + 119875sash + 119875plsh (11)

where 119875sapl 119875axsa 119875sash and 119875plsh correspond to the part ofthe power dissipated between the two satellite and planetarybevel pinions the satellite pinion and the satellite-carrieraxis between the two bevel heads of the satellite pinions andthe surrounding plastic shell and the two bevel heads ofthe planetary pinions and the surrounding plastic shell (seeFigure 3)

231 Power Dissipated between Two Satellite Bevel Pinionsand Two Planetary Bevel Pinions In order to account forthe power dissipated between the two satellite bevel pinionsand the two planetary bevel pinions in the differentialmechanism (giving four meshing contacts) we introducea yield 120578sapl (denoting the power transmitted in the gearbetween satelliteplanetary pinions) which can be written asfollows

120578sapl =119875rg minus 119875sapl

119875rg119867+(Δ120596) (12)

where 119875sapl denotes the power dissipated between the twosatelliteplanetary contacts and 119867+(sdot) is a Heaviside-likefunction (adapted in order to ensure that 119867+(119909) = 1 when119909 gt 0 and 119867(119909) = 0 when 119909 le 0) Note that a yield 120578saplwith a nonzero value (ie 120578sapl gt 0) occurs between thetwo satellite pinions and two planetary pinions only whenthe angular velocity differs between the driven wheels thatis Δ120596 gt 0

Based on (12) we can therefore write

119875sapl = (1 minus 120578sapl) 119862rg120596rg119867+(Δ120596) (13)

232 Power Dissipated between the Satellite Bevel Pinionand Satellite-Carrier Axis Neglecting the presence of an

over-centre mechanism between the satellite pinion andsatellite-carrier axis (Figure 5) the force exerted on thesatellite-carrier axis 119865ax upon the rotation of the differentialring gear (120596rg) can be written as

119865ax =119862rg

2119877119901

(14)

In (14) the point of application of the force 119865ax isassumed to take the same contact path as both the left andright meshing gears (Figure 7)

Adopting aCoulomb-type friction law the friction torqueapplied to the satellite-carrier axis 119862119891axsa is

119862119891

axsa = 120583axsa119865ax119889ax2 (15)

where 120583axsa denotes the friction coefficient (which is con-stant) between the satellite pinion and the satellite-carrier axisand 119889ax is the axis diameter (Figure 6)

Combining (14) and (15) the power dissipated betweenthese two assembly components reads

119875axsa = 2119862119891

axsa120596119904 = 120583axsa119862rg

4

119889ax119877119904

Δ120596 (16)

233 Power Dissipated between the Satellite Bevel Pinions andthe Plastic Shell The power dissipated between the two bevelsatellite pinions and the surrounding plastic shell 119875sash canbe written as

119875sash = 2119862119891

sash120596119904 = 119862119891

sash

119877119901

119877119904

Δ120596 (17)

where119862119891sash denotes the friction torque applied to the satellitebevel pinion Note that the friction torque 119862119891sash is assumedhere to be identical on the two satellite pinions

In order to write the equations giving the equilibrium ofeach satellite pinion which is meshed with the two planetarypinions associated with the driven wheels and in contact withthe satellite-carrier axis as well as with the plastic shell we usethe Fundamental Principle of Statics119879lwrarr119878119872 + 119879rwrarr119878119872 + 119879axrarr119878119872 + 119879shrarr119878119872 = 0

(18)where 119879

119894rarr 119878119872denotes the transmittable force torsor of solid

119894 (and 119894 = (lw rw ax sh) associated with the planetarypinions of the left and right wheels the satellite-carrier axisand the plastic shell resp) exerted on solid 119878 (the satellitepinion) at point119872 (see [15])

Looking only at the equilibrium of the resulting force ofeach torsor in the direction (119909 119910 119911) (see Figure 7) (18) writes

119865119905

lw119865119903

lw119865119886

lw

+

119865119905

rwminus119865119903

rw119865119886

rw

+

minus119865ax0

0

+

0

0

minus119883sa

=

0

0

0

(19)

lArrrArr

119865119905

lw + 119865119905

rw = 119865ax119865119903

lw = 119865119903

rw(119865119905

lw + 119865119905

rw) tan120572 sin 120575 = 119883sa

(20)

The Scientific World Journal 5

120595

Figure 5 Over-centre mechanism between the satellite pinion andsatellite-carrier axis 120595 denotes the angular eccentricity

2Rp

dax

Rs

Figure 6 Satellite bevel pinions and satellite-carrier axis

where 119865119905119895 119865119903119895 and 119865119886

119895(with 119895 = lw rw) are the tangential

radial and axial forces of 119895-planetary pinions (associatedwiththe left and rightwheels Figure 7 see [19]) respectively119883sa isthe resulting force applied to the satellite pinion generated bythe plastic shell 120575 is the half-angle pitch radius of the satellitepinion (see Figure 8(a)) and 120572 is the pressure angle betweenthe satellite and planetary pinions which is assumed hereto be identical between the satelliteplanetary meshing gears(see Figure 8(b))

y

x

z

y

x

z

Fax

Fax

Frlw

Falw Fa

rw

Frrw

Frlw Fr

rw

Ftlw Ft

rw

Psphsa

Psphsa

Figure 7 Forces applied to a satellite pinion

The pressure applied to the head surface of the satellitepinion 119875sphsa (its spherical part see Figure 9) due to the result-ing force119883sa (in line with (20)) matches the relationship (see[15])

119883sa = int2120587

0

int

1205751

1205750

119875sphsa (120579 120601) 119877

2

sa cos 120579 sin 120579 119889120579 119889120601 (21)

where 1205751and 120575

0are the angles defining the spherical part of

the head-satellite pinion and 119877sa is the radius of the innersphere of the plastic shell (where the contact with the headsurface of the satellite pinion and the plastic shell occurs)

The expression for the friction torque119862119891sash applied to thehead surface of the satellite pinion can be written (see [17]) as

119862119891

sash = int2120587

0

int

1205751

1205750

120591sphsa (120579 120601) 119877

3

sa sin 120579 119889120579 119889120601

= int

2120587

0

int

1205751

1205750

120583sash119875sphsa (120579 120601) 119877

3

sa sin 120579 119889120579 119889120601

(22)

where 120591sphsa (120579 120601) = 120583sash 119875sphsa (120579 120601) denotes the shear stress

exerted on the head surface of the satellite pinion since aCoulomb-type friction law is adopted (see Figure 9) and120583sash is the coefficient of friction (which is constant) betweenthe satellite pinion and the plastic shell

6 The Scientific World Journal

120575

1205751

1205750Rs

Rsa

Rsa

Rp

dax

1205759984000

1205752

120575

(a)

120572

(b)

Figure 8 Geometry of a differential gearbox

Psphj

120591sphj

Figure 9 Pressure 119875sph119895

(solid arrow) and shear 120591sph119895

(dashed arrow)fields on a head-pinion 119895 (with 119895 = (sa pl)) caused by the plasticshell

Combining (14) and (20)ndash(22) and assuming that ina first approximation the pressure field exerted on the

head-satellite pinion is uniform (ie 119875sphsa (120579 120601) = 119875sphsa ) (21)

and (22) reduce to

119883sa = 21205871198772

sa119875sphsa (

cos (21205750) minus cos (2120575

1)

4)

119862119891

sash = 120583sash119877sa119862rg

2119877119901

tan120572 sin 120575

times (cos (120575

1+ 1205750) sin (120575

1minus 1205750) minus (120575

1minus 1205750)

sin (1205750+ 1205751) sin (120575

0minus 1205751)

)

(23)

234 Power Dissipated between the Planetary Bevel Pinionsand the Plastic Shell The power dissipated between the twoplanetary pinions and the plastic shell 119875plsh can be writtenas

119875plsh = 119862119891(lw)plsh 120596

rslw + 119862

119891(rw)plsh 120596

rsrw (24)

where119862119891(119894)plsh are the frictional torque between the planetary 119894-pinion (119894 = (lw rw) refers to the left wheel (lw) and the rightwheel (rw)) and the plastic shell and 120596rs

119894= Δ1205962 denotes

the relative speed of the 119894-pinion in relation to the sumpdifferential

Assuming that 119862119891(lw)plsh cong 119862119891(rw)plsh = 119862

119891

plsh (24) reads

119875plsh = 119862119891

plshΔ120596 (25)

Using the sameprocedure as above namelymeshing eachof the planetary pinions with two satellite pinions which

The Scientific World Journal 7

means that the resulting force applied to the head-planetarypinion by the plastic shell119883pl is

119883pl = 119865ax tan120572 sin 120575 (26)

where 120575 is the half-angle pitch radius of the planetary pinion(see Figure 8(a)) Note that given the particular geometry ofthe mechanism the sum of the half-angle pitch radii of theplanetary and satellite pinions satisfies 120575 + 120575 = 1205872 and wetherefore obtain the following relationship sin 120575 = cos 120575

The expressions for both the resulting force 119883pl and thefriction torque119862119891plsh applied to the head-satellite pinion are

119883pl = int2120587

0

int

1205752

1205751015840

0

119875sphpl (120579 120601) 119877

2

sa cos 120579 sin 120579 119889120579 119889120601

119862119891

plsh = int2120587

0

int

1205752

1205751015840

0

120591sphpl (120579 120601) 119877

3

sa sin 120579 119889120579 119889120601

(27)

where 120591sphpl (120579 120601) = 120583plsh119875sphpl (120579 120601) denotes the shear stress

exerted on the outer surface of the planetary pinion with aCoulomb-type friction law (Figure 9) 120583plsh is the coefficientof friction (which is constant) between the satellite pinionand the plastic shell 119875sphpl is the pressure applied to the headsurface of the planetary pinion and 120575

2and 1205751015840

0are the angles

defining the spherical portion of the head-planetary pinionIt is again assumed here that in a first approximation the

pressure field 119875sphpl exerted on the head-planetary pinion isuniform (ie (120579 120601)-independent) and (27) reduce to

119883pl = 21205871198772

sa119875sphpl (

cos (212057510158400) minus cos (2120575

2)

4)

119862119891

plsh = 120583plsh119877sa119862rg

2119877119901

tan120572 cos 120575

times (cos (120575

2+ 1205751015840

0) sin (120575

2minus 1205751015840

0) minus (120575

2minus 1205751015840

0)

sin (12057510158400+ 1205752) sin (1205751015840

0minus 1205752)

)

(28)

235 Total Power Dissipated in the Differential GearboxBased on the above constitutive equations the total powerdissipated in the differential gearbox can bewritten as follows

119875ab = 119875sapl + 119875axsa + 119875sash + 119875plsh

= (1 minus 120578sapl) 119862rg120596rg119867+(Δ120596) + 120583axsa

119862rg

4

119889ax119877119904

Δ120596

+ 120583sash119877sa119862rg

2119877119904

tan120572 sin 120575

times (cos (120575

1+ 1205750) sin (120575

1minus 1205750) minus (120575

1minus 1205750)

sin (1205750+ 1205751) sin (120575

0minus 1205751)

)Δ120596

+ 120583plsh119877sa119862rg

2119877119901

tan120572 cos 120575

times (cos (120575

2+ 1205751015840

0) sin (120575

2minus 1205751015840

0) minus (120575

2minus 1205751015840

0)

sin (12057510158400+ 1205752) sin (1205751015840

0minus 1205752)

)Δ120596

(29)

Using (10) and (29) and assuming that in a first approx-imation 120583sash = 120583plsh = 120583sh (where 120583sh denotes thecoefficient of friction between each of the pinions in contactwith the plastic shell) the total differential gearbox yield canbe written as follows

120578di = 1 minussum119894

119897119894

= 1 minus

[[[[

[

119875sapl

119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897sapl

+119875axsa119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897axsa

+119875sash119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897sash

+119875plsh

119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897plsh

]]]]

]

(30)

with

119897sapl = (1 minus 120578sapl)119867+(Δ120596)

119897axsa = 120583axsa119889ax4119877119904

Δ120596

120596rg

119897sash = 120583sh119877sa2119877119904

tan120572 sin 120575

times (cos (120575

1+ 1205750) sin (120575

1minus 1205750) minus (120575

1minus 1205750)

sin (1205750+ 1205751) sin (120575

0minus 1205751)

)Δ120596

120596rg

119897plsh = 120583sh119877sa2119877119901

tan120572 cos 120575

times (cos (120575

2+ 1205751015840

0) sin (120575

2minus 1205751015840

0) minus (120575

2minus 1205751015840

0)

sin (12057510158400+ 1205752) sin (1205751015840

0minus 1205752)

)Δ120596

120596rg

(31)

where 119897119894= 119875119894119875rg denotes the loss ratio of the 119894-mechanism

(with 119894 = (sapl axsa sash plsh)) It is worth noting thatthe differential yield 120578di shows mechanical friction losseswhenever the rotational-speed differs between two of theplanetary pinions associated with the driven wheels that iswhen Δ120596 gt 0

3 Discussions

In the first part of this section we discuss the order of magni-tude of the constitutive parameters of the model presented inSection 2 In the second part we present a sensitivity analysis

8 The Scientific World Journal

0 05 1 15 2085

09

095

1D

iffer

entia

l gea

rbox

yie

ld120578

di

Δ120596120596rg

(a)

0 05 1 15 20

0005

001

0015

002

0025

003

0035

004

PiP

rg

Δ120596120596rg

(b)

Figure 10 (a) Yield of a differential gearbox 120578di as a function of the load parameterΔ120596120596rg for various sets of coefficients of friction (120583axsa 120583sh)(003 005) in a dashed line (005 007) in a dash-dot line (003 007) in a dotted line and in a solid line for (120583axsa 120583sh) and 120578sapl undergoinga concomitant linear increase from (003 005) to (005 007) and 098 to 096 respectively with Δ120596120596rg the thick solid line gives the yieldof the differential gearbox without any losses (which is constant) with the load parameter Δ120596120596rg that is 119897119894 = 119875119894119875rg = 0 forall119894 (b) Loss ratio119897119894= 119875119894119875rg of the 119894-mechanism (with 119894 = (sapl axsa sash plsh)) as a function of the load parameter Δ120596120596rg and with the same sets of

coefficients of friction as in (a) 119897sapl without (black solid line) or with (black dashed line) a linear decrease in 120578sapl from 098 to 096 119897axsa(red line) 119897sash (blue line) and 119897plsh (green line)

0 005 01 015 020

001

002

003

004

005

006

Cf iC

rg

Coefficient of friction 120583i

Figure 11 Friction torque ratio 119862119894119862rg as a function of coefficient of

friction 120583119894with 119894 = (axsa sash plsh) (while the other parameters

were taken to be constant) 119862axsa119862rg increased linearly with 120583axsa(red solid line) 119862sash119862rg increased linearly with 120583sash = 120583sh (bluesolid line)119862plsh119862rg increased linearly with 120583plsh = 120583sh (green solidline) in a range of variation 120583

119894isin [0 02]

in which the response of the model to a given load parameterwas examined in order to assess the effect of this responseon themechanical losses occurring in the differential gearboxmechanism

31 Order of Magnitude of the Constitutive Parameters Thisanalytical model involves thirteen parameters (120578sapl 120583axsa

0 200 400 600 800 10000

02

04

06

08

1

12

14

16

18

2

Time (s)

Δ120596120596

rg

Figure 12 Example of a load parameterΔ120596120596rg with a complex pathdepending on time 119905 isin [0 1000 s] when Δ120596120596rg = 0 (in the casewhere 120596lw = 0 and 120596rw = 0) then 120578di = 1 and 119897119894 = 119875119894119875rg = 0 forall119894 (whenthe vehiclewas travelling along a straight line) when 0 lt Δ120596120596rg lt 2

then 120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894 (when the vehicle was taking acurvilinear course) when Δ120596120596rg = 2 then 120578di lt 1 and 119897119894 = 119875119894119875rg gt0 forall119894 (when one of the driven wheels underwent slipping the vehiclewas stationary) on the plateau Δ120596120596rg = constant and |120596lw minus120596rw| =

constant (the vehicle was taking a constant curvilinear trajectory)

120583sh 119889ax 119877119904 119877119901 119877sa 120572 120575 1205750 1205751015840

0 1205751 1205752) consisting of (i) ten

geometrical parameters (119889ax 119877119904 119877119901 119877sa 120572 120575 1205750 1205751015840

0 1205751 1205752)

depending on the differential gearbox under consideration

The Scientific World Journal 9

0 200 400 600 800 1000091

092

093

094

095

096

097

098

099

1

Time (s)

Diff

eren

tial g

earb

ox y

ield

120578di

(a)

0 200 400 600 800 10000

0005

001

0015

002

0025

003

0035

Time (s)

PiP

rg

(b)

Figure 13 (a) Yield of a differential gearbox 120578119889119894with time 119905 isin [0 1000 s] in the case of the complex path presented in Figure 12 with various

sets of friction coefficients (120583axsa 120583sh) (003005) in a dashed line (005007) in a dash-dot line and (003007) in a dotted line (b) Lossratio 119897

119894= 119875119894119875rg of the 119894-mechanism (with 119894 = (sapl axsa sash plsh)) with time 119905 isin [0 1000 s] in the case of the complex path presented

in Figure 12 with the same sets of friction coefficients as in (a) 119897sapl = 096 which remained constant with time 119905 (black solid line) 119897axsa (redline) 119897sash (blue line) and 119897plsh (green line)

which can be identified fairly easily (ii) one yield param-eter (120578sapl) corresponding to the power transmitted by thegear between the satelliteplanetary pinions (iii) two fric-tion parameters (120583axsa 120583sh) which depend considerably onthe types of materials in contact and the surface conditionstheir state of lubrication and their temperature

Concerning (i) although these specific parameters de-pend on the differential gearbox under consideration theirorder of magnitude can be said to be 119889ax asymp 10ndash20mm 119877

119904asymp

20ndash30mm 119877119901asymp 20ndash30mm 119877sa asymp 30ndash50mm and for the

angle parameters 1205750asymp 10∘ndash15∘ 1205751015840

0asymp 15∘ndash25∘ 120575

1asymp 30∘ndash40∘

1205752asymp 35∘ndash45∘ 120572 asymp 20ndash25∘ 120575 asymp 30∘ndash40∘ readers can refer to

Fanchon [19] for further details about some of themConcerning (ii) the parameter 120578sapl which denotes the

power transmitted by the gear between satelliteplanetarypinions using a yield term has a value ranging between 092and 098 Since Fanchon [19] has reported that the yield inthe case of two meshed pinions is around 98 it can beconcluded that in a differential mechanism where there arefour pinions in contact the lowest value of 120578sapl asymp (098)

4cong

092 and the highest value is likely to be 120578sapl cong 098 Ittherefore seems reasonable to assume that the real yield 120578saplof this complex mechanism is approximately 096

The parameters that need to be investigated more closelyin order to determine their influence on themechanical lossesare the various coefficients of friction (iii) 120583axsa and 120583shThe sliding contacts in the differential gearbox mechanismcan be of very different kinds [20 21] since both thesatelliteplanetary bevel pinion and satellite-pinionsatellite-carrier axis in contact are in the mixed elastohydrodynamiclubrication (MEHL) mode [22ndash24] which is a mixed regimebetween (1) elastohydrodynamic lubrication (EHL) [25 26]where the heavy loads exerted at the surface of the two gears

(due to the presence of both high contact pressures and strongtransmitted torques) induce elastic strains in the two solidsin contact with thin lubricant films and (2) boundary lubri-cation (BL) [27] which may tend to constitute a completelydry frictional contact [28] leading to the development ofscuffing effects [29 30] The contacts between the bevel headpinion (satellites and planetaries) and the plastic shell arein the elastohydrodynamic lubrication mode (EHL) whichcan lead to the development of the hydrodynamic lubricationregime (HL) [31 32] where the lubricant film thickness islarge enough to completely separate the directly apposedsurfaces so that the loads applied between the surfaces arerestored and balanced under specific operating conditionsUnder the above conditions the contacts existing betweenthe satellite pinions and the satellite-carrier axis are of thesteelsteel contact with lubrication type 120583axsa isin [005 007]and those existing between the head pinions and the plasticshell are rather of the steelTeflon contact with lubricationtype 120583sh isin [003 005]

32 Sensitivity Analysis Assessment of the Modelrsquos Responseto a Given Load Parameter and Prediction of the MechanicalLosses In what follows it is proposed to test the modelrsquosresponse to a given load parameter and to determinethe influence of some parameters (120578sapl 120583axsa 120583sh) on themechanical losses occurring in a differential gearbox mecha-nism The expression for the load parameter Δ120596120596rg isin [0 2]means that (i) when Δ120596120596rg = 0 (in cases where 120596lw = 0 and120596rw = 0) the vehicle is travelling on a straight line 120578di = 1 and119897119894= 119875119894119875rg = 0 forall119894 (ii) when 0 lt Δ120596120596rg lt 2 the vehicle is

taking a curvilinear course 120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894(iii) when Δ120596120596rg = 2 one of the driven wheels undergoes

10 The Scientific World Journal

slipping (the extreme situation) the vehicle is stationary120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894

The values adopted for the other parameters are 119889ax =18 times 10

minus3m 119877119904= 22 times 10

minus3m 119877119901= 27 times 10

minus3m 119877sa =49 times 10

minus3m 1205750= 10∘ 12057510158400= 18∘ 1205751= 32∘ 1205752= 40∘ 120572 = 20∘

120575 = 39∘

Figure 10(a) shows the yield of a differential gearbox 120578didepending on the load parameter Δ120596120596rg with various setsof friction coefficients of friction (120583axsa 120583sh) and a lineardecrease in 120578sapl from 098 to 096 In the most gentle oper-ating range that is 0 le Δ120596120596rg le 1 120578di decreased from 098to around 0963 (120583axsa 120583sh) = (003 005) 120578di isin [098 0959](resp 120578di isin [098 0954]) when (120583axsa 120583sh) = (003 005)(resp (120583axsa 120583sh) = (005 007)) and 120578di decrease from 098to around 0949 when (120583axsa 120583sh) showed a concomitantlinear increase from (003 005) to (005 007) In the heavyoperating range that is 1 lt Δ120596120596rg le 2 the maximumdifferential gearbox yield 120578di isin ]0963 0946] was obtainedwith (120583axsa 120583sh) = (003 005) and the minimum yield(120578di isin ]0949 0909]) occurred when (120583axsa 120583sh) and 120578saplincreased linearly from (003 005) to (005 007) concomi-tantly with the load parameter Δ120596120596rg 120578di isin ]0959 0938](resp 120578di isin ]0954 0929]) when (120583axsa 120583sh) = (003 005)(resp (120583axsa 120583sh) = (005 007)) The results presentedin Figure 10(b) show the loss ratio 119897

119894= 119875119894119875rg associated

with 119894-mechanism (where 119894 = (sapl axsa sash plsh)) indecreasing order of influence 119897axsa (red line) 119897sapl (blackline) 119897plsh (green line) 119897sash (blue line) when 120578sapl had a con-stant value of 098 and in the case where 120578sapl decreased from098 to 096 the effects of the loss ratio 119897sapl became greaterthan those of the other parameters (119897axsa 119897axsa 119897sash 119897plsh)It should be noted that when 0 le Δ120596120596rg le 1 then119897axsa isin [0 00143] 119897plsh isin [0 00057] and 119897sash isin [0 00052]whereas when 1 lt Δ120596120596rg le 2 then 119897axsa isin [0 00286]119897plsh isin [0 00115] and 119897sash isin [0 00104] Figure 11 showsthe friction torque ratio 119862

119894119862rg as a function of the friction

coefficient 120583119894with 119894 = (axsa sash plsh) when the other

parameters were taken to be fixed constants The range ofvariation of the friction coefficient 120583

119894was taken to be between

0 and 02 (this extreme value may correspond to steelsteelcontact without any lubrification) In view of these resultsthe maximum 119862

119894119862rg ratio was obtained with 119862axsa119862rg and

the minimum one with 119862sash119862rg The friction torque ratios119862119894119862rg progressed linearly with the friction coefficient 120583

119894

(where 120583sh = 120583sash = 120583plsh) as predicted by the constitutiveequations in Section 2 Note that the results obtained in thisfirst approximation are likely to differ from those obtained bysimulating nonlinear behaviour between 119862

119894119862rg and 120583119894

A complex path of the load parameter Δ120596120596rg wasthen studied in order to test the ability of the model todescribe the differential gearbox yield 120578di and the loss ratios119897119894= 119875119894119875rg associated with the 119894-mechanism (where 119894 =

(sapl axsa sash plsh)) under more realistic conditionsinvolving a straight line situation (Δ120596120596rg = 0 with 120596lw = 0

and 120596rw = 0) a curved line situation (0 lt Δ120596120596rg lt 2)a constant curved line situation (Δ120596120596rg = constant) orwhere one of the drivenwheels undergoes slipping (Δ120596120596rg =2) Figure 12 shows the evolution of the load parameter

Δ120596120596rg as a function of the time 119905 isin [0 1000 s] Theresults obtained with a complex path in terms of the yieldof the differential gearbox (120578di) and the loss ratios (119897

119894) are

shown in Figure 13 Specifically Figure 13(a) gives the yieldof the differential gearbox 120578di as a function of the time119905 isin [0 1000 s] in the case of the complex path plotted inFigure 12 with various sets of friction coefficients (120583axsa 120583sh)(003005) in dashed line (005007) in dash-dot line and(003007) in dotted line During the whole complex path ofΔ120596120596rg the maximum (resp minimum) yield of differentialgearbox was reached with 120583axsa = 003 and 120583sh = 005

(resp 120583axsa = 005 and 120583sh = 007) On the 119896-plate (with119896 = 1 2 3 4) defined by the 119896-time interval [119905119894 119905119891]

119896 it can be

noted that for example (1) in [400 s 500 s]2 120578di asymp 09774with

(005007) 120578di asymp 09591 with (003007) and 120578di asymp 09547with (005007) (2) in [900 s 950 s]

4 120578di asymp 09464 with

(005007) 120578di asymp 09382 with (003007) and 120578di asymp 09294with (005007) The loss ratios due to friction 119897

119894= 119875119894119875rg are

shown in Figure 13(b) The maximum loss ratios obtained indecreasing order of magnitude (along the loading path) were119897sapl (black line) 119897axsa (red line) 119897plsh (green line) and 119897sash(blue line) except for the last loadwhere 119897axsa could be greaterthan 119897sapl with the pair of friction coefficients (005007)and (003007) It should be noted that (1) on the secondplate ([400 s 500 s]

2) 119897max

axsa asymp 00143 with (005007) 119897maxplsh asymp

00057 with (003007) and 120578di asymp 00052 with (005007)(2) on the fourth plate ([900 s 950 s]

4) 119897max

axsa asymp 00286 with(005007) 119897max

plsh asymp 00115 with (003007) and 120578di asymp 00104with (005007)

Although only a sensitivity analysis was performed on themodel the results obtained show quite clearly that this modelcan be used to assess and predict the mechanical frictionlosses occurring in a differential gearbox An experimentalstudy shall be conducted in order to obtain more realisticvalues for some of the parameters such as the frictioncoefficients (120583axsa and 120583sh) which significantly influenceboth the friction torques and the differential gearbox yield

4 Conclusion

In this paper an analytical model is presented for assessingand predicting the mechanical friction losses occurring indifferential gearboxes Some of the parameters involvedin this model can be determined quite easily (geometricparameters) while others which are more delicate dependdirectly on the type of friction occurring in the mechanism(and therefore on the friction coefficients) which affectsthe mechanical losses to a variable extent In order totest the influence of these parameters and determine theability of the model to predict any mechanical losses asensitivity analysis was conducted After this initial numericalapproach an experimental study shall be performed in orderto obtainmore realistic values for some of the parameters andconfirm some of the assumptionsmade here in themodellingprocedure

The Scientific World Journal 11

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is greatly indebted to Dr J-L Ligier for helpfuldiscussions comments and advices The author would liketo thank also Dr Jessica Blanc for her help with this paper

References

[1] S C Lee and H S Cheng ldquoScuffing theory modeling andexperimental correlationsrdquo Journal of Tribology vol 113 no 2pp 327ndash334 1991

[2] D Dowson G Dalmaz Childs M T H C Godet and C MTaylorThin Films in Tribology Elsevier Science 1993

[3] J C Enthoven P M Cann and H A Spikes ldquoTemperatureand scuffingrdquoTribology Transactions vol 36 no 2 pp 258ndash2661993

[4] D Dowson G Dalmaz T H C Childs and C M TaylorLubricants and Lubrication Elsevier Science 1995

[5] C Taylor L Flamand G Dalmaz et al Elastohydrodynamicsmdashrsquo96 Fundamentals and Applications in Lubrication and TractionElsevier Science 1997

[6] M Priest P Ehret L Flamand et al Lubrication at the Frontierthe Role of the Interface and Surface Layers in the Thin Film andBoundary Regime Elsevier Science 1999

[7] A Cornet and J P Deville Physique et Ingenierie des SurfacesEDP Sciences 1998

[8] W Piekoszewski M Szczerek and W Tuszynski ldquoThe actionof lubricants under extreme pressure conditions in a modifiedrdquoWear vol 249 no 3-4 pp 188ndash193 2001

[9] M Wisniewski M Szczerek and W Tuszynski ldquoThe tempera-tures at scuffing and seizure in a four-ball contactrdquo LubricationScience vol 16 no 3 pp 215ndash227 2004

[10] Q Wang ldquoSeizure failure of journal-bearing conformal con-tactsrdquoWear vol 210 no 1-2 pp 8ndash16 1997

[11] G Spinnler Conception Des MachinesmdashPrincipes Et Applica-tions vol 1 Presses Polytechniques et Universitaires Romandes2002

[12] L Flamand Fatigue Des Surface Techniques De LrsquoIngenieur1993

[13] J L Ligier Materiaux Pour Palier Lisses Techniques DeLrsquoIngenieur 1995

[14] J L Ligier Avaries En Lubrification Ecole Nationales SuperieureDu Petrole Et Des Moteurs Editions Technip 2007

[15] M Aublin R Boncompain M Boulaton et al SystemesMecaniquesmdashTheorie Et Dimensionnement Dunod 2005

[16] J L Ligier and R Gojon ldquoPrediction de lrsquousure et du grippagedrsquoun palier de moteur dieselrdquo Tech Rep SIA 93065 1993

[17] J L Ligier Lubrification des Paliers Moteurs Ecole NationalesSuperieure du Petrole et des Moteurs Editions Technip 1997

[18] S Picard Fonctionnement et Maintenance Du VehiculemdashTransmission amp Freinage Delta Press 1994

[19] J L Fanchon Guide des Sciences et Technologies IndustriellesNathan 2001

[20] J HMathieu E Bergmann and R GrasAnalyse et Technologiedes Surfaces Couches Minces et Tribologie Presses Polytech-niques et Universitaires Romandes Traite desMateriaux Tome2003

[21] V N Constantinescu Laminar Viscous Flow Mechanical Engi-neering Series Springer 1995

[22] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part IWithout consideration of contact adhering layer in the inletzonerdquo Journal of Molecular Liquids vol 130 no 1ndash3 pp 88ndash942007

[23] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part II-Considering the contact adhering layer effect in the inlet zonerdquoJournal of Molecular Liquids vol 130 no 1ndash3 pp 95ndash103 2007

[24] S Li and A Kahraman ldquoA fatigue model for contacts undermixed elastohydrodynamic lubrication conditionrdquo Interna-tional Journal of Fatigue vol 33 no 3 pp 427ndash436 2011

[25] K L Johnson J A Greenwood and S Y Poon ldquoA simple theoryof asperity contact in elastohydro-dynamic lubricationrdquo Wearvol 19 no 1 pp 91ndash108 1972

[26] G Ramsey Elastohydrodynamics Imperial College Press 2001[27] B J Hamrock S R Schmid and B O Jacobson Fundamentals

of Fluid Film Lubrication CRC Press 2004[28] K L Johnson Contact Mechanics Cambridge University Press

1987[29] R Brun Manuel Du Mecanicien et Du Thermicien Editions

Technip 2000[30] P Arques Diagnostic Predictif et Defaillances Des Machines

Theorie Traitement Analyse Reconnaissance Prediction Edi-tions Technip 2007

[31] J Frene D Nicolas B Degueurce D Berthe and M GodetLubrication Hydrodynamique Paliers Et Butees Eyrolles 1990

[32] J Frene ldquoButees et paliers hydrodynamiquerdquo Tech Rep B5055Techniques de lrsquoIngenieur 1995

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2 The Scientific World Journal

Sump

Left axle shaft

Right axle shaft

Figure 1 Differential gearbox (with its sump and the axle shafts)

2 Analysis of the Mechanical Friction LossesOccurring in Differential AutomobileGearboxes Constitutive Equations

21 General Description Generally speaking a differentialgearbox [18] (Figure 1) is a complex device involving gearswhich are connected to output shafts transmitting bothrotation and torque forces In the framework of motorvehicles differential systems of this kind make it possibleto deliver an equal or different speed to all the wheelsdepending on whether the vehicle is taking a straight orcurvilinear course when the vehicle is starting to turnthe inner wheels must rotate more slowly than those inthe outside because they have to cover a smaller distanceduring the same time span Differential systems consist offour bevel gears (two ldquoplanetaryrdquo pinions and two ldquosatelliterdquopinions) a satellite-carrier axis and a plastic shell (Figure 2)all these components are enclosed in a nonsymmetric sumpentrained by the differential ring gear which is itself drivenin a rotating manner by both the primary and secondarygearbox shafts which are made to rotate by the engine ofthe vehicle The two planetary bevel pinions connected toeach axle shaft placed opposite each other are meshed by thetwo smaller satellite pinions (Figure 3) When the vehicle ismoving straight ahead the two planetary pinions rotate atthe same speed (as does the axle shaft associated with each ofthe driven wheel) and the two satellite pinions are stationaryrelative to their axis that is the differential housing and these

pinions rotate at the same speed as the ring gear Howeverwhen the vehicle turns the rotational speed of the wheel (andthat of the corresponding planetary pinions) differs from thatof the two satellite pinions which start rotating around theiraxis in order to compensate for this difference

22 Power and Yield Equations The total gearbox yield 120578gbis obtained by dividing the sum of the output powers of eachof the driven wheels by the gearboxrsquos input power

120578gb =119875lw + 119875rw119875en

=119862lw120596lw + 119862rw120596rw

119862en120596en (1)

where 119875119894 119862119894 and 120596

119894(with 119894 = (lw rw en)) are powers

torques and angular velocities associated with the left wheelright wheel and engine respectively

Upon introducing both the primary and secondary gear-box shaft yields the ratio between the power of the differentialring gear and that of the engine is equal to the product of theseshaft yields that is

120578ps120578ss =119875rg

119875en=119862rg120596rg

119862en120596en (2)

where 120578ps and 120578ss denote the primary and secondary shaftyields respectively and 119875rg 119862rg and 120596rg are the power torqueand angular velocity associated with the differential ring gear(drive gear) respectively

On the other hand the total gearbox yield can be writtenas the product of the primary and secondary shaft yields andthe differential gearbox

120578gb = 120578ps120578ss120578di (3)

After combining (1)ndash(3) the total differential gearboxyield 120578di can be obtained by dividing the sum of the outputpowers of each of the driven wheels by the input power

120578di =119875lw + 119875rw119875rg

=119862lw120596lw + 119862rw120596rw

119862rg120596rg (4)

Since a differential gearbox can both transmit and dis-tribute the differential ring gear power (the input power) tothe output gears associated with each of the driven wheels itfollows that

119875rg = 119875lw + 119875rw + 119875ab (5)

where 119875ab is the power absorbed in the differential gearboxdue to the friction losses

In the equations governing the angular velocities and thetorques in the differential gearbox (Figure 4)

120596lw119877119901 = 120596rg119877119901 minus 120596119904119877119904 (6a)

120596rw119877119901 = 120596rg119877119901 + 120596119904119877119904 (6b)

119862rg = 119862lw + 119862rw (7)

The Scientific World Journal 3

(a) (b)

Figure 2 (a) Sump of a differential gearbox (b) Various components of a differential mechanism

Satellite

Satellite

Left wheel planetary Right wheel planetary

Satellite-carrier axis

Left axle shaft

Right axle shaft

Plastic shell

Figure 3 Components of a differential gearbox two planetary bevelpinions two satellite bevel pinions with their shaft and the plasticshell

where 120596119904denotes the angular velocity of satellite pinion and

119877119901and119877

119904are the pitch radii of planetary and satellite pinions

respectivelyAdding or subtracting (6a)-(6b) we obtain the classical

relations

120596rg =120596lw + 120596rw

2 (8a)

120596lw120596rw

120596rg

Clw Crw

Crg

120596s

120596s

Figure 4 Angular velocities and torques in the differential gearbox

120596119904=120596rw minus 120596lw

2

119877119901

119877119904

(8b)

Note that from now on Δ120596 = |120596lw minus 120596rw| ge 0 andΔ119862 = |119862rw minus 119862lw| ge 0 will be used to denote the differencein the angular velocity and torque between the driven wheelsrespectively (where | sdot | is the absolute-value function)

Combining (5) and (6a) (6b) and (7) the powerabsorbed in the differential can be written as

119875ab =1

2Δ119862Δ120596 (9)

Based on the above equations the total differential gear-box yield is

120578di = 1 minus119875ab119875rg

= 1 minus (Δ119862Δ120596

2119862rg120596rg) (10)

4 The Scientific World Journal

Comments

(i) Under straight driving conditions the angular veloc-ities of two satellite pinions in relation to the satelliteaxis are zero (Δ120596 = 0) and the speed of thedriven wheels is equal to the angular velocity of thedifferential ring gear that is 120596lw = 120596rw = 120596rg

(ii) In curvilinear driving situations the angular velocityof two satellite pinions in relation to the satellite-carrier axis is no longer zero (Δ120596 gt 0) and the speedsof the driven wheels are no longer equal that is 120596lw gt120596rw (right turn) or 120596lw lt 120596rw (left turn)

(iii) If one of the drivenwheels undergoes slipping [18]mdashifit is the left (resp right) wheel 120596lw = 0 (resp 120596rw =0) that is Δ120596 = 2120596rgmdashthe differential gearbox yieldreduces to 120578di = 1 minus (Δ119862119862rg)

23 Mechanical Losses Friction Torques Thepower absorbedin the differential gearbox is the sum of the power dissipatedat the various contact points existing in the overall mecha-nism which can be decomposed as follows

119875ab = 119875sapl + 119875axsa + 119875sash + 119875plsh (11)

where 119875sapl 119875axsa 119875sash and 119875plsh correspond to the part ofthe power dissipated between the two satellite and planetarybevel pinions the satellite pinion and the satellite-carrieraxis between the two bevel heads of the satellite pinions andthe surrounding plastic shell and the two bevel heads ofthe planetary pinions and the surrounding plastic shell (seeFigure 3)

231 Power Dissipated between Two Satellite Bevel Pinionsand Two Planetary Bevel Pinions In order to account forthe power dissipated between the two satellite bevel pinionsand the two planetary bevel pinions in the differentialmechanism (giving four meshing contacts) we introducea yield 120578sapl (denoting the power transmitted in the gearbetween satelliteplanetary pinions) which can be written asfollows

120578sapl =119875rg minus 119875sapl

119875rg119867+(Δ120596) (12)

where 119875sapl denotes the power dissipated between the twosatelliteplanetary contacts and 119867+(sdot) is a Heaviside-likefunction (adapted in order to ensure that 119867+(119909) = 1 when119909 gt 0 and 119867(119909) = 0 when 119909 le 0) Note that a yield 120578saplwith a nonzero value (ie 120578sapl gt 0) occurs between thetwo satellite pinions and two planetary pinions only whenthe angular velocity differs between the driven wheels thatis Δ120596 gt 0

Based on (12) we can therefore write

119875sapl = (1 minus 120578sapl) 119862rg120596rg119867+(Δ120596) (13)

232 Power Dissipated between the Satellite Bevel Pinionand Satellite-Carrier Axis Neglecting the presence of an

over-centre mechanism between the satellite pinion andsatellite-carrier axis (Figure 5) the force exerted on thesatellite-carrier axis 119865ax upon the rotation of the differentialring gear (120596rg) can be written as

119865ax =119862rg

2119877119901

(14)

In (14) the point of application of the force 119865ax isassumed to take the same contact path as both the left andright meshing gears (Figure 7)

Adopting aCoulomb-type friction law the friction torqueapplied to the satellite-carrier axis 119862119891axsa is

119862119891

axsa = 120583axsa119865ax119889ax2 (15)

where 120583axsa denotes the friction coefficient (which is con-stant) between the satellite pinion and the satellite-carrier axisand 119889ax is the axis diameter (Figure 6)

Combining (14) and (15) the power dissipated betweenthese two assembly components reads

119875axsa = 2119862119891

axsa120596119904 = 120583axsa119862rg

4

119889ax119877119904

Δ120596 (16)

233 Power Dissipated between the Satellite Bevel Pinions andthe Plastic Shell The power dissipated between the two bevelsatellite pinions and the surrounding plastic shell 119875sash canbe written as

119875sash = 2119862119891

sash120596119904 = 119862119891

sash

119877119901

119877119904

Δ120596 (17)

where119862119891sash denotes the friction torque applied to the satellitebevel pinion Note that the friction torque 119862119891sash is assumedhere to be identical on the two satellite pinions

In order to write the equations giving the equilibrium ofeach satellite pinion which is meshed with the two planetarypinions associated with the driven wheels and in contact withthe satellite-carrier axis as well as with the plastic shell we usethe Fundamental Principle of Statics119879lwrarr119878119872 + 119879rwrarr119878119872 + 119879axrarr119878119872 + 119879shrarr119878119872 = 0

(18)where 119879

119894rarr 119878119872denotes the transmittable force torsor of solid

119894 (and 119894 = (lw rw ax sh) associated with the planetarypinions of the left and right wheels the satellite-carrier axisand the plastic shell resp) exerted on solid 119878 (the satellitepinion) at point119872 (see [15])

Looking only at the equilibrium of the resulting force ofeach torsor in the direction (119909 119910 119911) (see Figure 7) (18) writes

119865119905

lw119865119903

lw119865119886

lw

+

119865119905

rwminus119865119903

rw119865119886

rw

+

minus119865ax0

0

+

0

0

minus119883sa

=

0

0

0

(19)

lArrrArr

119865119905

lw + 119865119905

rw = 119865ax119865119903

lw = 119865119903

rw(119865119905

lw + 119865119905

rw) tan120572 sin 120575 = 119883sa

(20)

The Scientific World Journal 5

120595

Figure 5 Over-centre mechanism between the satellite pinion andsatellite-carrier axis 120595 denotes the angular eccentricity

2Rp

dax

Rs

Figure 6 Satellite bevel pinions and satellite-carrier axis

where 119865119905119895 119865119903119895 and 119865119886

119895(with 119895 = lw rw) are the tangential

radial and axial forces of 119895-planetary pinions (associatedwiththe left and rightwheels Figure 7 see [19]) respectively119883sa isthe resulting force applied to the satellite pinion generated bythe plastic shell 120575 is the half-angle pitch radius of the satellitepinion (see Figure 8(a)) and 120572 is the pressure angle betweenthe satellite and planetary pinions which is assumed hereto be identical between the satelliteplanetary meshing gears(see Figure 8(b))

y

x

z

y

x

z

Fax

Fax

Frlw

Falw Fa

rw

Frrw

Frlw Fr

rw

Ftlw Ft

rw

Psphsa

Psphsa

Figure 7 Forces applied to a satellite pinion

The pressure applied to the head surface of the satellitepinion 119875sphsa (its spherical part see Figure 9) due to the result-ing force119883sa (in line with (20)) matches the relationship (see[15])

119883sa = int2120587

0

int

1205751

1205750

119875sphsa (120579 120601) 119877

2

sa cos 120579 sin 120579 119889120579 119889120601 (21)

where 1205751and 120575

0are the angles defining the spherical part of

the head-satellite pinion and 119877sa is the radius of the innersphere of the plastic shell (where the contact with the headsurface of the satellite pinion and the plastic shell occurs)

The expression for the friction torque119862119891sash applied to thehead surface of the satellite pinion can be written (see [17]) as

119862119891

sash = int2120587

0

int

1205751

1205750

120591sphsa (120579 120601) 119877

3

sa sin 120579 119889120579 119889120601

= int

2120587

0

int

1205751

1205750

120583sash119875sphsa (120579 120601) 119877

3

sa sin 120579 119889120579 119889120601

(22)

where 120591sphsa (120579 120601) = 120583sash 119875sphsa (120579 120601) denotes the shear stress

exerted on the head surface of the satellite pinion since aCoulomb-type friction law is adopted (see Figure 9) and120583sash is the coefficient of friction (which is constant) betweenthe satellite pinion and the plastic shell

6 The Scientific World Journal

120575

1205751

1205750Rs

Rsa

Rsa

Rp

dax

1205759984000

1205752

120575

(a)

120572

(b)

Figure 8 Geometry of a differential gearbox

Psphj

120591sphj

Figure 9 Pressure 119875sph119895

(solid arrow) and shear 120591sph119895

(dashed arrow)fields on a head-pinion 119895 (with 119895 = (sa pl)) caused by the plasticshell

Combining (14) and (20)ndash(22) and assuming that ina first approximation the pressure field exerted on the

head-satellite pinion is uniform (ie 119875sphsa (120579 120601) = 119875sphsa ) (21)

and (22) reduce to

119883sa = 21205871198772

sa119875sphsa (

cos (21205750) minus cos (2120575

1)

4)

119862119891

sash = 120583sash119877sa119862rg

2119877119901

tan120572 sin 120575

times (cos (120575

1+ 1205750) sin (120575

1minus 1205750) minus (120575

1minus 1205750)

sin (1205750+ 1205751) sin (120575

0minus 1205751)

)

(23)

234 Power Dissipated between the Planetary Bevel Pinionsand the Plastic Shell The power dissipated between the twoplanetary pinions and the plastic shell 119875plsh can be writtenas

119875plsh = 119862119891(lw)plsh 120596

rslw + 119862

119891(rw)plsh 120596

rsrw (24)

where119862119891(119894)plsh are the frictional torque between the planetary 119894-pinion (119894 = (lw rw) refers to the left wheel (lw) and the rightwheel (rw)) and the plastic shell and 120596rs

119894= Δ1205962 denotes

the relative speed of the 119894-pinion in relation to the sumpdifferential

Assuming that 119862119891(lw)plsh cong 119862119891(rw)plsh = 119862

119891

plsh (24) reads

119875plsh = 119862119891

plshΔ120596 (25)

Using the sameprocedure as above namelymeshing eachof the planetary pinions with two satellite pinions which

The Scientific World Journal 7

means that the resulting force applied to the head-planetarypinion by the plastic shell119883pl is

119883pl = 119865ax tan120572 sin 120575 (26)

where 120575 is the half-angle pitch radius of the planetary pinion(see Figure 8(a)) Note that given the particular geometry ofthe mechanism the sum of the half-angle pitch radii of theplanetary and satellite pinions satisfies 120575 + 120575 = 1205872 and wetherefore obtain the following relationship sin 120575 = cos 120575

The expressions for both the resulting force 119883pl and thefriction torque119862119891plsh applied to the head-satellite pinion are

119883pl = int2120587

0

int

1205752

1205751015840

0

119875sphpl (120579 120601) 119877

2

sa cos 120579 sin 120579 119889120579 119889120601

119862119891

plsh = int2120587

0

int

1205752

1205751015840

0

120591sphpl (120579 120601) 119877

3

sa sin 120579 119889120579 119889120601

(27)

where 120591sphpl (120579 120601) = 120583plsh119875sphpl (120579 120601) denotes the shear stress

exerted on the outer surface of the planetary pinion with aCoulomb-type friction law (Figure 9) 120583plsh is the coefficientof friction (which is constant) between the satellite pinionand the plastic shell 119875sphpl is the pressure applied to the headsurface of the planetary pinion and 120575

2and 1205751015840

0are the angles

defining the spherical portion of the head-planetary pinionIt is again assumed here that in a first approximation the

pressure field 119875sphpl exerted on the head-planetary pinion isuniform (ie (120579 120601)-independent) and (27) reduce to

119883pl = 21205871198772

sa119875sphpl (

cos (212057510158400) minus cos (2120575

2)

4)

119862119891

plsh = 120583plsh119877sa119862rg

2119877119901

tan120572 cos 120575

times (cos (120575

2+ 1205751015840

0) sin (120575

2minus 1205751015840

0) minus (120575

2minus 1205751015840

0)

sin (12057510158400+ 1205752) sin (1205751015840

0minus 1205752)

)

(28)

235 Total Power Dissipated in the Differential GearboxBased on the above constitutive equations the total powerdissipated in the differential gearbox can bewritten as follows

119875ab = 119875sapl + 119875axsa + 119875sash + 119875plsh

= (1 minus 120578sapl) 119862rg120596rg119867+(Δ120596) + 120583axsa

119862rg

4

119889ax119877119904

Δ120596

+ 120583sash119877sa119862rg

2119877119904

tan120572 sin 120575

times (cos (120575

1+ 1205750) sin (120575

1minus 1205750) minus (120575

1minus 1205750)

sin (1205750+ 1205751) sin (120575

0minus 1205751)

)Δ120596

+ 120583plsh119877sa119862rg

2119877119901

tan120572 cos 120575

times (cos (120575

2+ 1205751015840

0) sin (120575

2minus 1205751015840

0) minus (120575

2minus 1205751015840

0)

sin (12057510158400+ 1205752) sin (1205751015840

0minus 1205752)

)Δ120596

(29)

Using (10) and (29) and assuming that in a first approx-imation 120583sash = 120583plsh = 120583sh (where 120583sh denotes thecoefficient of friction between each of the pinions in contactwith the plastic shell) the total differential gearbox yield canbe written as follows

120578di = 1 minussum119894

119897119894

= 1 minus

[[[[

[

119875sapl

119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897sapl

+119875axsa119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897axsa

+119875sash119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897sash

+119875plsh

119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897plsh

]]]]

]

(30)

with

119897sapl = (1 minus 120578sapl)119867+(Δ120596)

119897axsa = 120583axsa119889ax4119877119904

Δ120596

120596rg

119897sash = 120583sh119877sa2119877119904

tan120572 sin 120575

times (cos (120575

1+ 1205750) sin (120575

1minus 1205750) minus (120575

1minus 1205750)

sin (1205750+ 1205751) sin (120575

0minus 1205751)

)Δ120596

120596rg

119897plsh = 120583sh119877sa2119877119901

tan120572 cos 120575

times (cos (120575

2+ 1205751015840

0) sin (120575

2minus 1205751015840

0) minus (120575

2minus 1205751015840

0)

sin (12057510158400+ 1205752) sin (1205751015840

0minus 1205752)

)Δ120596

120596rg

(31)

where 119897119894= 119875119894119875rg denotes the loss ratio of the 119894-mechanism

(with 119894 = (sapl axsa sash plsh)) It is worth noting thatthe differential yield 120578di shows mechanical friction losseswhenever the rotational-speed differs between two of theplanetary pinions associated with the driven wheels that iswhen Δ120596 gt 0

3 Discussions

In the first part of this section we discuss the order of magni-tude of the constitutive parameters of the model presented inSection 2 In the second part we present a sensitivity analysis

8 The Scientific World Journal

0 05 1 15 2085

09

095

1D

iffer

entia

l gea

rbox

yie

ld120578

di

Δ120596120596rg

(a)

0 05 1 15 20

0005

001

0015

002

0025

003

0035

004

PiP

rg

Δ120596120596rg

(b)

Figure 10 (a) Yield of a differential gearbox 120578di as a function of the load parameterΔ120596120596rg for various sets of coefficients of friction (120583axsa 120583sh)(003 005) in a dashed line (005 007) in a dash-dot line (003 007) in a dotted line and in a solid line for (120583axsa 120583sh) and 120578sapl undergoinga concomitant linear increase from (003 005) to (005 007) and 098 to 096 respectively with Δ120596120596rg the thick solid line gives the yieldof the differential gearbox without any losses (which is constant) with the load parameter Δ120596120596rg that is 119897119894 = 119875119894119875rg = 0 forall119894 (b) Loss ratio119897119894= 119875119894119875rg of the 119894-mechanism (with 119894 = (sapl axsa sash plsh)) as a function of the load parameter Δ120596120596rg and with the same sets of

coefficients of friction as in (a) 119897sapl without (black solid line) or with (black dashed line) a linear decrease in 120578sapl from 098 to 096 119897axsa(red line) 119897sash (blue line) and 119897plsh (green line)

0 005 01 015 020

001

002

003

004

005

006

Cf iC

rg

Coefficient of friction 120583i

Figure 11 Friction torque ratio 119862119894119862rg as a function of coefficient of

friction 120583119894with 119894 = (axsa sash plsh) (while the other parameters

were taken to be constant) 119862axsa119862rg increased linearly with 120583axsa(red solid line) 119862sash119862rg increased linearly with 120583sash = 120583sh (bluesolid line)119862plsh119862rg increased linearly with 120583plsh = 120583sh (green solidline) in a range of variation 120583

119894isin [0 02]

in which the response of the model to a given load parameterwas examined in order to assess the effect of this responseon themechanical losses occurring in the differential gearboxmechanism

31 Order of Magnitude of the Constitutive Parameters Thisanalytical model involves thirteen parameters (120578sapl 120583axsa

0 200 400 600 800 10000

02

04

06

08

1

12

14

16

18

2

Time (s)

Δ120596120596

rg

Figure 12 Example of a load parameterΔ120596120596rg with a complex pathdepending on time 119905 isin [0 1000 s] when Δ120596120596rg = 0 (in the casewhere 120596lw = 0 and 120596rw = 0) then 120578di = 1 and 119897119894 = 119875119894119875rg = 0 forall119894 (whenthe vehiclewas travelling along a straight line) when 0 lt Δ120596120596rg lt 2

then 120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894 (when the vehicle was taking acurvilinear course) when Δ120596120596rg = 2 then 120578di lt 1 and 119897119894 = 119875119894119875rg gt0 forall119894 (when one of the driven wheels underwent slipping the vehiclewas stationary) on the plateau Δ120596120596rg = constant and |120596lw minus120596rw| =

constant (the vehicle was taking a constant curvilinear trajectory)

120583sh 119889ax 119877119904 119877119901 119877sa 120572 120575 1205750 1205751015840

0 1205751 1205752) consisting of (i) ten

geometrical parameters (119889ax 119877119904 119877119901 119877sa 120572 120575 1205750 1205751015840

0 1205751 1205752)

depending on the differential gearbox under consideration

The Scientific World Journal 9

0 200 400 600 800 1000091

092

093

094

095

096

097

098

099

1

Time (s)

Diff

eren

tial g

earb

ox y

ield

120578di

(a)

0 200 400 600 800 10000

0005

001

0015

002

0025

003

0035

Time (s)

PiP

rg

(b)

Figure 13 (a) Yield of a differential gearbox 120578119889119894with time 119905 isin [0 1000 s] in the case of the complex path presented in Figure 12 with various

sets of friction coefficients (120583axsa 120583sh) (003005) in a dashed line (005007) in a dash-dot line and (003007) in a dotted line (b) Lossratio 119897

119894= 119875119894119875rg of the 119894-mechanism (with 119894 = (sapl axsa sash plsh)) with time 119905 isin [0 1000 s] in the case of the complex path presented

in Figure 12 with the same sets of friction coefficients as in (a) 119897sapl = 096 which remained constant with time 119905 (black solid line) 119897axsa (redline) 119897sash (blue line) and 119897plsh (green line)

which can be identified fairly easily (ii) one yield param-eter (120578sapl) corresponding to the power transmitted by thegear between the satelliteplanetary pinions (iii) two fric-tion parameters (120583axsa 120583sh) which depend considerably onthe types of materials in contact and the surface conditionstheir state of lubrication and their temperature

Concerning (i) although these specific parameters de-pend on the differential gearbox under consideration theirorder of magnitude can be said to be 119889ax asymp 10ndash20mm 119877

119904asymp

20ndash30mm 119877119901asymp 20ndash30mm 119877sa asymp 30ndash50mm and for the

angle parameters 1205750asymp 10∘ndash15∘ 1205751015840

0asymp 15∘ndash25∘ 120575

1asymp 30∘ndash40∘

1205752asymp 35∘ndash45∘ 120572 asymp 20ndash25∘ 120575 asymp 30∘ndash40∘ readers can refer to

Fanchon [19] for further details about some of themConcerning (ii) the parameter 120578sapl which denotes the

power transmitted by the gear between satelliteplanetarypinions using a yield term has a value ranging between 092and 098 Since Fanchon [19] has reported that the yield inthe case of two meshed pinions is around 98 it can beconcluded that in a differential mechanism where there arefour pinions in contact the lowest value of 120578sapl asymp (098)

4cong

092 and the highest value is likely to be 120578sapl cong 098 Ittherefore seems reasonable to assume that the real yield 120578saplof this complex mechanism is approximately 096

The parameters that need to be investigated more closelyin order to determine their influence on themechanical lossesare the various coefficients of friction (iii) 120583axsa and 120583shThe sliding contacts in the differential gearbox mechanismcan be of very different kinds [20 21] since both thesatelliteplanetary bevel pinion and satellite-pinionsatellite-carrier axis in contact are in the mixed elastohydrodynamiclubrication (MEHL) mode [22ndash24] which is a mixed regimebetween (1) elastohydrodynamic lubrication (EHL) [25 26]where the heavy loads exerted at the surface of the two gears

(due to the presence of both high contact pressures and strongtransmitted torques) induce elastic strains in the two solidsin contact with thin lubricant films and (2) boundary lubri-cation (BL) [27] which may tend to constitute a completelydry frictional contact [28] leading to the development ofscuffing effects [29 30] The contacts between the bevel headpinion (satellites and planetaries) and the plastic shell arein the elastohydrodynamic lubrication mode (EHL) whichcan lead to the development of the hydrodynamic lubricationregime (HL) [31 32] where the lubricant film thickness islarge enough to completely separate the directly apposedsurfaces so that the loads applied between the surfaces arerestored and balanced under specific operating conditionsUnder the above conditions the contacts existing betweenthe satellite pinions and the satellite-carrier axis are of thesteelsteel contact with lubrication type 120583axsa isin [005 007]and those existing between the head pinions and the plasticshell are rather of the steelTeflon contact with lubricationtype 120583sh isin [003 005]

32 Sensitivity Analysis Assessment of the Modelrsquos Responseto a Given Load Parameter and Prediction of the MechanicalLosses In what follows it is proposed to test the modelrsquosresponse to a given load parameter and to determinethe influence of some parameters (120578sapl 120583axsa 120583sh) on themechanical losses occurring in a differential gearbox mecha-nism The expression for the load parameter Δ120596120596rg isin [0 2]means that (i) when Δ120596120596rg = 0 (in cases where 120596lw = 0 and120596rw = 0) the vehicle is travelling on a straight line 120578di = 1 and119897119894= 119875119894119875rg = 0 forall119894 (ii) when 0 lt Δ120596120596rg lt 2 the vehicle is

taking a curvilinear course 120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894(iii) when Δ120596120596rg = 2 one of the driven wheels undergoes

10 The Scientific World Journal

slipping (the extreme situation) the vehicle is stationary120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894

The values adopted for the other parameters are 119889ax =18 times 10

minus3m 119877119904= 22 times 10

minus3m 119877119901= 27 times 10

minus3m 119877sa =49 times 10

minus3m 1205750= 10∘ 12057510158400= 18∘ 1205751= 32∘ 1205752= 40∘ 120572 = 20∘

120575 = 39∘

Figure 10(a) shows the yield of a differential gearbox 120578didepending on the load parameter Δ120596120596rg with various setsof friction coefficients of friction (120583axsa 120583sh) and a lineardecrease in 120578sapl from 098 to 096 In the most gentle oper-ating range that is 0 le Δ120596120596rg le 1 120578di decreased from 098to around 0963 (120583axsa 120583sh) = (003 005) 120578di isin [098 0959](resp 120578di isin [098 0954]) when (120583axsa 120583sh) = (003 005)(resp (120583axsa 120583sh) = (005 007)) and 120578di decrease from 098to around 0949 when (120583axsa 120583sh) showed a concomitantlinear increase from (003 005) to (005 007) In the heavyoperating range that is 1 lt Δ120596120596rg le 2 the maximumdifferential gearbox yield 120578di isin ]0963 0946] was obtainedwith (120583axsa 120583sh) = (003 005) and the minimum yield(120578di isin ]0949 0909]) occurred when (120583axsa 120583sh) and 120578saplincreased linearly from (003 005) to (005 007) concomi-tantly with the load parameter Δ120596120596rg 120578di isin ]0959 0938](resp 120578di isin ]0954 0929]) when (120583axsa 120583sh) = (003 005)(resp (120583axsa 120583sh) = (005 007)) The results presentedin Figure 10(b) show the loss ratio 119897

119894= 119875119894119875rg associated

with 119894-mechanism (where 119894 = (sapl axsa sash plsh)) indecreasing order of influence 119897axsa (red line) 119897sapl (blackline) 119897plsh (green line) 119897sash (blue line) when 120578sapl had a con-stant value of 098 and in the case where 120578sapl decreased from098 to 096 the effects of the loss ratio 119897sapl became greaterthan those of the other parameters (119897axsa 119897axsa 119897sash 119897plsh)It should be noted that when 0 le Δ120596120596rg le 1 then119897axsa isin [0 00143] 119897plsh isin [0 00057] and 119897sash isin [0 00052]whereas when 1 lt Δ120596120596rg le 2 then 119897axsa isin [0 00286]119897plsh isin [0 00115] and 119897sash isin [0 00104] Figure 11 showsthe friction torque ratio 119862

119894119862rg as a function of the friction

coefficient 120583119894with 119894 = (axsa sash plsh) when the other

parameters were taken to be fixed constants The range ofvariation of the friction coefficient 120583

119894was taken to be between

0 and 02 (this extreme value may correspond to steelsteelcontact without any lubrification) In view of these resultsthe maximum 119862

119894119862rg ratio was obtained with 119862axsa119862rg and

the minimum one with 119862sash119862rg The friction torque ratios119862119894119862rg progressed linearly with the friction coefficient 120583

119894

(where 120583sh = 120583sash = 120583plsh) as predicted by the constitutiveequations in Section 2 Note that the results obtained in thisfirst approximation are likely to differ from those obtained bysimulating nonlinear behaviour between 119862

119894119862rg and 120583119894

A complex path of the load parameter Δ120596120596rg wasthen studied in order to test the ability of the model todescribe the differential gearbox yield 120578di and the loss ratios119897119894= 119875119894119875rg associated with the 119894-mechanism (where 119894 =

(sapl axsa sash plsh)) under more realistic conditionsinvolving a straight line situation (Δ120596120596rg = 0 with 120596lw = 0

and 120596rw = 0) a curved line situation (0 lt Δ120596120596rg lt 2)a constant curved line situation (Δ120596120596rg = constant) orwhere one of the drivenwheels undergoes slipping (Δ120596120596rg =2) Figure 12 shows the evolution of the load parameter

Δ120596120596rg as a function of the time 119905 isin [0 1000 s] Theresults obtained with a complex path in terms of the yieldof the differential gearbox (120578di) and the loss ratios (119897

119894) are

shown in Figure 13 Specifically Figure 13(a) gives the yieldof the differential gearbox 120578di as a function of the time119905 isin [0 1000 s] in the case of the complex path plotted inFigure 12 with various sets of friction coefficients (120583axsa 120583sh)(003005) in dashed line (005007) in dash-dot line and(003007) in dotted line During the whole complex path ofΔ120596120596rg the maximum (resp minimum) yield of differentialgearbox was reached with 120583axsa = 003 and 120583sh = 005

(resp 120583axsa = 005 and 120583sh = 007) On the 119896-plate (with119896 = 1 2 3 4) defined by the 119896-time interval [119905119894 119905119891]

119896 it can be

noted that for example (1) in [400 s 500 s]2 120578di asymp 09774with

(005007) 120578di asymp 09591 with (003007) and 120578di asymp 09547with (005007) (2) in [900 s 950 s]

4 120578di asymp 09464 with

(005007) 120578di asymp 09382 with (003007) and 120578di asymp 09294with (005007) The loss ratios due to friction 119897

119894= 119875119894119875rg are

shown in Figure 13(b) The maximum loss ratios obtained indecreasing order of magnitude (along the loading path) were119897sapl (black line) 119897axsa (red line) 119897plsh (green line) and 119897sash(blue line) except for the last loadwhere 119897axsa could be greaterthan 119897sapl with the pair of friction coefficients (005007)and (003007) It should be noted that (1) on the secondplate ([400 s 500 s]

2) 119897max

axsa asymp 00143 with (005007) 119897maxplsh asymp

00057 with (003007) and 120578di asymp 00052 with (005007)(2) on the fourth plate ([900 s 950 s]

4) 119897max

axsa asymp 00286 with(005007) 119897max

plsh asymp 00115 with (003007) and 120578di asymp 00104with (005007)

Although only a sensitivity analysis was performed on themodel the results obtained show quite clearly that this modelcan be used to assess and predict the mechanical frictionlosses occurring in a differential gearbox An experimentalstudy shall be conducted in order to obtain more realisticvalues for some of the parameters such as the frictioncoefficients (120583axsa and 120583sh) which significantly influenceboth the friction torques and the differential gearbox yield

4 Conclusion

In this paper an analytical model is presented for assessingand predicting the mechanical friction losses occurring indifferential gearboxes Some of the parameters involvedin this model can be determined quite easily (geometricparameters) while others which are more delicate dependdirectly on the type of friction occurring in the mechanism(and therefore on the friction coefficients) which affectsthe mechanical losses to a variable extent In order totest the influence of these parameters and determine theability of the model to predict any mechanical losses asensitivity analysis was conducted After this initial numericalapproach an experimental study shall be performed in orderto obtainmore realistic values for some of the parameters andconfirm some of the assumptionsmade here in themodellingprocedure

The Scientific World Journal 11

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is greatly indebted to Dr J-L Ligier for helpfuldiscussions comments and advices The author would liketo thank also Dr Jessica Blanc for her help with this paper

References

[1] S C Lee and H S Cheng ldquoScuffing theory modeling andexperimental correlationsrdquo Journal of Tribology vol 113 no 2pp 327ndash334 1991

[2] D Dowson G Dalmaz Childs M T H C Godet and C MTaylorThin Films in Tribology Elsevier Science 1993

[3] J C Enthoven P M Cann and H A Spikes ldquoTemperatureand scuffingrdquoTribology Transactions vol 36 no 2 pp 258ndash2661993

[4] D Dowson G Dalmaz T H C Childs and C M TaylorLubricants and Lubrication Elsevier Science 1995

[5] C Taylor L Flamand G Dalmaz et al Elastohydrodynamicsmdashrsquo96 Fundamentals and Applications in Lubrication and TractionElsevier Science 1997

[6] M Priest P Ehret L Flamand et al Lubrication at the Frontierthe Role of the Interface and Surface Layers in the Thin Film andBoundary Regime Elsevier Science 1999

[7] A Cornet and J P Deville Physique et Ingenierie des SurfacesEDP Sciences 1998

[8] W Piekoszewski M Szczerek and W Tuszynski ldquoThe actionof lubricants under extreme pressure conditions in a modifiedrdquoWear vol 249 no 3-4 pp 188ndash193 2001

[9] M Wisniewski M Szczerek and W Tuszynski ldquoThe tempera-tures at scuffing and seizure in a four-ball contactrdquo LubricationScience vol 16 no 3 pp 215ndash227 2004

[10] Q Wang ldquoSeizure failure of journal-bearing conformal con-tactsrdquoWear vol 210 no 1-2 pp 8ndash16 1997

[11] G Spinnler Conception Des MachinesmdashPrincipes Et Applica-tions vol 1 Presses Polytechniques et Universitaires Romandes2002

[12] L Flamand Fatigue Des Surface Techniques De LrsquoIngenieur1993

[13] J L Ligier Materiaux Pour Palier Lisses Techniques DeLrsquoIngenieur 1995

[14] J L Ligier Avaries En Lubrification Ecole Nationales SuperieureDu Petrole Et Des Moteurs Editions Technip 2007

[15] M Aublin R Boncompain M Boulaton et al SystemesMecaniquesmdashTheorie Et Dimensionnement Dunod 2005

[16] J L Ligier and R Gojon ldquoPrediction de lrsquousure et du grippagedrsquoun palier de moteur dieselrdquo Tech Rep SIA 93065 1993

[17] J L Ligier Lubrification des Paliers Moteurs Ecole NationalesSuperieure du Petrole et des Moteurs Editions Technip 1997

[18] S Picard Fonctionnement et Maintenance Du VehiculemdashTransmission amp Freinage Delta Press 1994

[19] J L Fanchon Guide des Sciences et Technologies IndustriellesNathan 2001

[20] J HMathieu E Bergmann and R GrasAnalyse et Technologiedes Surfaces Couches Minces et Tribologie Presses Polytech-niques et Universitaires Romandes Traite desMateriaux Tome2003

[21] V N Constantinescu Laminar Viscous Flow Mechanical Engi-neering Series Springer 1995

[22] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part IWithout consideration of contact adhering layer in the inletzonerdquo Journal of Molecular Liquids vol 130 no 1ndash3 pp 88ndash942007

[23] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part II-Considering the contact adhering layer effect in the inlet zonerdquoJournal of Molecular Liquids vol 130 no 1ndash3 pp 95ndash103 2007

[24] S Li and A Kahraman ldquoA fatigue model for contacts undermixed elastohydrodynamic lubrication conditionrdquo Interna-tional Journal of Fatigue vol 33 no 3 pp 427ndash436 2011

[25] K L Johnson J A Greenwood and S Y Poon ldquoA simple theoryof asperity contact in elastohydro-dynamic lubricationrdquo Wearvol 19 no 1 pp 91ndash108 1972

[26] G Ramsey Elastohydrodynamics Imperial College Press 2001[27] B J Hamrock S R Schmid and B O Jacobson Fundamentals

of Fluid Film Lubrication CRC Press 2004[28] K L Johnson Contact Mechanics Cambridge University Press

1987[29] R Brun Manuel Du Mecanicien et Du Thermicien Editions

Technip 2000[30] P Arques Diagnostic Predictif et Defaillances Des Machines

Theorie Traitement Analyse Reconnaissance Prediction Edi-tions Technip 2007

[31] J Frene D Nicolas B Degueurce D Berthe and M GodetLubrication Hydrodynamique Paliers Et Butees Eyrolles 1990

[32] J Frene ldquoButees et paliers hydrodynamiquerdquo Tech Rep B5055Techniques de lrsquoIngenieur 1995

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The Scientific World Journal 3

(a) (b)

Figure 2 (a) Sump of a differential gearbox (b) Various components of a differential mechanism

Satellite

Satellite

Left wheel planetary Right wheel planetary

Satellite-carrier axis

Left axle shaft

Right axle shaft

Plastic shell

Figure 3 Components of a differential gearbox two planetary bevelpinions two satellite bevel pinions with their shaft and the plasticshell

where 120596119904denotes the angular velocity of satellite pinion and

119877119901and119877

119904are the pitch radii of planetary and satellite pinions

respectivelyAdding or subtracting (6a)-(6b) we obtain the classical

relations

120596rg =120596lw + 120596rw

2 (8a)

120596lw120596rw

120596rg

Clw Crw

Crg

120596s

120596s

Figure 4 Angular velocities and torques in the differential gearbox

120596119904=120596rw minus 120596lw

2

119877119901

119877119904

(8b)

Note that from now on Δ120596 = |120596lw minus 120596rw| ge 0 andΔ119862 = |119862rw minus 119862lw| ge 0 will be used to denote the differencein the angular velocity and torque between the driven wheelsrespectively (where | sdot | is the absolute-value function)

Combining (5) and (6a) (6b) and (7) the powerabsorbed in the differential can be written as

119875ab =1

2Δ119862Δ120596 (9)

Based on the above equations the total differential gear-box yield is

120578di = 1 minus119875ab119875rg

= 1 minus (Δ119862Δ120596

2119862rg120596rg) (10)

4 The Scientific World Journal

Comments

(i) Under straight driving conditions the angular veloc-ities of two satellite pinions in relation to the satelliteaxis are zero (Δ120596 = 0) and the speed of thedriven wheels is equal to the angular velocity of thedifferential ring gear that is 120596lw = 120596rw = 120596rg

(ii) In curvilinear driving situations the angular velocityof two satellite pinions in relation to the satellite-carrier axis is no longer zero (Δ120596 gt 0) and the speedsof the driven wheels are no longer equal that is 120596lw gt120596rw (right turn) or 120596lw lt 120596rw (left turn)

(iii) If one of the drivenwheels undergoes slipping [18]mdashifit is the left (resp right) wheel 120596lw = 0 (resp 120596rw =0) that is Δ120596 = 2120596rgmdashthe differential gearbox yieldreduces to 120578di = 1 minus (Δ119862119862rg)

23 Mechanical Losses Friction Torques Thepower absorbedin the differential gearbox is the sum of the power dissipatedat the various contact points existing in the overall mecha-nism which can be decomposed as follows

119875ab = 119875sapl + 119875axsa + 119875sash + 119875plsh (11)

where 119875sapl 119875axsa 119875sash and 119875plsh correspond to the part ofthe power dissipated between the two satellite and planetarybevel pinions the satellite pinion and the satellite-carrieraxis between the two bevel heads of the satellite pinions andthe surrounding plastic shell and the two bevel heads ofthe planetary pinions and the surrounding plastic shell (seeFigure 3)

231 Power Dissipated between Two Satellite Bevel Pinionsand Two Planetary Bevel Pinions In order to account forthe power dissipated between the two satellite bevel pinionsand the two planetary bevel pinions in the differentialmechanism (giving four meshing contacts) we introducea yield 120578sapl (denoting the power transmitted in the gearbetween satelliteplanetary pinions) which can be written asfollows

120578sapl =119875rg minus 119875sapl

119875rg119867+(Δ120596) (12)

where 119875sapl denotes the power dissipated between the twosatelliteplanetary contacts and 119867+(sdot) is a Heaviside-likefunction (adapted in order to ensure that 119867+(119909) = 1 when119909 gt 0 and 119867(119909) = 0 when 119909 le 0) Note that a yield 120578saplwith a nonzero value (ie 120578sapl gt 0) occurs between thetwo satellite pinions and two planetary pinions only whenthe angular velocity differs between the driven wheels thatis Δ120596 gt 0

Based on (12) we can therefore write

119875sapl = (1 minus 120578sapl) 119862rg120596rg119867+(Δ120596) (13)

232 Power Dissipated between the Satellite Bevel Pinionand Satellite-Carrier Axis Neglecting the presence of an

over-centre mechanism between the satellite pinion andsatellite-carrier axis (Figure 5) the force exerted on thesatellite-carrier axis 119865ax upon the rotation of the differentialring gear (120596rg) can be written as

119865ax =119862rg

2119877119901

(14)

In (14) the point of application of the force 119865ax isassumed to take the same contact path as both the left andright meshing gears (Figure 7)

Adopting aCoulomb-type friction law the friction torqueapplied to the satellite-carrier axis 119862119891axsa is

119862119891

axsa = 120583axsa119865ax119889ax2 (15)

where 120583axsa denotes the friction coefficient (which is con-stant) between the satellite pinion and the satellite-carrier axisand 119889ax is the axis diameter (Figure 6)

Combining (14) and (15) the power dissipated betweenthese two assembly components reads

119875axsa = 2119862119891

axsa120596119904 = 120583axsa119862rg

4

119889ax119877119904

Δ120596 (16)

233 Power Dissipated between the Satellite Bevel Pinions andthe Plastic Shell The power dissipated between the two bevelsatellite pinions and the surrounding plastic shell 119875sash canbe written as

119875sash = 2119862119891

sash120596119904 = 119862119891

sash

119877119901

119877119904

Δ120596 (17)

where119862119891sash denotes the friction torque applied to the satellitebevel pinion Note that the friction torque 119862119891sash is assumedhere to be identical on the two satellite pinions

In order to write the equations giving the equilibrium ofeach satellite pinion which is meshed with the two planetarypinions associated with the driven wheels and in contact withthe satellite-carrier axis as well as with the plastic shell we usethe Fundamental Principle of Statics119879lwrarr119878119872 + 119879rwrarr119878119872 + 119879axrarr119878119872 + 119879shrarr119878119872 = 0

(18)where 119879

119894rarr 119878119872denotes the transmittable force torsor of solid

119894 (and 119894 = (lw rw ax sh) associated with the planetarypinions of the left and right wheels the satellite-carrier axisand the plastic shell resp) exerted on solid 119878 (the satellitepinion) at point119872 (see [15])

Looking only at the equilibrium of the resulting force ofeach torsor in the direction (119909 119910 119911) (see Figure 7) (18) writes

119865119905

lw119865119903

lw119865119886

lw

+

119865119905

rwminus119865119903

rw119865119886

rw

+

minus119865ax0

0

+

0

0

minus119883sa

=

0

0

0

(19)

lArrrArr

119865119905

lw + 119865119905

rw = 119865ax119865119903

lw = 119865119903

rw(119865119905

lw + 119865119905

rw) tan120572 sin 120575 = 119883sa

(20)

The Scientific World Journal 5

120595

Figure 5 Over-centre mechanism between the satellite pinion andsatellite-carrier axis 120595 denotes the angular eccentricity

2Rp

dax

Rs

Figure 6 Satellite bevel pinions and satellite-carrier axis

where 119865119905119895 119865119903119895 and 119865119886

119895(with 119895 = lw rw) are the tangential

radial and axial forces of 119895-planetary pinions (associatedwiththe left and rightwheels Figure 7 see [19]) respectively119883sa isthe resulting force applied to the satellite pinion generated bythe plastic shell 120575 is the half-angle pitch radius of the satellitepinion (see Figure 8(a)) and 120572 is the pressure angle betweenthe satellite and planetary pinions which is assumed hereto be identical between the satelliteplanetary meshing gears(see Figure 8(b))

y

x

z

y

x

z

Fax

Fax

Frlw

Falw Fa

rw

Frrw

Frlw Fr

rw

Ftlw Ft

rw

Psphsa

Psphsa

Figure 7 Forces applied to a satellite pinion

The pressure applied to the head surface of the satellitepinion 119875sphsa (its spherical part see Figure 9) due to the result-ing force119883sa (in line with (20)) matches the relationship (see[15])

119883sa = int2120587

0

int

1205751

1205750

119875sphsa (120579 120601) 119877

2

sa cos 120579 sin 120579 119889120579 119889120601 (21)

where 1205751and 120575

0are the angles defining the spherical part of

the head-satellite pinion and 119877sa is the radius of the innersphere of the plastic shell (where the contact with the headsurface of the satellite pinion and the plastic shell occurs)

The expression for the friction torque119862119891sash applied to thehead surface of the satellite pinion can be written (see [17]) as

119862119891

sash = int2120587

0

int

1205751

1205750

120591sphsa (120579 120601) 119877

3

sa sin 120579 119889120579 119889120601

= int

2120587

0

int

1205751

1205750

120583sash119875sphsa (120579 120601) 119877

3

sa sin 120579 119889120579 119889120601

(22)

where 120591sphsa (120579 120601) = 120583sash 119875sphsa (120579 120601) denotes the shear stress

exerted on the head surface of the satellite pinion since aCoulomb-type friction law is adopted (see Figure 9) and120583sash is the coefficient of friction (which is constant) betweenthe satellite pinion and the plastic shell

6 The Scientific World Journal

120575

1205751

1205750Rs

Rsa

Rsa

Rp

dax

1205759984000

1205752

120575

(a)

120572

(b)

Figure 8 Geometry of a differential gearbox

Psphj

120591sphj

Figure 9 Pressure 119875sph119895

(solid arrow) and shear 120591sph119895

(dashed arrow)fields on a head-pinion 119895 (with 119895 = (sa pl)) caused by the plasticshell

Combining (14) and (20)ndash(22) and assuming that ina first approximation the pressure field exerted on the

head-satellite pinion is uniform (ie 119875sphsa (120579 120601) = 119875sphsa ) (21)

and (22) reduce to

119883sa = 21205871198772

sa119875sphsa (

cos (21205750) minus cos (2120575

1)

4)

119862119891

sash = 120583sash119877sa119862rg

2119877119901

tan120572 sin 120575

times (cos (120575

1+ 1205750) sin (120575

1minus 1205750) minus (120575

1minus 1205750)

sin (1205750+ 1205751) sin (120575

0minus 1205751)

)

(23)

234 Power Dissipated between the Planetary Bevel Pinionsand the Plastic Shell The power dissipated between the twoplanetary pinions and the plastic shell 119875plsh can be writtenas

119875plsh = 119862119891(lw)plsh 120596

rslw + 119862

119891(rw)plsh 120596

rsrw (24)

where119862119891(119894)plsh are the frictional torque between the planetary 119894-pinion (119894 = (lw rw) refers to the left wheel (lw) and the rightwheel (rw)) and the plastic shell and 120596rs

119894= Δ1205962 denotes

the relative speed of the 119894-pinion in relation to the sumpdifferential

Assuming that 119862119891(lw)plsh cong 119862119891(rw)plsh = 119862

119891

plsh (24) reads

119875plsh = 119862119891

plshΔ120596 (25)

Using the sameprocedure as above namelymeshing eachof the planetary pinions with two satellite pinions which

The Scientific World Journal 7

means that the resulting force applied to the head-planetarypinion by the plastic shell119883pl is

119883pl = 119865ax tan120572 sin 120575 (26)

where 120575 is the half-angle pitch radius of the planetary pinion(see Figure 8(a)) Note that given the particular geometry ofthe mechanism the sum of the half-angle pitch radii of theplanetary and satellite pinions satisfies 120575 + 120575 = 1205872 and wetherefore obtain the following relationship sin 120575 = cos 120575

The expressions for both the resulting force 119883pl and thefriction torque119862119891plsh applied to the head-satellite pinion are

119883pl = int2120587

0

int

1205752

1205751015840

0

119875sphpl (120579 120601) 119877

2

sa cos 120579 sin 120579 119889120579 119889120601

119862119891

plsh = int2120587

0

int

1205752

1205751015840

0

120591sphpl (120579 120601) 119877

3

sa sin 120579 119889120579 119889120601

(27)

where 120591sphpl (120579 120601) = 120583plsh119875sphpl (120579 120601) denotes the shear stress

exerted on the outer surface of the planetary pinion with aCoulomb-type friction law (Figure 9) 120583plsh is the coefficientof friction (which is constant) between the satellite pinionand the plastic shell 119875sphpl is the pressure applied to the headsurface of the planetary pinion and 120575

2and 1205751015840

0are the angles

defining the spherical portion of the head-planetary pinionIt is again assumed here that in a first approximation the

pressure field 119875sphpl exerted on the head-planetary pinion isuniform (ie (120579 120601)-independent) and (27) reduce to

119883pl = 21205871198772

sa119875sphpl (

cos (212057510158400) minus cos (2120575

2)

4)

119862119891

plsh = 120583plsh119877sa119862rg

2119877119901

tan120572 cos 120575

times (cos (120575

2+ 1205751015840

0) sin (120575

2minus 1205751015840

0) minus (120575

2minus 1205751015840

0)

sin (12057510158400+ 1205752) sin (1205751015840

0minus 1205752)

)

(28)

235 Total Power Dissipated in the Differential GearboxBased on the above constitutive equations the total powerdissipated in the differential gearbox can bewritten as follows

119875ab = 119875sapl + 119875axsa + 119875sash + 119875plsh

= (1 minus 120578sapl) 119862rg120596rg119867+(Δ120596) + 120583axsa

119862rg

4

119889ax119877119904

Δ120596

+ 120583sash119877sa119862rg

2119877119904

tan120572 sin 120575

times (cos (120575

1+ 1205750) sin (120575

1minus 1205750) minus (120575

1minus 1205750)

sin (1205750+ 1205751) sin (120575

0minus 1205751)

)Δ120596

+ 120583plsh119877sa119862rg

2119877119901

tan120572 cos 120575

times (cos (120575

2+ 1205751015840

0) sin (120575

2minus 1205751015840

0) minus (120575

2minus 1205751015840

0)

sin (12057510158400+ 1205752) sin (1205751015840

0minus 1205752)

)Δ120596

(29)

Using (10) and (29) and assuming that in a first approx-imation 120583sash = 120583plsh = 120583sh (where 120583sh denotes thecoefficient of friction between each of the pinions in contactwith the plastic shell) the total differential gearbox yield canbe written as follows

120578di = 1 minussum119894

119897119894

= 1 minus

[[[[

[

119875sapl

119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897sapl

+119875axsa119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897axsa

+119875sash119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897sash

+119875plsh

119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897plsh

]]]]

]

(30)

with

119897sapl = (1 minus 120578sapl)119867+(Δ120596)

119897axsa = 120583axsa119889ax4119877119904

Δ120596

120596rg

119897sash = 120583sh119877sa2119877119904

tan120572 sin 120575

times (cos (120575

1+ 1205750) sin (120575

1minus 1205750) minus (120575

1minus 1205750)

sin (1205750+ 1205751) sin (120575

0minus 1205751)

)Δ120596

120596rg

119897plsh = 120583sh119877sa2119877119901

tan120572 cos 120575

times (cos (120575

2+ 1205751015840

0) sin (120575

2minus 1205751015840

0) minus (120575

2minus 1205751015840

0)

sin (12057510158400+ 1205752) sin (1205751015840

0minus 1205752)

)Δ120596

120596rg

(31)

where 119897119894= 119875119894119875rg denotes the loss ratio of the 119894-mechanism

(with 119894 = (sapl axsa sash plsh)) It is worth noting thatthe differential yield 120578di shows mechanical friction losseswhenever the rotational-speed differs between two of theplanetary pinions associated with the driven wheels that iswhen Δ120596 gt 0

3 Discussions

In the first part of this section we discuss the order of magni-tude of the constitutive parameters of the model presented inSection 2 In the second part we present a sensitivity analysis

8 The Scientific World Journal

0 05 1 15 2085

09

095

1D

iffer

entia

l gea

rbox

yie

ld120578

di

Δ120596120596rg

(a)

0 05 1 15 20

0005

001

0015

002

0025

003

0035

004

PiP

rg

Δ120596120596rg

(b)

Figure 10 (a) Yield of a differential gearbox 120578di as a function of the load parameterΔ120596120596rg for various sets of coefficients of friction (120583axsa 120583sh)(003 005) in a dashed line (005 007) in a dash-dot line (003 007) in a dotted line and in a solid line for (120583axsa 120583sh) and 120578sapl undergoinga concomitant linear increase from (003 005) to (005 007) and 098 to 096 respectively with Δ120596120596rg the thick solid line gives the yieldof the differential gearbox without any losses (which is constant) with the load parameter Δ120596120596rg that is 119897119894 = 119875119894119875rg = 0 forall119894 (b) Loss ratio119897119894= 119875119894119875rg of the 119894-mechanism (with 119894 = (sapl axsa sash plsh)) as a function of the load parameter Δ120596120596rg and with the same sets of

coefficients of friction as in (a) 119897sapl without (black solid line) or with (black dashed line) a linear decrease in 120578sapl from 098 to 096 119897axsa(red line) 119897sash (blue line) and 119897plsh (green line)

0 005 01 015 020

001

002

003

004

005

006

Cf iC

rg

Coefficient of friction 120583i

Figure 11 Friction torque ratio 119862119894119862rg as a function of coefficient of

friction 120583119894with 119894 = (axsa sash plsh) (while the other parameters

were taken to be constant) 119862axsa119862rg increased linearly with 120583axsa(red solid line) 119862sash119862rg increased linearly with 120583sash = 120583sh (bluesolid line)119862plsh119862rg increased linearly with 120583plsh = 120583sh (green solidline) in a range of variation 120583

119894isin [0 02]

in which the response of the model to a given load parameterwas examined in order to assess the effect of this responseon themechanical losses occurring in the differential gearboxmechanism

31 Order of Magnitude of the Constitutive Parameters Thisanalytical model involves thirteen parameters (120578sapl 120583axsa

0 200 400 600 800 10000

02

04

06

08

1

12

14

16

18

2

Time (s)

Δ120596120596

rg

Figure 12 Example of a load parameterΔ120596120596rg with a complex pathdepending on time 119905 isin [0 1000 s] when Δ120596120596rg = 0 (in the casewhere 120596lw = 0 and 120596rw = 0) then 120578di = 1 and 119897119894 = 119875119894119875rg = 0 forall119894 (whenthe vehiclewas travelling along a straight line) when 0 lt Δ120596120596rg lt 2

then 120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894 (when the vehicle was taking acurvilinear course) when Δ120596120596rg = 2 then 120578di lt 1 and 119897119894 = 119875119894119875rg gt0 forall119894 (when one of the driven wheels underwent slipping the vehiclewas stationary) on the plateau Δ120596120596rg = constant and |120596lw minus120596rw| =

constant (the vehicle was taking a constant curvilinear trajectory)

120583sh 119889ax 119877119904 119877119901 119877sa 120572 120575 1205750 1205751015840

0 1205751 1205752) consisting of (i) ten

geometrical parameters (119889ax 119877119904 119877119901 119877sa 120572 120575 1205750 1205751015840

0 1205751 1205752)

depending on the differential gearbox under consideration

The Scientific World Journal 9

0 200 400 600 800 1000091

092

093

094

095

096

097

098

099

1

Time (s)

Diff

eren

tial g

earb

ox y

ield

120578di

(a)

0 200 400 600 800 10000

0005

001

0015

002

0025

003

0035

Time (s)

PiP

rg

(b)

Figure 13 (a) Yield of a differential gearbox 120578119889119894with time 119905 isin [0 1000 s] in the case of the complex path presented in Figure 12 with various

sets of friction coefficients (120583axsa 120583sh) (003005) in a dashed line (005007) in a dash-dot line and (003007) in a dotted line (b) Lossratio 119897

119894= 119875119894119875rg of the 119894-mechanism (with 119894 = (sapl axsa sash plsh)) with time 119905 isin [0 1000 s] in the case of the complex path presented

in Figure 12 with the same sets of friction coefficients as in (a) 119897sapl = 096 which remained constant with time 119905 (black solid line) 119897axsa (redline) 119897sash (blue line) and 119897plsh (green line)

which can be identified fairly easily (ii) one yield param-eter (120578sapl) corresponding to the power transmitted by thegear between the satelliteplanetary pinions (iii) two fric-tion parameters (120583axsa 120583sh) which depend considerably onthe types of materials in contact and the surface conditionstheir state of lubrication and their temperature

Concerning (i) although these specific parameters de-pend on the differential gearbox under consideration theirorder of magnitude can be said to be 119889ax asymp 10ndash20mm 119877

119904asymp

20ndash30mm 119877119901asymp 20ndash30mm 119877sa asymp 30ndash50mm and for the

angle parameters 1205750asymp 10∘ndash15∘ 1205751015840

0asymp 15∘ndash25∘ 120575

1asymp 30∘ndash40∘

1205752asymp 35∘ndash45∘ 120572 asymp 20ndash25∘ 120575 asymp 30∘ndash40∘ readers can refer to

Fanchon [19] for further details about some of themConcerning (ii) the parameter 120578sapl which denotes the

power transmitted by the gear between satelliteplanetarypinions using a yield term has a value ranging between 092and 098 Since Fanchon [19] has reported that the yield inthe case of two meshed pinions is around 98 it can beconcluded that in a differential mechanism where there arefour pinions in contact the lowest value of 120578sapl asymp (098)

4cong

092 and the highest value is likely to be 120578sapl cong 098 Ittherefore seems reasonable to assume that the real yield 120578saplof this complex mechanism is approximately 096

The parameters that need to be investigated more closelyin order to determine their influence on themechanical lossesare the various coefficients of friction (iii) 120583axsa and 120583shThe sliding contacts in the differential gearbox mechanismcan be of very different kinds [20 21] since both thesatelliteplanetary bevel pinion and satellite-pinionsatellite-carrier axis in contact are in the mixed elastohydrodynamiclubrication (MEHL) mode [22ndash24] which is a mixed regimebetween (1) elastohydrodynamic lubrication (EHL) [25 26]where the heavy loads exerted at the surface of the two gears

(due to the presence of both high contact pressures and strongtransmitted torques) induce elastic strains in the two solidsin contact with thin lubricant films and (2) boundary lubri-cation (BL) [27] which may tend to constitute a completelydry frictional contact [28] leading to the development ofscuffing effects [29 30] The contacts between the bevel headpinion (satellites and planetaries) and the plastic shell arein the elastohydrodynamic lubrication mode (EHL) whichcan lead to the development of the hydrodynamic lubricationregime (HL) [31 32] where the lubricant film thickness islarge enough to completely separate the directly apposedsurfaces so that the loads applied between the surfaces arerestored and balanced under specific operating conditionsUnder the above conditions the contacts existing betweenthe satellite pinions and the satellite-carrier axis are of thesteelsteel contact with lubrication type 120583axsa isin [005 007]and those existing between the head pinions and the plasticshell are rather of the steelTeflon contact with lubricationtype 120583sh isin [003 005]

32 Sensitivity Analysis Assessment of the Modelrsquos Responseto a Given Load Parameter and Prediction of the MechanicalLosses In what follows it is proposed to test the modelrsquosresponse to a given load parameter and to determinethe influence of some parameters (120578sapl 120583axsa 120583sh) on themechanical losses occurring in a differential gearbox mecha-nism The expression for the load parameter Δ120596120596rg isin [0 2]means that (i) when Δ120596120596rg = 0 (in cases where 120596lw = 0 and120596rw = 0) the vehicle is travelling on a straight line 120578di = 1 and119897119894= 119875119894119875rg = 0 forall119894 (ii) when 0 lt Δ120596120596rg lt 2 the vehicle is

taking a curvilinear course 120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894(iii) when Δ120596120596rg = 2 one of the driven wheels undergoes

10 The Scientific World Journal

slipping (the extreme situation) the vehicle is stationary120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894

The values adopted for the other parameters are 119889ax =18 times 10

minus3m 119877119904= 22 times 10

minus3m 119877119901= 27 times 10

minus3m 119877sa =49 times 10

minus3m 1205750= 10∘ 12057510158400= 18∘ 1205751= 32∘ 1205752= 40∘ 120572 = 20∘

120575 = 39∘

Figure 10(a) shows the yield of a differential gearbox 120578didepending on the load parameter Δ120596120596rg with various setsof friction coefficients of friction (120583axsa 120583sh) and a lineardecrease in 120578sapl from 098 to 096 In the most gentle oper-ating range that is 0 le Δ120596120596rg le 1 120578di decreased from 098to around 0963 (120583axsa 120583sh) = (003 005) 120578di isin [098 0959](resp 120578di isin [098 0954]) when (120583axsa 120583sh) = (003 005)(resp (120583axsa 120583sh) = (005 007)) and 120578di decrease from 098to around 0949 when (120583axsa 120583sh) showed a concomitantlinear increase from (003 005) to (005 007) In the heavyoperating range that is 1 lt Δ120596120596rg le 2 the maximumdifferential gearbox yield 120578di isin ]0963 0946] was obtainedwith (120583axsa 120583sh) = (003 005) and the minimum yield(120578di isin ]0949 0909]) occurred when (120583axsa 120583sh) and 120578saplincreased linearly from (003 005) to (005 007) concomi-tantly with the load parameter Δ120596120596rg 120578di isin ]0959 0938](resp 120578di isin ]0954 0929]) when (120583axsa 120583sh) = (003 005)(resp (120583axsa 120583sh) = (005 007)) The results presentedin Figure 10(b) show the loss ratio 119897

119894= 119875119894119875rg associated

with 119894-mechanism (where 119894 = (sapl axsa sash plsh)) indecreasing order of influence 119897axsa (red line) 119897sapl (blackline) 119897plsh (green line) 119897sash (blue line) when 120578sapl had a con-stant value of 098 and in the case where 120578sapl decreased from098 to 096 the effects of the loss ratio 119897sapl became greaterthan those of the other parameters (119897axsa 119897axsa 119897sash 119897plsh)It should be noted that when 0 le Δ120596120596rg le 1 then119897axsa isin [0 00143] 119897plsh isin [0 00057] and 119897sash isin [0 00052]whereas when 1 lt Δ120596120596rg le 2 then 119897axsa isin [0 00286]119897plsh isin [0 00115] and 119897sash isin [0 00104] Figure 11 showsthe friction torque ratio 119862

119894119862rg as a function of the friction

coefficient 120583119894with 119894 = (axsa sash plsh) when the other

parameters were taken to be fixed constants The range ofvariation of the friction coefficient 120583

119894was taken to be between

0 and 02 (this extreme value may correspond to steelsteelcontact without any lubrification) In view of these resultsthe maximum 119862

119894119862rg ratio was obtained with 119862axsa119862rg and

the minimum one with 119862sash119862rg The friction torque ratios119862119894119862rg progressed linearly with the friction coefficient 120583

119894

(where 120583sh = 120583sash = 120583plsh) as predicted by the constitutiveequations in Section 2 Note that the results obtained in thisfirst approximation are likely to differ from those obtained bysimulating nonlinear behaviour between 119862

119894119862rg and 120583119894

A complex path of the load parameter Δ120596120596rg wasthen studied in order to test the ability of the model todescribe the differential gearbox yield 120578di and the loss ratios119897119894= 119875119894119875rg associated with the 119894-mechanism (where 119894 =

(sapl axsa sash plsh)) under more realistic conditionsinvolving a straight line situation (Δ120596120596rg = 0 with 120596lw = 0

and 120596rw = 0) a curved line situation (0 lt Δ120596120596rg lt 2)a constant curved line situation (Δ120596120596rg = constant) orwhere one of the drivenwheels undergoes slipping (Δ120596120596rg =2) Figure 12 shows the evolution of the load parameter

Δ120596120596rg as a function of the time 119905 isin [0 1000 s] Theresults obtained with a complex path in terms of the yieldof the differential gearbox (120578di) and the loss ratios (119897

119894) are

shown in Figure 13 Specifically Figure 13(a) gives the yieldof the differential gearbox 120578di as a function of the time119905 isin [0 1000 s] in the case of the complex path plotted inFigure 12 with various sets of friction coefficients (120583axsa 120583sh)(003005) in dashed line (005007) in dash-dot line and(003007) in dotted line During the whole complex path ofΔ120596120596rg the maximum (resp minimum) yield of differentialgearbox was reached with 120583axsa = 003 and 120583sh = 005

(resp 120583axsa = 005 and 120583sh = 007) On the 119896-plate (with119896 = 1 2 3 4) defined by the 119896-time interval [119905119894 119905119891]

119896 it can be

noted that for example (1) in [400 s 500 s]2 120578di asymp 09774with

(005007) 120578di asymp 09591 with (003007) and 120578di asymp 09547with (005007) (2) in [900 s 950 s]

4 120578di asymp 09464 with

(005007) 120578di asymp 09382 with (003007) and 120578di asymp 09294with (005007) The loss ratios due to friction 119897

119894= 119875119894119875rg are

shown in Figure 13(b) The maximum loss ratios obtained indecreasing order of magnitude (along the loading path) were119897sapl (black line) 119897axsa (red line) 119897plsh (green line) and 119897sash(blue line) except for the last loadwhere 119897axsa could be greaterthan 119897sapl with the pair of friction coefficients (005007)and (003007) It should be noted that (1) on the secondplate ([400 s 500 s]

2) 119897max

axsa asymp 00143 with (005007) 119897maxplsh asymp

00057 with (003007) and 120578di asymp 00052 with (005007)(2) on the fourth plate ([900 s 950 s]

4) 119897max

axsa asymp 00286 with(005007) 119897max

plsh asymp 00115 with (003007) and 120578di asymp 00104with (005007)

Although only a sensitivity analysis was performed on themodel the results obtained show quite clearly that this modelcan be used to assess and predict the mechanical frictionlosses occurring in a differential gearbox An experimentalstudy shall be conducted in order to obtain more realisticvalues for some of the parameters such as the frictioncoefficients (120583axsa and 120583sh) which significantly influenceboth the friction torques and the differential gearbox yield

4 Conclusion

In this paper an analytical model is presented for assessingand predicting the mechanical friction losses occurring indifferential gearboxes Some of the parameters involvedin this model can be determined quite easily (geometricparameters) while others which are more delicate dependdirectly on the type of friction occurring in the mechanism(and therefore on the friction coefficients) which affectsthe mechanical losses to a variable extent In order totest the influence of these parameters and determine theability of the model to predict any mechanical losses asensitivity analysis was conducted After this initial numericalapproach an experimental study shall be performed in orderto obtainmore realistic values for some of the parameters andconfirm some of the assumptionsmade here in themodellingprocedure

The Scientific World Journal 11

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is greatly indebted to Dr J-L Ligier for helpfuldiscussions comments and advices The author would liketo thank also Dr Jessica Blanc for her help with this paper

References

[1] S C Lee and H S Cheng ldquoScuffing theory modeling andexperimental correlationsrdquo Journal of Tribology vol 113 no 2pp 327ndash334 1991

[2] D Dowson G Dalmaz Childs M T H C Godet and C MTaylorThin Films in Tribology Elsevier Science 1993

[3] J C Enthoven P M Cann and H A Spikes ldquoTemperatureand scuffingrdquoTribology Transactions vol 36 no 2 pp 258ndash2661993

[4] D Dowson G Dalmaz T H C Childs and C M TaylorLubricants and Lubrication Elsevier Science 1995

[5] C Taylor L Flamand G Dalmaz et al Elastohydrodynamicsmdashrsquo96 Fundamentals and Applications in Lubrication and TractionElsevier Science 1997

[6] M Priest P Ehret L Flamand et al Lubrication at the Frontierthe Role of the Interface and Surface Layers in the Thin Film andBoundary Regime Elsevier Science 1999

[7] A Cornet and J P Deville Physique et Ingenierie des SurfacesEDP Sciences 1998

[8] W Piekoszewski M Szczerek and W Tuszynski ldquoThe actionof lubricants under extreme pressure conditions in a modifiedrdquoWear vol 249 no 3-4 pp 188ndash193 2001

[9] M Wisniewski M Szczerek and W Tuszynski ldquoThe tempera-tures at scuffing and seizure in a four-ball contactrdquo LubricationScience vol 16 no 3 pp 215ndash227 2004

[10] Q Wang ldquoSeizure failure of journal-bearing conformal con-tactsrdquoWear vol 210 no 1-2 pp 8ndash16 1997

[11] G Spinnler Conception Des MachinesmdashPrincipes Et Applica-tions vol 1 Presses Polytechniques et Universitaires Romandes2002

[12] L Flamand Fatigue Des Surface Techniques De LrsquoIngenieur1993

[13] J L Ligier Materiaux Pour Palier Lisses Techniques DeLrsquoIngenieur 1995

[14] J L Ligier Avaries En Lubrification Ecole Nationales SuperieureDu Petrole Et Des Moteurs Editions Technip 2007

[15] M Aublin R Boncompain M Boulaton et al SystemesMecaniquesmdashTheorie Et Dimensionnement Dunod 2005

[16] J L Ligier and R Gojon ldquoPrediction de lrsquousure et du grippagedrsquoun palier de moteur dieselrdquo Tech Rep SIA 93065 1993

[17] J L Ligier Lubrification des Paliers Moteurs Ecole NationalesSuperieure du Petrole et des Moteurs Editions Technip 1997

[18] S Picard Fonctionnement et Maintenance Du VehiculemdashTransmission amp Freinage Delta Press 1994

[19] J L Fanchon Guide des Sciences et Technologies IndustriellesNathan 2001

[20] J HMathieu E Bergmann and R GrasAnalyse et Technologiedes Surfaces Couches Minces et Tribologie Presses Polytech-niques et Universitaires Romandes Traite desMateriaux Tome2003

[21] V N Constantinescu Laminar Viscous Flow Mechanical Engi-neering Series Springer 1995

[22] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part IWithout consideration of contact adhering layer in the inletzonerdquo Journal of Molecular Liquids vol 130 no 1ndash3 pp 88ndash942007

[23] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part II-Considering the contact adhering layer effect in the inlet zonerdquoJournal of Molecular Liquids vol 130 no 1ndash3 pp 95ndash103 2007

[24] S Li and A Kahraman ldquoA fatigue model for contacts undermixed elastohydrodynamic lubrication conditionrdquo Interna-tional Journal of Fatigue vol 33 no 3 pp 427ndash436 2011

[25] K L Johnson J A Greenwood and S Y Poon ldquoA simple theoryof asperity contact in elastohydro-dynamic lubricationrdquo Wearvol 19 no 1 pp 91ndash108 1972

[26] G Ramsey Elastohydrodynamics Imperial College Press 2001[27] B J Hamrock S R Schmid and B O Jacobson Fundamentals

of Fluid Film Lubrication CRC Press 2004[28] K L Johnson Contact Mechanics Cambridge University Press

1987[29] R Brun Manuel Du Mecanicien et Du Thermicien Editions

Technip 2000[30] P Arques Diagnostic Predictif et Defaillances Des Machines

Theorie Traitement Analyse Reconnaissance Prediction Edi-tions Technip 2007

[31] J Frene D Nicolas B Degueurce D Berthe and M GodetLubrication Hydrodynamique Paliers Et Butees Eyrolles 1990

[32] J Frene ldquoButees et paliers hydrodynamiquerdquo Tech Rep B5055Techniques de lrsquoIngenieur 1995

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DistributedSensor Networks

International Journal of

4 The Scientific World Journal

Comments

(i) Under straight driving conditions the angular veloc-ities of two satellite pinions in relation to the satelliteaxis are zero (Δ120596 = 0) and the speed of thedriven wheels is equal to the angular velocity of thedifferential ring gear that is 120596lw = 120596rw = 120596rg

(ii) In curvilinear driving situations the angular velocityof two satellite pinions in relation to the satellite-carrier axis is no longer zero (Δ120596 gt 0) and the speedsof the driven wheels are no longer equal that is 120596lw gt120596rw (right turn) or 120596lw lt 120596rw (left turn)

(iii) If one of the drivenwheels undergoes slipping [18]mdashifit is the left (resp right) wheel 120596lw = 0 (resp 120596rw =0) that is Δ120596 = 2120596rgmdashthe differential gearbox yieldreduces to 120578di = 1 minus (Δ119862119862rg)

23 Mechanical Losses Friction Torques Thepower absorbedin the differential gearbox is the sum of the power dissipatedat the various contact points existing in the overall mecha-nism which can be decomposed as follows

119875ab = 119875sapl + 119875axsa + 119875sash + 119875plsh (11)

where 119875sapl 119875axsa 119875sash and 119875plsh correspond to the part ofthe power dissipated between the two satellite and planetarybevel pinions the satellite pinion and the satellite-carrieraxis between the two bevel heads of the satellite pinions andthe surrounding plastic shell and the two bevel heads ofthe planetary pinions and the surrounding plastic shell (seeFigure 3)

231 Power Dissipated between Two Satellite Bevel Pinionsand Two Planetary Bevel Pinions In order to account forthe power dissipated between the two satellite bevel pinionsand the two planetary bevel pinions in the differentialmechanism (giving four meshing contacts) we introducea yield 120578sapl (denoting the power transmitted in the gearbetween satelliteplanetary pinions) which can be written asfollows

120578sapl =119875rg minus 119875sapl

119875rg119867+(Δ120596) (12)

where 119875sapl denotes the power dissipated between the twosatelliteplanetary contacts and 119867+(sdot) is a Heaviside-likefunction (adapted in order to ensure that 119867+(119909) = 1 when119909 gt 0 and 119867(119909) = 0 when 119909 le 0) Note that a yield 120578saplwith a nonzero value (ie 120578sapl gt 0) occurs between thetwo satellite pinions and two planetary pinions only whenthe angular velocity differs between the driven wheels thatis Δ120596 gt 0

Based on (12) we can therefore write

119875sapl = (1 minus 120578sapl) 119862rg120596rg119867+(Δ120596) (13)

232 Power Dissipated between the Satellite Bevel Pinionand Satellite-Carrier Axis Neglecting the presence of an

over-centre mechanism between the satellite pinion andsatellite-carrier axis (Figure 5) the force exerted on thesatellite-carrier axis 119865ax upon the rotation of the differentialring gear (120596rg) can be written as

119865ax =119862rg

2119877119901

(14)

In (14) the point of application of the force 119865ax isassumed to take the same contact path as both the left andright meshing gears (Figure 7)

Adopting aCoulomb-type friction law the friction torqueapplied to the satellite-carrier axis 119862119891axsa is

119862119891

axsa = 120583axsa119865ax119889ax2 (15)

where 120583axsa denotes the friction coefficient (which is con-stant) between the satellite pinion and the satellite-carrier axisand 119889ax is the axis diameter (Figure 6)

Combining (14) and (15) the power dissipated betweenthese two assembly components reads

119875axsa = 2119862119891

axsa120596119904 = 120583axsa119862rg

4

119889ax119877119904

Δ120596 (16)

233 Power Dissipated between the Satellite Bevel Pinions andthe Plastic Shell The power dissipated between the two bevelsatellite pinions and the surrounding plastic shell 119875sash canbe written as

119875sash = 2119862119891

sash120596119904 = 119862119891

sash

119877119901

119877119904

Δ120596 (17)

where119862119891sash denotes the friction torque applied to the satellitebevel pinion Note that the friction torque 119862119891sash is assumedhere to be identical on the two satellite pinions

In order to write the equations giving the equilibrium ofeach satellite pinion which is meshed with the two planetarypinions associated with the driven wheels and in contact withthe satellite-carrier axis as well as with the plastic shell we usethe Fundamental Principle of Statics119879lwrarr119878119872 + 119879rwrarr119878119872 + 119879axrarr119878119872 + 119879shrarr119878119872 = 0

(18)where 119879

119894rarr 119878119872denotes the transmittable force torsor of solid

119894 (and 119894 = (lw rw ax sh) associated with the planetarypinions of the left and right wheels the satellite-carrier axisand the plastic shell resp) exerted on solid 119878 (the satellitepinion) at point119872 (see [15])

Looking only at the equilibrium of the resulting force ofeach torsor in the direction (119909 119910 119911) (see Figure 7) (18) writes

119865119905

lw119865119903

lw119865119886

lw

+

119865119905

rwminus119865119903

rw119865119886

rw

+

minus119865ax0

0

+

0

0

minus119883sa

=

0

0

0

(19)

lArrrArr

119865119905

lw + 119865119905

rw = 119865ax119865119903

lw = 119865119903

rw(119865119905

lw + 119865119905

rw) tan120572 sin 120575 = 119883sa

(20)

The Scientific World Journal 5

120595

Figure 5 Over-centre mechanism between the satellite pinion andsatellite-carrier axis 120595 denotes the angular eccentricity

2Rp

dax

Rs

Figure 6 Satellite bevel pinions and satellite-carrier axis

where 119865119905119895 119865119903119895 and 119865119886

119895(with 119895 = lw rw) are the tangential

radial and axial forces of 119895-planetary pinions (associatedwiththe left and rightwheels Figure 7 see [19]) respectively119883sa isthe resulting force applied to the satellite pinion generated bythe plastic shell 120575 is the half-angle pitch radius of the satellitepinion (see Figure 8(a)) and 120572 is the pressure angle betweenthe satellite and planetary pinions which is assumed hereto be identical between the satelliteplanetary meshing gears(see Figure 8(b))

y

x

z

y

x

z

Fax

Fax

Frlw

Falw Fa

rw

Frrw

Frlw Fr

rw

Ftlw Ft

rw

Psphsa

Psphsa

Figure 7 Forces applied to a satellite pinion

The pressure applied to the head surface of the satellitepinion 119875sphsa (its spherical part see Figure 9) due to the result-ing force119883sa (in line with (20)) matches the relationship (see[15])

119883sa = int2120587

0

int

1205751

1205750

119875sphsa (120579 120601) 119877

2

sa cos 120579 sin 120579 119889120579 119889120601 (21)

where 1205751and 120575

0are the angles defining the spherical part of

the head-satellite pinion and 119877sa is the radius of the innersphere of the plastic shell (where the contact with the headsurface of the satellite pinion and the plastic shell occurs)

The expression for the friction torque119862119891sash applied to thehead surface of the satellite pinion can be written (see [17]) as

119862119891

sash = int2120587

0

int

1205751

1205750

120591sphsa (120579 120601) 119877

3

sa sin 120579 119889120579 119889120601

= int

2120587

0

int

1205751

1205750

120583sash119875sphsa (120579 120601) 119877

3

sa sin 120579 119889120579 119889120601

(22)

where 120591sphsa (120579 120601) = 120583sash 119875sphsa (120579 120601) denotes the shear stress

exerted on the head surface of the satellite pinion since aCoulomb-type friction law is adopted (see Figure 9) and120583sash is the coefficient of friction (which is constant) betweenthe satellite pinion and the plastic shell

6 The Scientific World Journal

120575

1205751

1205750Rs

Rsa

Rsa

Rp

dax

1205759984000

1205752

120575

(a)

120572

(b)

Figure 8 Geometry of a differential gearbox

Psphj

120591sphj

Figure 9 Pressure 119875sph119895

(solid arrow) and shear 120591sph119895

(dashed arrow)fields on a head-pinion 119895 (with 119895 = (sa pl)) caused by the plasticshell

Combining (14) and (20)ndash(22) and assuming that ina first approximation the pressure field exerted on the

head-satellite pinion is uniform (ie 119875sphsa (120579 120601) = 119875sphsa ) (21)

and (22) reduce to

119883sa = 21205871198772

sa119875sphsa (

cos (21205750) minus cos (2120575

1)

4)

119862119891

sash = 120583sash119877sa119862rg

2119877119901

tan120572 sin 120575

times (cos (120575

1+ 1205750) sin (120575

1minus 1205750) minus (120575

1minus 1205750)

sin (1205750+ 1205751) sin (120575

0minus 1205751)

)

(23)

234 Power Dissipated between the Planetary Bevel Pinionsand the Plastic Shell The power dissipated between the twoplanetary pinions and the plastic shell 119875plsh can be writtenas

119875plsh = 119862119891(lw)plsh 120596

rslw + 119862

119891(rw)plsh 120596

rsrw (24)

where119862119891(119894)plsh are the frictional torque between the planetary 119894-pinion (119894 = (lw rw) refers to the left wheel (lw) and the rightwheel (rw)) and the plastic shell and 120596rs

119894= Δ1205962 denotes

the relative speed of the 119894-pinion in relation to the sumpdifferential

Assuming that 119862119891(lw)plsh cong 119862119891(rw)plsh = 119862

119891

plsh (24) reads

119875plsh = 119862119891

plshΔ120596 (25)

Using the sameprocedure as above namelymeshing eachof the planetary pinions with two satellite pinions which

The Scientific World Journal 7

means that the resulting force applied to the head-planetarypinion by the plastic shell119883pl is

119883pl = 119865ax tan120572 sin 120575 (26)

where 120575 is the half-angle pitch radius of the planetary pinion(see Figure 8(a)) Note that given the particular geometry ofthe mechanism the sum of the half-angle pitch radii of theplanetary and satellite pinions satisfies 120575 + 120575 = 1205872 and wetherefore obtain the following relationship sin 120575 = cos 120575

The expressions for both the resulting force 119883pl and thefriction torque119862119891plsh applied to the head-satellite pinion are

119883pl = int2120587

0

int

1205752

1205751015840

0

119875sphpl (120579 120601) 119877

2

sa cos 120579 sin 120579 119889120579 119889120601

119862119891

plsh = int2120587

0

int

1205752

1205751015840

0

120591sphpl (120579 120601) 119877

3

sa sin 120579 119889120579 119889120601

(27)

where 120591sphpl (120579 120601) = 120583plsh119875sphpl (120579 120601) denotes the shear stress

exerted on the outer surface of the planetary pinion with aCoulomb-type friction law (Figure 9) 120583plsh is the coefficientof friction (which is constant) between the satellite pinionand the plastic shell 119875sphpl is the pressure applied to the headsurface of the planetary pinion and 120575

2and 1205751015840

0are the angles

defining the spherical portion of the head-planetary pinionIt is again assumed here that in a first approximation the

pressure field 119875sphpl exerted on the head-planetary pinion isuniform (ie (120579 120601)-independent) and (27) reduce to

119883pl = 21205871198772

sa119875sphpl (

cos (212057510158400) minus cos (2120575

2)

4)

119862119891

plsh = 120583plsh119877sa119862rg

2119877119901

tan120572 cos 120575

times (cos (120575

2+ 1205751015840

0) sin (120575

2minus 1205751015840

0) minus (120575

2minus 1205751015840

0)

sin (12057510158400+ 1205752) sin (1205751015840

0minus 1205752)

)

(28)

235 Total Power Dissipated in the Differential GearboxBased on the above constitutive equations the total powerdissipated in the differential gearbox can bewritten as follows

119875ab = 119875sapl + 119875axsa + 119875sash + 119875plsh

= (1 minus 120578sapl) 119862rg120596rg119867+(Δ120596) + 120583axsa

119862rg

4

119889ax119877119904

Δ120596

+ 120583sash119877sa119862rg

2119877119904

tan120572 sin 120575

times (cos (120575

1+ 1205750) sin (120575

1minus 1205750) minus (120575

1minus 1205750)

sin (1205750+ 1205751) sin (120575

0minus 1205751)

)Δ120596

+ 120583plsh119877sa119862rg

2119877119901

tan120572 cos 120575

times (cos (120575

2+ 1205751015840

0) sin (120575

2minus 1205751015840

0) minus (120575

2minus 1205751015840

0)

sin (12057510158400+ 1205752) sin (1205751015840

0minus 1205752)

)Δ120596

(29)

Using (10) and (29) and assuming that in a first approx-imation 120583sash = 120583plsh = 120583sh (where 120583sh denotes thecoefficient of friction between each of the pinions in contactwith the plastic shell) the total differential gearbox yield canbe written as follows

120578di = 1 minussum119894

119897119894

= 1 minus

[[[[

[

119875sapl

119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897sapl

+119875axsa119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897axsa

+119875sash119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897sash

+119875plsh

119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897plsh

]]]]

]

(30)

with

119897sapl = (1 minus 120578sapl)119867+(Δ120596)

119897axsa = 120583axsa119889ax4119877119904

Δ120596

120596rg

119897sash = 120583sh119877sa2119877119904

tan120572 sin 120575

times (cos (120575

1+ 1205750) sin (120575

1minus 1205750) minus (120575

1minus 1205750)

sin (1205750+ 1205751) sin (120575

0minus 1205751)

)Δ120596

120596rg

119897plsh = 120583sh119877sa2119877119901

tan120572 cos 120575

times (cos (120575

2+ 1205751015840

0) sin (120575

2minus 1205751015840

0) minus (120575

2minus 1205751015840

0)

sin (12057510158400+ 1205752) sin (1205751015840

0minus 1205752)

)Δ120596

120596rg

(31)

where 119897119894= 119875119894119875rg denotes the loss ratio of the 119894-mechanism

(with 119894 = (sapl axsa sash plsh)) It is worth noting thatthe differential yield 120578di shows mechanical friction losseswhenever the rotational-speed differs between two of theplanetary pinions associated with the driven wheels that iswhen Δ120596 gt 0

3 Discussions

In the first part of this section we discuss the order of magni-tude of the constitutive parameters of the model presented inSection 2 In the second part we present a sensitivity analysis

8 The Scientific World Journal

0 05 1 15 2085

09

095

1D

iffer

entia

l gea

rbox

yie

ld120578

di

Δ120596120596rg

(a)

0 05 1 15 20

0005

001

0015

002

0025

003

0035

004

PiP

rg

Δ120596120596rg

(b)

Figure 10 (a) Yield of a differential gearbox 120578di as a function of the load parameterΔ120596120596rg for various sets of coefficients of friction (120583axsa 120583sh)(003 005) in a dashed line (005 007) in a dash-dot line (003 007) in a dotted line and in a solid line for (120583axsa 120583sh) and 120578sapl undergoinga concomitant linear increase from (003 005) to (005 007) and 098 to 096 respectively with Δ120596120596rg the thick solid line gives the yieldof the differential gearbox without any losses (which is constant) with the load parameter Δ120596120596rg that is 119897119894 = 119875119894119875rg = 0 forall119894 (b) Loss ratio119897119894= 119875119894119875rg of the 119894-mechanism (with 119894 = (sapl axsa sash plsh)) as a function of the load parameter Δ120596120596rg and with the same sets of

coefficients of friction as in (a) 119897sapl without (black solid line) or with (black dashed line) a linear decrease in 120578sapl from 098 to 096 119897axsa(red line) 119897sash (blue line) and 119897plsh (green line)

0 005 01 015 020

001

002

003

004

005

006

Cf iC

rg

Coefficient of friction 120583i

Figure 11 Friction torque ratio 119862119894119862rg as a function of coefficient of

friction 120583119894with 119894 = (axsa sash plsh) (while the other parameters

were taken to be constant) 119862axsa119862rg increased linearly with 120583axsa(red solid line) 119862sash119862rg increased linearly with 120583sash = 120583sh (bluesolid line)119862plsh119862rg increased linearly with 120583plsh = 120583sh (green solidline) in a range of variation 120583

119894isin [0 02]

in which the response of the model to a given load parameterwas examined in order to assess the effect of this responseon themechanical losses occurring in the differential gearboxmechanism

31 Order of Magnitude of the Constitutive Parameters Thisanalytical model involves thirteen parameters (120578sapl 120583axsa

0 200 400 600 800 10000

02

04

06

08

1

12

14

16

18

2

Time (s)

Δ120596120596

rg

Figure 12 Example of a load parameterΔ120596120596rg with a complex pathdepending on time 119905 isin [0 1000 s] when Δ120596120596rg = 0 (in the casewhere 120596lw = 0 and 120596rw = 0) then 120578di = 1 and 119897119894 = 119875119894119875rg = 0 forall119894 (whenthe vehiclewas travelling along a straight line) when 0 lt Δ120596120596rg lt 2

then 120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894 (when the vehicle was taking acurvilinear course) when Δ120596120596rg = 2 then 120578di lt 1 and 119897119894 = 119875119894119875rg gt0 forall119894 (when one of the driven wheels underwent slipping the vehiclewas stationary) on the plateau Δ120596120596rg = constant and |120596lw minus120596rw| =

constant (the vehicle was taking a constant curvilinear trajectory)

120583sh 119889ax 119877119904 119877119901 119877sa 120572 120575 1205750 1205751015840

0 1205751 1205752) consisting of (i) ten

geometrical parameters (119889ax 119877119904 119877119901 119877sa 120572 120575 1205750 1205751015840

0 1205751 1205752)

depending on the differential gearbox under consideration

The Scientific World Journal 9

0 200 400 600 800 1000091

092

093

094

095

096

097

098

099

1

Time (s)

Diff

eren

tial g

earb

ox y

ield

120578di

(a)

0 200 400 600 800 10000

0005

001

0015

002

0025

003

0035

Time (s)

PiP

rg

(b)

Figure 13 (a) Yield of a differential gearbox 120578119889119894with time 119905 isin [0 1000 s] in the case of the complex path presented in Figure 12 with various

sets of friction coefficients (120583axsa 120583sh) (003005) in a dashed line (005007) in a dash-dot line and (003007) in a dotted line (b) Lossratio 119897

119894= 119875119894119875rg of the 119894-mechanism (with 119894 = (sapl axsa sash plsh)) with time 119905 isin [0 1000 s] in the case of the complex path presented

in Figure 12 with the same sets of friction coefficients as in (a) 119897sapl = 096 which remained constant with time 119905 (black solid line) 119897axsa (redline) 119897sash (blue line) and 119897plsh (green line)

which can be identified fairly easily (ii) one yield param-eter (120578sapl) corresponding to the power transmitted by thegear between the satelliteplanetary pinions (iii) two fric-tion parameters (120583axsa 120583sh) which depend considerably onthe types of materials in contact and the surface conditionstheir state of lubrication and their temperature

Concerning (i) although these specific parameters de-pend on the differential gearbox under consideration theirorder of magnitude can be said to be 119889ax asymp 10ndash20mm 119877

119904asymp

20ndash30mm 119877119901asymp 20ndash30mm 119877sa asymp 30ndash50mm and for the

angle parameters 1205750asymp 10∘ndash15∘ 1205751015840

0asymp 15∘ndash25∘ 120575

1asymp 30∘ndash40∘

1205752asymp 35∘ndash45∘ 120572 asymp 20ndash25∘ 120575 asymp 30∘ndash40∘ readers can refer to

Fanchon [19] for further details about some of themConcerning (ii) the parameter 120578sapl which denotes the

power transmitted by the gear between satelliteplanetarypinions using a yield term has a value ranging between 092and 098 Since Fanchon [19] has reported that the yield inthe case of two meshed pinions is around 98 it can beconcluded that in a differential mechanism where there arefour pinions in contact the lowest value of 120578sapl asymp (098)

4cong

092 and the highest value is likely to be 120578sapl cong 098 Ittherefore seems reasonable to assume that the real yield 120578saplof this complex mechanism is approximately 096

The parameters that need to be investigated more closelyin order to determine their influence on themechanical lossesare the various coefficients of friction (iii) 120583axsa and 120583shThe sliding contacts in the differential gearbox mechanismcan be of very different kinds [20 21] since both thesatelliteplanetary bevel pinion and satellite-pinionsatellite-carrier axis in contact are in the mixed elastohydrodynamiclubrication (MEHL) mode [22ndash24] which is a mixed regimebetween (1) elastohydrodynamic lubrication (EHL) [25 26]where the heavy loads exerted at the surface of the two gears

(due to the presence of both high contact pressures and strongtransmitted torques) induce elastic strains in the two solidsin contact with thin lubricant films and (2) boundary lubri-cation (BL) [27] which may tend to constitute a completelydry frictional contact [28] leading to the development ofscuffing effects [29 30] The contacts between the bevel headpinion (satellites and planetaries) and the plastic shell arein the elastohydrodynamic lubrication mode (EHL) whichcan lead to the development of the hydrodynamic lubricationregime (HL) [31 32] where the lubricant film thickness islarge enough to completely separate the directly apposedsurfaces so that the loads applied between the surfaces arerestored and balanced under specific operating conditionsUnder the above conditions the contacts existing betweenthe satellite pinions and the satellite-carrier axis are of thesteelsteel contact with lubrication type 120583axsa isin [005 007]and those existing between the head pinions and the plasticshell are rather of the steelTeflon contact with lubricationtype 120583sh isin [003 005]

32 Sensitivity Analysis Assessment of the Modelrsquos Responseto a Given Load Parameter and Prediction of the MechanicalLosses In what follows it is proposed to test the modelrsquosresponse to a given load parameter and to determinethe influence of some parameters (120578sapl 120583axsa 120583sh) on themechanical losses occurring in a differential gearbox mecha-nism The expression for the load parameter Δ120596120596rg isin [0 2]means that (i) when Δ120596120596rg = 0 (in cases where 120596lw = 0 and120596rw = 0) the vehicle is travelling on a straight line 120578di = 1 and119897119894= 119875119894119875rg = 0 forall119894 (ii) when 0 lt Δ120596120596rg lt 2 the vehicle is

taking a curvilinear course 120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894(iii) when Δ120596120596rg = 2 one of the driven wheels undergoes

10 The Scientific World Journal

slipping (the extreme situation) the vehicle is stationary120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894

The values adopted for the other parameters are 119889ax =18 times 10

minus3m 119877119904= 22 times 10

minus3m 119877119901= 27 times 10

minus3m 119877sa =49 times 10

minus3m 1205750= 10∘ 12057510158400= 18∘ 1205751= 32∘ 1205752= 40∘ 120572 = 20∘

120575 = 39∘

Figure 10(a) shows the yield of a differential gearbox 120578didepending on the load parameter Δ120596120596rg with various setsof friction coefficients of friction (120583axsa 120583sh) and a lineardecrease in 120578sapl from 098 to 096 In the most gentle oper-ating range that is 0 le Δ120596120596rg le 1 120578di decreased from 098to around 0963 (120583axsa 120583sh) = (003 005) 120578di isin [098 0959](resp 120578di isin [098 0954]) when (120583axsa 120583sh) = (003 005)(resp (120583axsa 120583sh) = (005 007)) and 120578di decrease from 098to around 0949 when (120583axsa 120583sh) showed a concomitantlinear increase from (003 005) to (005 007) In the heavyoperating range that is 1 lt Δ120596120596rg le 2 the maximumdifferential gearbox yield 120578di isin ]0963 0946] was obtainedwith (120583axsa 120583sh) = (003 005) and the minimum yield(120578di isin ]0949 0909]) occurred when (120583axsa 120583sh) and 120578saplincreased linearly from (003 005) to (005 007) concomi-tantly with the load parameter Δ120596120596rg 120578di isin ]0959 0938](resp 120578di isin ]0954 0929]) when (120583axsa 120583sh) = (003 005)(resp (120583axsa 120583sh) = (005 007)) The results presentedin Figure 10(b) show the loss ratio 119897

119894= 119875119894119875rg associated

with 119894-mechanism (where 119894 = (sapl axsa sash plsh)) indecreasing order of influence 119897axsa (red line) 119897sapl (blackline) 119897plsh (green line) 119897sash (blue line) when 120578sapl had a con-stant value of 098 and in the case where 120578sapl decreased from098 to 096 the effects of the loss ratio 119897sapl became greaterthan those of the other parameters (119897axsa 119897axsa 119897sash 119897plsh)It should be noted that when 0 le Δ120596120596rg le 1 then119897axsa isin [0 00143] 119897plsh isin [0 00057] and 119897sash isin [0 00052]whereas when 1 lt Δ120596120596rg le 2 then 119897axsa isin [0 00286]119897plsh isin [0 00115] and 119897sash isin [0 00104] Figure 11 showsthe friction torque ratio 119862

119894119862rg as a function of the friction

coefficient 120583119894with 119894 = (axsa sash plsh) when the other

parameters were taken to be fixed constants The range ofvariation of the friction coefficient 120583

119894was taken to be between

0 and 02 (this extreme value may correspond to steelsteelcontact without any lubrification) In view of these resultsthe maximum 119862

119894119862rg ratio was obtained with 119862axsa119862rg and

the minimum one with 119862sash119862rg The friction torque ratios119862119894119862rg progressed linearly with the friction coefficient 120583

119894

(where 120583sh = 120583sash = 120583plsh) as predicted by the constitutiveequations in Section 2 Note that the results obtained in thisfirst approximation are likely to differ from those obtained bysimulating nonlinear behaviour between 119862

119894119862rg and 120583119894

A complex path of the load parameter Δ120596120596rg wasthen studied in order to test the ability of the model todescribe the differential gearbox yield 120578di and the loss ratios119897119894= 119875119894119875rg associated with the 119894-mechanism (where 119894 =

(sapl axsa sash plsh)) under more realistic conditionsinvolving a straight line situation (Δ120596120596rg = 0 with 120596lw = 0

and 120596rw = 0) a curved line situation (0 lt Δ120596120596rg lt 2)a constant curved line situation (Δ120596120596rg = constant) orwhere one of the drivenwheels undergoes slipping (Δ120596120596rg =2) Figure 12 shows the evolution of the load parameter

Δ120596120596rg as a function of the time 119905 isin [0 1000 s] Theresults obtained with a complex path in terms of the yieldof the differential gearbox (120578di) and the loss ratios (119897

119894) are

shown in Figure 13 Specifically Figure 13(a) gives the yieldof the differential gearbox 120578di as a function of the time119905 isin [0 1000 s] in the case of the complex path plotted inFigure 12 with various sets of friction coefficients (120583axsa 120583sh)(003005) in dashed line (005007) in dash-dot line and(003007) in dotted line During the whole complex path ofΔ120596120596rg the maximum (resp minimum) yield of differentialgearbox was reached with 120583axsa = 003 and 120583sh = 005

(resp 120583axsa = 005 and 120583sh = 007) On the 119896-plate (with119896 = 1 2 3 4) defined by the 119896-time interval [119905119894 119905119891]

119896 it can be

noted that for example (1) in [400 s 500 s]2 120578di asymp 09774with

(005007) 120578di asymp 09591 with (003007) and 120578di asymp 09547with (005007) (2) in [900 s 950 s]

4 120578di asymp 09464 with

(005007) 120578di asymp 09382 with (003007) and 120578di asymp 09294with (005007) The loss ratios due to friction 119897

119894= 119875119894119875rg are

shown in Figure 13(b) The maximum loss ratios obtained indecreasing order of magnitude (along the loading path) were119897sapl (black line) 119897axsa (red line) 119897plsh (green line) and 119897sash(blue line) except for the last loadwhere 119897axsa could be greaterthan 119897sapl with the pair of friction coefficients (005007)and (003007) It should be noted that (1) on the secondplate ([400 s 500 s]

2) 119897max

axsa asymp 00143 with (005007) 119897maxplsh asymp

00057 with (003007) and 120578di asymp 00052 with (005007)(2) on the fourth plate ([900 s 950 s]

4) 119897max

axsa asymp 00286 with(005007) 119897max

plsh asymp 00115 with (003007) and 120578di asymp 00104with (005007)

Although only a sensitivity analysis was performed on themodel the results obtained show quite clearly that this modelcan be used to assess and predict the mechanical frictionlosses occurring in a differential gearbox An experimentalstudy shall be conducted in order to obtain more realisticvalues for some of the parameters such as the frictioncoefficients (120583axsa and 120583sh) which significantly influenceboth the friction torques and the differential gearbox yield

4 Conclusion

In this paper an analytical model is presented for assessingand predicting the mechanical friction losses occurring indifferential gearboxes Some of the parameters involvedin this model can be determined quite easily (geometricparameters) while others which are more delicate dependdirectly on the type of friction occurring in the mechanism(and therefore on the friction coefficients) which affectsthe mechanical losses to a variable extent In order totest the influence of these parameters and determine theability of the model to predict any mechanical losses asensitivity analysis was conducted After this initial numericalapproach an experimental study shall be performed in orderto obtainmore realistic values for some of the parameters andconfirm some of the assumptionsmade here in themodellingprocedure

The Scientific World Journal 11

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is greatly indebted to Dr J-L Ligier for helpfuldiscussions comments and advices The author would liketo thank also Dr Jessica Blanc for her help with this paper

References

[1] S C Lee and H S Cheng ldquoScuffing theory modeling andexperimental correlationsrdquo Journal of Tribology vol 113 no 2pp 327ndash334 1991

[2] D Dowson G Dalmaz Childs M T H C Godet and C MTaylorThin Films in Tribology Elsevier Science 1993

[3] J C Enthoven P M Cann and H A Spikes ldquoTemperatureand scuffingrdquoTribology Transactions vol 36 no 2 pp 258ndash2661993

[4] D Dowson G Dalmaz T H C Childs and C M TaylorLubricants and Lubrication Elsevier Science 1995

[5] C Taylor L Flamand G Dalmaz et al Elastohydrodynamicsmdashrsquo96 Fundamentals and Applications in Lubrication and TractionElsevier Science 1997

[6] M Priest P Ehret L Flamand et al Lubrication at the Frontierthe Role of the Interface and Surface Layers in the Thin Film andBoundary Regime Elsevier Science 1999

[7] A Cornet and J P Deville Physique et Ingenierie des SurfacesEDP Sciences 1998

[8] W Piekoszewski M Szczerek and W Tuszynski ldquoThe actionof lubricants under extreme pressure conditions in a modifiedrdquoWear vol 249 no 3-4 pp 188ndash193 2001

[9] M Wisniewski M Szczerek and W Tuszynski ldquoThe tempera-tures at scuffing and seizure in a four-ball contactrdquo LubricationScience vol 16 no 3 pp 215ndash227 2004

[10] Q Wang ldquoSeizure failure of journal-bearing conformal con-tactsrdquoWear vol 210 no 1-2 pp 8ndash16 1997

[11] G Spinnler Conception Des MachinesmdashPrincipes Et Applica-tions vol 1 Presses Polytechniques et Universitaires Romandes2002

[12] L Flamand Fatigue Des Surface Techniques De LrsquoIngenieur1993

[13] J L Ligier Materiaux Pour Palier Lisses Techniques DeLrsquoIngenieur 1995

[14] J L Ligier Avaries En Lubrification Ecole Nationales SuperieureDu Petrole Et Des Moteurs Editions Technip 2007

[15] M Aublin R Boncompain M Boulaton et al SystemesMecaniquesmdashTheorie Et Dimensionnement Dunod 2005

[16] J L Ligier and R Gojon ldquoPrediction de lrsquousure et du grippagedrsquoun palier de moteur dieselrdquo Tech Rep SIA 93065 1993

[17] J L Ligier Lubrification des Paliers Moteurs Ecole NationalesSuperieure du Petrole et des Moteurs Editions Technip 1997

[18] S Picard Fonctionnement et Maintenance Du VehiculemdashTransmission amp Freinage Delta Press 1994

[19] J L Fanchon Guide des Sciences et Technologies IndustriellesNathan 2001

[20] J HMathieu E Bergmann and R GrasAnalyse et Technologiedes Surfaces Couches Minces et Tribologie Presses Polytech-niques et Universitaires Romandes Traite desMateriaux Tome2003

[21] V N Constantinescu Laminar Viscous Flow Mechanical Engi-neering Series Springer 1995

[22] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part IWithout consideration of contact adhering layer in the inletzonerdquo Journal of Molecular Liquids vol 130 no 1ndash3 pp 88ndash942007

[23] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part II-Considering the contact adhering layer effect in the inlet zonerdquoJournal of Molecular Liquids vol 130 no 1ndash3 pp 95ndash103 2007

[24] S Li and A Kahraman ldquoA fatigue model for contacts undermixed elastohydrodynamic lubrication conditionrdquo Interna-tional Journal of Fatigue vol 33 no 3 pp 427ndash436 2011

[25] K L Johnson J A Greenwood and S Y Poon ldquoA simple theoryof asperity contact in elastohydro-dynamic lubricationrdquo Wearvol 19 no 1 pp 91ndash108 1972

[26] G Ramsey Elastohydrodynamics Imperial College Press 2001[27] B J Hamrock S R Schmid and B O Jacobson Fundamentals

of Fluid Film Lubrication CRC Press 2004[28] K L Johnson Contact Mechanics Cambridge University Press

1987[29] R Brun Manuel Du Mecanicien et Du Thermicien Editions

Technip 2000[30] P Arques Diagnostic Predictif et Defaillances Des Machines

Theorie Traitement Analyse Reconnaissance Prediction Edi-tions Technip 2007

[31] J Frene D Nicolas B Degueurce D Berthe and M GodetLubrication Hydrodynamique Paliers Et Butees Eyrolles 1990

[32] J Frene ldquoButees et paliers hydrodynamiquerdquo Tech Rep B5055Techniques de lrsquoIngenieur 1995

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International Journal of

The Scientific World Journal 5

120595

Figure 5 Over-centre mechanism between the satellite pinion andsatellite-carrier axis 120595 denotes the angular eccentricity

2Rp

dax

Rs

Figure 6 Satellite bevel pinions and satellite-carrier axis

where 119865119905119895 119865119903119895 and 119865119886

119895(with 119895 = lw rw) are the tangential

radial and axial forces of 119895-planetary pinions (associatedwiththe left and rightwheels Figure 7 see [19]) respectively119883sa isthe resulting force applied to the satellite pinion generated bythe plastic shell 120575 is the half-angle pitch radius of the satellitepinion (see Figure 8(a)) and 120572 is the pressure angle betweenthe satellite and planetary pinions which is assumed hereto be identical between the satelliteplanetary meshing gears(see Figure 8(b))

y

x

z

y

x

z

Fax

Fax

Frlw

Falw Fa

rw

Frrw

Frlw Fr

rw

Ftlw Ft

rw

Psphsa

Psphsa

Figure 7 Forces applied to a satellite pinion

The pressure applied to the head surface of the satellitepinion 119875sphsa (its spherical part see Figure 9) due to the result-ing force119883sa (in line with (20)) matches the relationship (see[15])

119883sa = int2120587

0

int

1205751

1205750

119875sphsa (120579 120601) 119877

2

sa cos 120579 sin 120579 119889120579 119889120601 (21)

where 1205751and 120575

0are the angles defining the spherical part of

the head-satellite pinion and 119877sa is the radius of the innersphere of the plastic shell (where the contact with the headsurface of the satellite pinion and the plastic shell occurs)

The expression for the friction torque119862119891sash applied to thehead surface of the satellite pinion can be written (see [17]) as

119862119891

sash = int2120587

0

int

1205751

1205750

120591sphsa (120579 120601) 119877

3

sa sin 120579 119889120579 119889120601

= int

2120587

0

int

1205751

1205750

120583sash119875sphsa (120579 120601) 119877

3

sa sin 120579 119889120579 119889120601

(22)

where 120591sphsa (120579 120601) = 120583sash 119875sphsa (120579 120601) denotes the shear stress

exerted on the head surface of the satellite pinion since aCoulomb-type friction law is adopted (see Figure 9) and120583sash is the coefficient of friction (which is constant) betweenthe satellite pinion and the plastic shell

6 The Scientific World Journal

120575

1205751

1205750Rs

Rsa

Rsa

Rp

dax

1205759984000

1205752

120575

(a)

120572

(b)

Figure 8 Geometry of a differential gearbox

Psphj

120591sphj

Figure 9 Pressure 119875sph119895

(solid arrow) and shear 120591sph119895

(dashed arrow)fields on a head-pinion 119895 (with 119895 = (sa pl)) caused by the plasticshell

Combining (14) and (20)ndash(22) and assuming that ina first approximation the pressure field exerted on the

head-satellite pinion is uniform (ie 119875sphsa (120579 120601) = 119875sphsa ) (21)

and (22) reduce to

119883sa = 21205871198772

sa119875sphsa (

cos (21205750) minus cos (2120575

1)

4)

119862119891

sash = 120583sash119877sa119862rg

2119877119901

tan120572 sin 120575

times (cos (120575

1+ 1205750) sin (120575

1minus 1205750) minus (120575

1minus 1205750)

sin (1205750+ 1205751) sin (120575

0minus 1205751)

)

(23)

234 Power Dissipated between the Planetary Bevel Pinionsand the Plastic Shell The power dissipated between the twoplanetary pinions and the plastic shell 119875plsh can be writtenas

119875plsh = 119862119891(lw)plsh 120596

rslw + 119862

119891(rw)plsh 120596

rsrw (24)

where119862119891(119894)plsh are the frictional torque between the planetary 119894-pinion (119894 = (lw rw) refers to the left wheel (lw) and the rightwheel (rw)) and the plastic shell and 120596rs

119894= Δ1205962 denotes

the relative speed of the 119894-pinion in relation to the sumpdifferential

Assuming that 119862119891(lw)plsh cong 119862119891(rw)plsh = 119862

119891

plsh (24) reads

119875plsh = 119862119891

plshΔ120596 (25)

Using the sameprocedure as above namelymeshing eachof the planetary pinions with two satellite pinions which

The Scientific World Journal 7

means that the resulting force applied to the head-planetarypinion by the plastic shell119883pl is

119883pl = 119865ax tan120572 sin 120575 (26)

where 120575 is the half-angle pitch radius of the planetary pinion(see Figure 8(a)) Note that given the particular geometry ofthe mechanism the sum of the half-angle pitch radii of theplanetary and satellite pinions satisfies 120575 + 120575 = 1205872 and wetherefore obtain the following relationship sin 120575 = cos 120575

The expressions for both the resulting force 119883pl and thefriction torque119862119891plsh applied to the head-satellite pinion are

119883pl = int2120587

0

int

1205752

1205751015840

0

119875sphpl (120579 120601) 119877

2

sa cos 120579 sin 120579 119889120579 119889120601

119862119891

plsh = int2120587

0

int

1205752

1205751015840

0

120591sphpl (120579 120601) 119877

3

sa sin 120579 119889120579 119889120601

(27)

where 120591sphpl (120579 120601) = 120583plsh119875sphpl (120579 120601) denotes the shear stress

exerted on the outer surface of the planetary pinion with aCoulomb-type friction law (Figure 9) 120583plsh is the coefficientof friction (which is constant) between the satellite pinionand the plastic shell 119875sphpl is the pressure applied to the headsurface of the planetary pinion and 120575

2and 1205751015840

0are the angles

defining the spherical portion of the head-planetary pinionIt is again assumed here that in a first approximation the

pressure field 119875sphpl exerted on the head-planetary pinion isuniform (ie (120579 120601)-independent) and (27) reduce to

119883pl = 21205871198772

sa119875sphpl (

cos (212057510158400) minus cos (2120575

2)

4)

119862119891

plsh = 120583plsh119877sa119862rg

2119877119901

tan120572 cos 120575

times (cos (120575

2+ 1205751015840

0) sin (120575

2minus 1205751015840

0) minus (120575

2minus 1205751015840

0)

sin (12057510158400+ 1205752) sin (1205751015840

0minus 1205752)

)

(28)

235 Total Power Dissipated in the Differential GearboxBased on the above constitutive equations the total powerdissipated in the differential gearbox can bewritten as follows

119875ab = 119875sapl + 119875axsa + 119875sash + 119875plsh

= (1 minus 120578sapl) 119862rg120596rg119867+(Δ120596) + 120583axsa

119862rg

4

119889ax119877119904

Δ120596

+ 120583sash119877sa119862rg

2119877119904

tan120572 sin 120575

times (cos (120575

1+ 1205750) sin (120575

1minus 1205750) minus (120575

1minus 1205750)

sin (1205750+ 1205751) sin (120575

0minus 1205751)

)Δ120596

+ 120583plsh119877sa119862rg

2119877119901

tan120572 cos 120575

times (cos (120575

2+ 1205751015840

0) sin (120575

2minus 1205751015840

0) minus (120575

2minus 1205751015840

0)

sin (12057510158400+ 1205752) sin (1205751015840

0minus 1205752)

)Δ120596

(29)

Using (10) and (29) and assuming that in a first approx-imation 120583sash = 120583plsh = 120583sh (where 120583sh denotes thecoefficient of friction between each of the pinions in contactwith the plastic shell) the total differential gearbox yield canbe written as follows

120578di = 1 minussum119894

119897119894

= 1 minus

[[[[

[

119875sapl

119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897sapl

+119875axsa119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897axsa

+119875sash119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897sash

+119875plsh

119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897plsh

]]]]

]

(30)

with

119897sapl = (1 minus 120578sapl)119867+(Δ120596)

119897axsa = 120583axsa119889ax4119877119904

Δ120596

120596rg

119897sash = 120583sh119877sa2119877119904

tan120572 sin 120575

times (cos (120575

1+ 1205750) sin (120575

1minus 1205750) minus (120575

1minus 1205750)

sin (1205750+ 1205751) sin (120575

0minus 1205751)

)Δ120596

120596rg

119897plsh = 120583sh119877sa2119877119901

tan120572 cos 120575

times (cos (120575

2+ 1205751015840

0) sin (120575

2minus 1205751015840

0) minus (120575

2minus 1205751015840

0)

sin (12057510158400+ 1205752) sin (1205751015840

0minus 1205752)

)Δ120596

120596rg

(31)

where 119897119894= 119875119894119875rg denotes the loss ratio of the 119894-mechanism

(with 119894 = (sapl axsa sash plsh)) It is worth noting thatthe differential yield 120578di shows mechanical friction losseswhenever the rotational-speed differs between two of theplanetary pinions associated with the driven wheels that iswhen Δ120596 gt 0

3 Discussions

In the first part of this section we discuss the order of magni-tude of the constitutive parameters of the model presented inSection 2 In the second part we present a sensitivity analysis

8 The Scientific World Journal

0 05 1 15 2085

09

095

1D

iffer

entia

l gea

rbox

yie

ld120578

di

Δ120596120596rg

(a)

0 05 1 15 20

0005

001

0015

002

0025

003

0035

004

PiP

rg

Δ120596120596rg

(b)

Figure 10 (a) Yield of a differential gearbox 120578di as a function of the load parameterΔ120596120596rg for various sets of coefficients of friction (120583axsa 120583sh)(003 005) in a dashed line (005 007) in a dash-dot line (003 007) in a dotted line and in a solid line for (120583axsa 120583sh) and 120578sapl undergoinga concomitant linear increase from (003 005) to (005 007) and 098 to 096 respectively with Δ120596120596rg the thick solid line gives the yieldof the differential gearbox without any losses (which is constant) with the load parameter Δ120596120596rg that is 119897119894 = 119875119894119875rg = 0 forall119894 (b) Loss ratio119897119894= 119875119894119875rg of the 119894-mechanism (with 119894 = (sapl axsa sash plsh)) as a function of the load parameter Δ120596120596rg and with the same sets of

coefficients of friction as in (a) 119897sapl without (black solid line) or with (black dashed line) a linear decrease in 120578sapl from 098 to 096 119897axsa(red line) 119897sash (blue line) and 119897plsh (green line)

0 005 01 015 020

001

002

003

004

005

006

Cf iC

rg

Coefficient of friction 120583i

Figure 11 Friction torque ratio 119862119894119862rg as a function of coefficient of

friction 120583119894with 119894 = (axsa sash plsh) (while the other parameters

were taken to be constant) 119862axsa119862rg increased linearly with 120583axsa(red solid line) 119862sash119862rg increased linearly with 120583sash = 120583sh (bluesolid line)119862plsh119862rg increased linearly with 120583plsh = 120583sh (green solidline) in a range of variation 120583

119894isin [0 02]

in which the response of the model to a given load parameterwas examined in order to assess the effect of this responseon themechanical losses occurring in the differential gearboxmechanism

31 Order of Magnitude of the Constitutive Parameters Thisanalytical model involves thirteen parameters (120578sapl 120583axsa

0 200 400 600 800 10000

02

04

06

08

1

12

14

16

18

2

Time (s)

Δ120596120596

rg

Figure 12 Example of a load parameterΔ120596120596rg with a complex pathdepending on time 119905 isin [0 1000 s] when Δ120596120596rg = 0 (in the casewhere 120596lw = 0 and 120596rw = 0) then 120578di = 1 and 119897119894 = 119875119894119875rg = 0 forall119894 (whenthe vehiclewas travelling along a straight line) when 0 lt Δ120596120596rg lt 2

then 120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894 (when the vehicle was taking acurvilinear course) when Δ120596120596rg = 2 then 120578di lt 1 and 119897119894 = 119875119894119875rg gt0 forall119894 (when one of the driven wheels underwent slipping the vehiclewas stationary) on the plateau Δ120596120596rg = constant and |120596lw minus120596rw| =

constant (the vehicle was taking a constant curvilinear trajectory)

120583sh 119889ax 119877119904 119877119901 119877sa 120572 120575 1205750 1205751015840

0 1205751 1205752) consisting of (i) ten

geometrical parameters (119889ax 119877119904 119877119901 119877sa 120572 120575 1205750 1205751015840

0 1205751 1205752)

depending on the differential gearbox under consideration

The Scientific World Journal 9

0 200 400 600 800 1000091

092

093

094

095

096

097

098

099

1

Time (s)

Diff

eren

tial g

earb

ox y

ield

120578di

(a)

0 200 400 600 800 10000

0005

001

0015

002

0025

003

0035

Time (s)

PiP

rg

(b)

Figure 13 (a) Yield of a differential gearbox 120578119889119894with time 119905 isin [0 1000 s] in the case of the complex path presented in Figure 12 with various

sets of friction coefficients (120583axsa 120583sh) (003005) in a dashed line (005007) in a dash-dot line and (003007) in a dotted line (b) Lossratio 119897

119894= 119875119894119875rg of the 119894-mechanism (with 119894 = (sapl axsa sash plsh)) with time 119905 isin [0 1000 s] in the case of the complex path presented

in Figure 12 with the same sets of friction coefficients as in (a) 119897sapl = 096 which remained constant with time 119905 (black solid line) 119897axsa (redline) 119897sash (blue line) and 119897plsh (green line)

which can be identified fairly easily (ii) one yield param-eter (120578sapl) corresponding to the power transmitted by thegear between the satelliteplanetary pinions (iii) two fric-tion parameters (120583axsa 120583sh) which depend considerably onthe types of materials in contact and the surface conditionstheir state of lubrication and their temperature

Concerning (i) although these specific parameters de-pend on the differential gearbox under consideration theirorder of magnitude can be said to be 119889ax asymp 10ndash20mm 119877

119904asymp

20ndash30mm 119877119901asymp 20ndash30mm 119877sa asymp 30ndash50mm and for the

angle parameters 1205750asymp 10∘ndash15∘ 1205751015840

0asymp 15∘ndash25∘ 120575

1asymp 30∘ndash40∘

1205752asymp 35∘ndash45∘ 120572 asymp 20ndash25∘ 120575 asymp 30∘ndash40∘ readers can refer to

Fanchon [19] for further details about some of themConcerning (ii) the parameter 120578sapl which denotes the

power transmitted by the gear between satelliteplanetarypinions using a yield term has a value ranging between 092and 098 Since Fanchon [19] has reported that the yield inthe case of two meshed pinions is around 98 it can beconcluded that in a differential mechanism where there arefour pinions in contact the lowest value of 120578sapl asymp (098)

4cong

092 and the highest value is likely to be 120578sapl cong 098 Ittherefore seems reasonable to assume that the real yield 120578saplof this complex mechanism is approximately 096

The parameters that need to be investigated more closelyin order to determine their influence on themechanical lossesare the various coefficients of friction (iii) 120583axsa and 120583shThe sliding contacts in the differential gearbox mechanismcan be of very different kinds [20 21] since both thesatelliteplanetary bevel pinion and satellite-pinionsatellite-carrier axis in contact are in the mixed elastohydrodynamiclubrication (MEHL) mode [22ndash24] which is a mixed regimebetween (1) elastohydrodynamic lubrication (EHL) [25 26]where the heavy loads exerted at the surface of the two gears

(due to the presence of both high contact pressures and strongtransmitted torques) induce elastic strains in the two solidsin contact with thin lubricant films and (2) boundary lubri-cation (BL) [27] which may tend to constitute a completelydry frictional contact [28] leading to the development ofscuffing effects [29 30] The contacts between the bevel headpinion (satellites and planetaries) and the plastic shell arein the elastohydrodynamic lubrication mode (EHL) whichcan lead to the development of the hydrodynamic lubricationregime (HL) [31 32] where the lubricant film thickness islarge enough to completely separate the directly apposedsurfaces so that the loads applied between the surfaces arerestored and balanced under specific operating conditionsUnder the above conditions the contacts existing betweenthe satellite pinions and the satellite-carrier axis are of thesteelsteel contact with lubrication type 120583axsa isin [005 007]and those existing between the head pinions and the plasticshell are rather of the steelTeflon contact with lubricationtype 120583sh isin [003 005]

32 Sensitivity Analysis Assessment of the Modelrsquos Responseto a Given Load Parameter and Prediction of the MechanicalLosses In what follows it is proposed to test the modelrsquosresponse to a given load parameter and to determinethe influence of some parameters (120578sapl 120583axsa 120583sh) on themechanical losses occurring in a differential gearbox mecha-nism The expression for the load parameter Δ120596120596rg isin [0 2]means that (i) when Δ120596120596rg = 0 (in cases where 120596lw = 0 and120596rw = 0) the vehicle is travelling on a straight line 120578di = 1 and119897119894= 119875119894119875rg = 0 forall119894 (ii) when 0 lt Δ120596120596rg lt 2 the vehicle is

taking a curvilinear course 120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894(iii) when Δ120596120596rg = 2 one of the driven wheels undergoes

10 The Scientific World Journal

slipping (the extreme situation) the vehicle is stationary120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894

The values adopted for the other parameters are 119889ax =18 times 10

minus3m 119877119904= 22 times 10

minus3m 119877119901= 27 times 10

minus3m 119877sa =49 times 10

minus3m 1205750= 10∘ 12057510158400= 18∘ 1205751= 32∘ 1205752= 40∘ 120572 = 20∘

120575 = 39∘

Figure 10(a) shows the yield of a differential gearbox 120578didepending on the load parameter Δ120596120596rg with various setsof friction coefficients of friction (120583axsa 120583sh) and a lineardecrease in 120578sapl from 098 to 096 In the most gentle oper-ating range that is 0 le Δ120596120596rg le 1 120578di decreased from 098to around 0963 (120583axsa 120583sh) = (003 005) 120578di isin [098 0959](resp 120578di isin [098 0954]) when (120583axsa 120583sh) = (003 005)(resp (120583axsa 120583sh) = (005 007)) and 120578di decrease from 098to around 0949 when (120583axsa 120583sh) showed a concomitantlinear increase from (003 005) to (005 007) In the heavyoperating range that is 1 lt Δ120596120596rg le 2 the maximumdifferential gearbox yield 120578di isin ]0963 0946] was obtainedwith (120583axsa 120583sh) = (003 005) and the minimum yield(120578di isin ]0949 0909]) occurred when (120583axsa 120583sh) and 120578saplincreased linearly from (003 005) to (005 007) concomi-tantly with the load parameter Δ120596120596rg 120578di isin ]0959 0938](resp 120578di isin ]0954 0929]) when (120583axsa 120583sh) = (003 005)(resp (120583axsa 120583sh) = (005 007)) The results presentedin Figure 10(b) show the loss ratio 119897

119894= 119875119894119875rg associated

with 119894-mechanism (where 119894 = (sapl axsa sash plsh)) indecreasing order of influence 119897axsa (red line) 119897sapl (blackline) 119897plsh (green line) 119897sash (blue line) when 120578sapl had a con-stant value of 098 and in the case where 120578sapl decreased from098 to 096 the effects of the loss ratio 119897sapl became greaterthan those of the other parameters (119897axsa 119897axsa 119897sash 119897plsh)It should be noted that when 0 le Δ120596120596rg le 1 then119897axsa isin [0 00143] 119897plsh isin [0 00057] and 119897sash isin [0 00052]whereas when 1 lt Δ120596120596rg le 2 then 119897axsa isin [0 00286]119897plsh isin [0 00115] and 119897sash isin [0 00104] Figure 11 showsthe friction torque ratio 119862

119894119862rg as a function of the friction

coefficient 120583119894with 119894 = (axsa sash plsh) when the other

parameters were taken to be fixed constants The range ofvariation of the friction coefficient 120583

119894was taken to be between

0 and 02 (this extreme value may correspond to steelsteelcontact without any lubrification) In view of these resultsthe maximum 119862

119894119862rg ratio was obtained with 119862axsa119862rg and

the minimum one with 119862sash119862rg The friction torque ratios119862119894119862rg progressed linearly with the friction coefficient 120583

119894

(where 120583sh = 120583sash = 120583plsh) as predicted by the constitutiveequations in Section 2 Note that the results obtained in thisfirst approximation are likely to differ from those obtained bysimulating nonlinear behaviour between 119862

119894119862rg and 120583119894

A complex path of the load parameter Δ120596120596rg wasthen studied in order to test the ability of the model todescribe the differential gearbox yield 120578di and the loss ratios119897119894= 119875119894119875rg associated with the 119894-mechanism (where 119894 =

(sapl axsa sash plsh)) under more realistic conditionsinvolving a straight line situation (Δ120596120596rg = 0 with 120596lw = 0

and 120596rw = 0) a curved line situation (0 lt Δ120596120596rg lt 2)a constant curved line situation (Δ120596120596rg = constant) orwhere one of the drivenwheels undergoes slipping (Δ120596120596rg =2) Figure 12 shows the evolution of the load parameter

Δ120596120596rg as a function of the time 119905 isin [0 1000 s] Theresults obtained with a complex path in terms of the yieldof the differential gearbox (120578di) and the loss ratios (119897

119894) are

shown in Figure 13 Specifically Figure 13(a) gives the yieldof the differential gearbox 120578di as a function of the time119905 isin [0 1000 s] in the case of the complex path plotted inFigure 12 with various sets of friction coefficients (120583axsa 120583sh)(003005) in dashed line (005007) in dash-dot line and(003007) in dotted line During the whole complex path ofΔ120596120596rg the maximum (resp minimum) yield of differentialgearbox was reached with 120583axsa = 003 and 120583sh = 005

(resp 120583axsa = 005 and 120583sh = 007) On the 119896-plate (with119896 = 1 2 3 4) defined by the 119896-time interval [119905119894 119905119891]

119896 it can be

noted that for example (1) in [400 s 500 s]2 120578di asymp 09774with

(005007) 120578di asymp 09591 with (003007) and 120578di asymp 09547with (005007) (2) in [900 s 950 s]

4 120578di asymp 09464 with

(005007) 120578di asymp 09382 with (003007) and 120578di asymp 09294with (005007) The loss ratios due to friction 119897

119894= 119875119894119875rg are

shown in Figure 13(b) The maximum loss ratios obtained indecreasing order of magnitude (along the loading path) were119897sapl (black line) 119897axsa (red line) 119897plsh (green line) and 119897sash(blue line) except for the last loadwhere 119897axsa could be greaterthan 119897sapl with the pair of friction coefficients (005007)and (003007) It should be noted that (1) on the secondplate ([400 s 500 s]

2) 119897max

axsa asymp 00143 with (005007) 119897maxplsh asymp

00057 with (003007) and 120578di asymp 00052 with (005007)(2) on the fourth plate ([900 s 950 s]

4) 119897max

axsa asymp 00286 with(005007) 119897max

plsh asymp 00115 with (003007) and 120578di asymp 00104with (005007)

Although only a sensitivity analysis was performed on themodel the results obtained show quite clearly that this modelcan be used to assess and predict the mechanical frictionlosses occurring in a differential gearbox An experimentalstudy shall be conducted in order to obtain more realisticvalues for some of the parameters such as the frictioncoefficients (120583axsa and 120583sh) which significantly influenceboth the friction torques and the differential gearbox yield

4 Conclusion

In this paper an analytical model is presented for assessingand predicting the mechanical friction losses occurring indifferential gearboxes Some of the parameters involvedin this model can be determined quite easily (geometricparameters) while others which are more delicate dependdirectly on the type of friction occurring in the mechanism(and therefore on the friction coefficients) which affectsthe mechanical losses to a variable extent In order totest the influence of these parameters and determine theability of the model to predict any mechanical losses asensitivity analysis was conducted After this initial numericalapproach an experimental study shall be performed in orderto obtainmore realistic values for some of the parameters andconfirm some of the assumptionsmade here in themodellingprocedure

The Scientific World Journal 11

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is greatly indebted to Dr J-L Ligier for helpfuldiscussions comments and advices The author would liketo thank also Dr Jessica Blanc for her help with this paper

References

[1] S C Lee and H S Cheng ldquoScuffing theory modeling andexperimental correlationsrdquo Journal of Tribology vol 113 no 2pp 327ndash334 1991

[2] D Dowson G Dalmaz Childs M T H C Godet and C MTaylorThin Films in Tribology Elsevier Science 1993

[3] J C Enthoven P M Cann and H A Spikes ldquoTemperatureand scuffingrdquoTribology Transactions vol 36 no 2 pp 258ndash2661993

[4] D Dowson G Dalmaz T H C Childs and C M TaylorLubricants and Lubrication Elsevier Science 1995

[5] C Taylor L Flamand G Dalmaz et al Elastohydrodynamicsmdashrsquo96 Fundamentals and Applications in Lubrication and TractionElsevier Science 1997

[6] M Priest P Ehret L Flamand et al Lubrication at the Frontierthe Role of the Interface and Surface Layers in the Thin Film andBoundary Regime Elsevier Science 1999

[7] A Cornet and J P Deville Physique et Ingenierie des SurfacesEDP Sciences 1998

[8] W Piekoszewski M Szczerek and W Tuszynski ldquoThe actionof lubricants under extreme pressure conditions in a modifiedrdquoWear vol 249 no 3-4 pp 188ndash193 2001

[9] M Wisniewski M Szczerek and W Tuszynski ldquoThe tempera-tures at scuffing and seizure in a four-ball contactrdquo LubricationScience vol 16 no 3 pp 215ndash227 2004

[10] Q Wang ldquoSeizure failure of journal-bearing conformal con-tactsrdquoWear vol 210 no 1-2 pp 8ndash16 1997

[11] G Spinnler Conception Des MachinesmdashPrincipes Et Applica-tions vol 1 Presses Polytechniques et Universitaires Romandes2002

[12] L Flamand Fatigue Des Surface Techniques De LrsquoIngenieur1993

[13] J L Ligier Materiaux Pour Palier Lisses Techniques DeLrsquoIngenieur 1995

[14] J L Ligier Avaries En Lubrification Ecole Nationales SuperieureDu Petrole Et Des Moteurs Editions Technip 2007

[15] M Aublin R Boncompain M Boulaton et al SystemesMecaniquesmdashTheorie Et Dimensionnement Dunod 2005

[16] J L Ligier and R Gojon ldquoPrediction de lrsquousure et du grippagedrsquoun palier de moteur dieselrdquo Tech Rep SIA 93065 1993

[17] J L Ligier Lubrification des Paliers Moteurs Ecole NationalesSuperieure du Petrole et des Moteurs Editions Technip 1997

[18] S Picard Fonctionnement et Maintenance Du VehiculemdashTransmission amp Freinage Delta Press 1994

[19] J L Fanchon Guide des Sciences et Technologies IndustriellesNathan 2001

[20] J HMathieu E Bergmann and R GrasAnalyse et Technologiedes Surfaces Couches Minces et Tribologie Presses Polytech-niques et Universitaires Romandes Traite desMateriaux Tome2003

[21] V N Constantinescu Laminar Viscous Flow Mechanical Engi-neering Series Springer 1995

[22] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part IWithout consideration of contact adhering layer in the inletzonerdquo Journal of Molecular Liquids vol 130 no 1ndash3 pp 88ndash942007

[23] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part II-Considering the contact adhering layer effect in the inlet zonerdquoJournal of Molecular Liquids vol 130 no 1ndash3 pp 95ndash103 2007

[24] S Li and A Kahraman ldquoA fatigue model for contacts undermixed elastohydrodynamic lubrication conditionrdquo Interna-tional Journal of Fatigue vol 33 no 3 pp 427ndash436 2011

[25] K L Johnson J A Greenwood and S Y Poon ldquoA simple theoryof asperity contact in elastohydro-dynamic lubricationrdquo Wearvol 19 no 1 pp 91ndash108 1972

[26] G Ramsey Elastohydrodynamics Imperial College Press 2001[27] B J Hamrock S R Schmid and B O Jacobson Fundamentals

of Fluid Film Lubrication CRC Press 2004[28] K L Johnson Contact Mechanics Cambridge University Press

1987[29] R Brun Manuel Du Mecanicien et Du Thermicien Editions

Technip 2000[30] P Arques Diagnostic Predictif et Defaillances Des Machines

Theorie Traitement Analyse Reconnaissance Prediction Edi-tions Technip 2007

[31] J Frene D Nicolas B Degueurce D Berthe and M GodetLubrication Hydrodynamique Paliers Et Butees Eyrolles 1990

[32] J Frene ldquoButees et paliers hydrodynamiquerdquo Tech Rep B5055Techniques de lrsquoIngenieur 1995

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International Journal of

6 The Scientific World Journal

120575

1205751

1205750Rs

Rsa

Rsa

Rp

dax

1205759984000

1205752

120575

(a)

120572

(b)

Figure 8 Geometry of a differential gearbox

Psphj

120591sphj

Figure 9 Pressure 119875sph119895

(solid arrow) and shear 120591sph119895

(dashed arrow)fields on a head-pinion 119895 (with 119895 = (sa pl)) caused by the plasticshell

Combining (14) and (20)ndash(22) and assuming that ina first approximation the pressure field exerted on the

head-satellite pinion is uniform (ie 119875sphsa (120579 120601) = 119875sphsa ) (21)

and (22) reduce to

119883sa = 21205871198772

sa119875sphsa (

cos (21205750) minus cos (2120575

1)

4)

119862119891

sash = 120583sash119877sa119862rg

2119877119901

tan120572 sin 120575

times (cos (120575

1+ 1205750) sin (120575

1minus 1205750) minus (120575

1minus 1205750)

sin (1205750+ 1205751) sin (120575

0minus 1205751)

)

(23)

234 Power Dissipated between the Planetary Bevel Pinionsand the Plastic Shell The power dissipated between the twoplanetary pinions and the plastic shell 119875plsh can be writtenas

119875plsh = 119862119891(lw)plsh 120596

rslw + 119862

119891(rw)plsh 120596

rsrw (24)

where119862119891(119894)plsh are the frictional torque between the planetary 119894-pinion (119894 = (lw rw) refers to the left wheel (lw) and the rightwheel (rw)) and the plastic shell and 120596rs

119894= Δ1205962 denotes

the relative speed of the 119894-pinion in relation to the sumpdifferential

Assuming that 119862119891(lw)plsh cong 119862119891(rw)plsh = 119862

119891

plsh (24) reads

119875plsh = 119862119891

plshΔ120596 (25)

Using the sameprocedure as above namelymeshing eachof the planetary pinions with two satellite pinions which

The Scientific World Journal 7

means that the resulting force applied to the head-planetarypinion by the plastic shell119883pl is

119883pl = 119865ax tan120572 sin 120575 (26)

where 120575 is the half-angle pitch radius of the planetary pinion(see Figure 8(a)) Note that given the particular geometry ofthe mechanism the sum of the half-angle pitch radii of theplanetary and satellite pinions satisfies 120575 + 120575 = 1205872 and wetherefore obtain the following relationship sin 120575 = cos 120575

The expressions for both the resulting force 119883pl and thefriction torque119862119891plsh applied to the head-satellite pinion are

119883pl = int2120587

0

int

1205752

1205751015840

0

119875sphpl (120579 120601) 119877

2

sa cos 120579 sin 120579 119889120579 119889120601

119862119891

plsh = int2120587

0

int

1205752

1205751015840

0

120591sphpl (120579 120601) 119877

3

sa sin 120579 119889120579 119889120601

(27)

where 120591sphpl (120579 120601) = 120583plsh119875sphpl (120579 120601) denotes the shear stress

exerted on the outer surface of the planetary pinion with aCoulomb-type friction law (Figure 9) 120583plsh is the coefficientof friction (which is constant) between the satellite pinionand the plastic shell 119875sphpl is the pressure applied to the headsurface of the planetary pinion and 120575

2and 1205751015840

0are the angles

defining the spherical portion of the head-planetary pinionIt is again assumed here that in a first approximation the

pressure field 119875sphpl exerted on the head-planetary pinion isuniform (ie (120579 120601)-independent) and (27) reduce to

119883pl = 21205871198772

sa119875sphpl (

cos (212057510158400) minus cos (2120575

2)

4)

119862119891

plsh = 120583plsh119877sa119862rg

2119877119901

tan120572 cos 120575

times (cos (120575

2+ 1205751015840

0) sin (120575

2minus 1205751015840

0) minus (120575

2minus 1205751015840

0)

sin (12057510158400+ 1205752) sin (1205751015840

0minus 1205752)

)

(28)

235 Total Power Dissipated in the Differential GearboxBased on the above constitutive equations the total powerdissipated in the differential gearbox can bewritten as follows

119875ab = 119875sapl + 119875axsa + 119875sash + 119875plsh

= (1 minus 120578sapl) 119862rg120596rg119867+(Δ120596) + 120583axsa

119862rg

4

119889ax119877119904

Δ120596

+ 120583sash119877sa119862rg

2119877119904

tan120572 sin 120575

times (cos (120575

1+ 1205750) sin (120575

1minus 1205750) minus (120575

1minus 1205750)

sin (1205750+ 1205751) sin (120575

0minus 1205751)

)Δ120596

+ 120583plsh119877sa119862rg

2119877119901

tan120572 cos 120575

times (cos (120575

2+ 1205751015840

0) sin (120575

2minus 1205751015840

0) minus (120575

2minus 1205751015840

0)

sin (12057510158400+ 1205752) sin (1205751015840

0minus 1205752)

)Δ120596

(29)

Using (10) and (29) and assuming that in a first approx-imation 120583sash = 120583plsh = 120583sh (where 120583sh denotes thecoefficient of friction between each of the pinions in contactwith the plastic shell) the total differential gearbox yield canbe written as follows

120578di = 1 minussum119894

119897119894

= 1 minus

[[[[

[

119875sapl

119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897sapl

+119875axsa119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897axsa

+119875sash119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897sash

+119875plsh

119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897plsh

]]]]

]

(30)

with

119897sapl = (1 minus 120578sapl)119867+(Δ120596)

119897axsa = 120583axsa119889ax4119877119904

Δ120596

120596rg

119897sash = 120583sh119877sa2119877119904

tan120572 sin 120575

times (cos (120575

1+ 1205750) sin (120575

1minus 1205750) minus (120575

1minus 1205750)

sin (1205750+ 1205751) sin (120575

0minus 1205751)

)Δ120596

120596rg

119897plsh = 120583sh119877sa2119877119901

tan120572 cos 120575

times (cos (120575

2+ 1205751015840

0) sin (120575

2minus 1205751015840

0) minus (120575

2minus 1205751015840

0)

sin (12057510158400+ 1205752) sin (1205751015840

0minus 1205752)

)Δ120596

120596rg

(31)

where 119897119894= 119875119894119875rg denotes the loss ratio of the 119894-mechanism

(with 119894 = (sapl axsa sash plsh)) It is worth noting thatthe differential yield 120578di shows mechanical friction losseswhenever the rotational-speed differs between two of theplanetary pinions associated with the driven wheels that iswhen Δ120596 gt 0

3 Discussions

In the first part of this section we discuss the order of magni-tude of the constitutive parameters of the model presented inSection 2 In the second part we present a sensitivity analysis

8 The Scientific World Journal

0 05 1 15 2085

09

095

1D

iffer

entia

l gea

rbox

yie

ld120578

di

Δ120596120596rg

(a)

0 05 1 15 20

0005

001

0015

002

0025

003

0035

004

PiP

rg

Δ120596120596rg

(b)

Figure 10 (a) Yield of a differential gearbox 120578di as a function of the load parameterΔ120596120596rg for various sets of coefficients of friction (120583axsa 120583sh)(003 005) in a dashed line (005 007) in a dash-dot line (003 007) in a dotted line and in a solid line for (120583axsa 120583sh) and 120578sapl undergoinga concomitant linear increase from (003 005) to (005 007) and 098 to 096 respectively with Δ120596120596rg the thick solid line gives the yieldof the differential gearbox without any losses (which is constant) with the load parameter Δ120596120596rg that is 119897119894 = 119875119894119875rg = 0 forall119894 (b) Loss ratio119897119894= 119875119894119875rg of the 119894-mechanism (with 119894 = (sapl axsa sash plsh)) as a function of the load parameter Δ120596120596rg and with the same sets of

coefficients of friction as in (a) 119897sapl without (black solid line) or with (black dashed line) a linear decrease in 120578sapl from 098 to 096 119897axsa(red line) 119897sash (blue line) and 119897plsh (green line)

0 005 01 015 020

001

002

003

004

005

006

Cf iC

rg

Coefficient of friction 120583i

Figure 11 Friction torque ratio 119862119894119862rg as a function of coefficient of

friction 120583119894with 119894 = (axsa sash plsh) (while the other parameters

were taken to be constant) 119862axsa119862rg increased linearly with 120583axsa(red solid line) 119862sash119862rg increased linearly with 120583sash = 120583sh (bluesolid line)119862plsh119862rg increased linearly with 120583plsh = 120583sh (green solidline) in a range of variation 120583

119894isin [0 02]

in which the response of the model to a given load parameterwas examined in order to assess the effect of this responseon themechanical losses occurring in the differential gearboxmechanism

31 Order of Magnitude of the Constitutive Parameters Thisanalytical model involves thirteen parameters (120578sapl 120583axsa

0 200 400 600 800 10000

02

04

06

08

1

12

14

16

18

2

Time (s)

Δ120596120596

rg

Figure 12 Example of a load parameterΔ120596120596rg with a complex pathdepending on time 119905 isin [0 1000 s] when Δ120596120596rg = 0 (in the casewhere 120596lw = 0 and 120596rw = 0) then 120578di = 1 and 119897119894 = 119875119894119875rg = 0 forall119894 (whenthe vehiclewas travelling along a straight line) when 0 lt Δ120596120596rg lt 2

then 120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894 (when the vehicle was taking acurvilinear course) when Δ120596120596rg = 2 then 120578di lt 1 and 119897119894 = 119875119894119875rg gt0 forall119894 (when one of the driven wheels underwent slipping the vehiclewas stationary) on the plateau Δ120596120596rg = constant and |120596lw minus120596rw| =

constant (the vehicle was taking a constant curvilinear trajectory)

120583sh 119889ax 119877119904 119877119901 119877sa 120572 120575 1205750 1205751015840

0 1205751 1205752) consisting of (i) ten

geometrical parameters (119889ax 119877119904 119877119901 119877sa 120572 120575 1205750 1205751015840

0 1205751 1205752)

depending on the differential gearbox under consideration

The Scientific World Journal 9

0 200 400 600 800 1000091

092

093

094

095

096

097

098

099

1

Time (s)

Diff

eren

tial g

earb

ox y

ield

120578di

(a)

0 200 400 600 800 10000

0005

001

0015

002

0025

003

0035

Time (s)

PiP

rg

(b)

Figure 13 (a) Yield of a differential gearbox 120578119889119894with time 119905 isin [0 1000 s] in the case of the complex path presented in Figure 12 with various

sets of friction coefficients (120583axsa 120583sh) (003005) in a dashed line (005007) in a dash-dot line and (003007) in a dotted line (b) Lossratio 119897

119894= 119875119894119875rg of the 119894-mechanism (with 119894 = (sapl axsa sash plsh)) with time 119905 isin [0 1000 s] in the case of the complex path presented

in Figure 12 with the same sets of friction coefficients as in (a) 119897sapl = 096 which remained constant with time 119905 (black solid line) 119897axsa (redline) 119897sash (blue line) and 119897plsh (green line)

which can be identified fairly easily (ii) one yield param-eter (120578sapl) corresponding to the power transmitted by thegear between the satelliteplanetary pinions (iii) two fric-tion parameters (120583axsa 120583sh) which depend considerably onthe types of materials in contact and the surface conditionstheir state of lubrication and their temperature

Concerning (i) although these specific parameters de-pend on the differential gearbox under consideration theirorder of magnitude can be said to be 119889ax asymp 10ndash20mm 119877

119904asymp

20ndash30mm 119877119901asymp 20ndash30mm 119877sa asymp 30ndash50mm and for the

angle parameters 1205750asymp 10∘ndash15∘ 1205751015840

0asymp 15∘ndash25∘ 120575

1asymp 30∘ndash40∘

1205752asymp 35∘ndash45∘ 120572 asymp 20ndash25∘ 120575 asymp 30∘ndash40∘ readers can refer to

Fanchon [19] for further details about some of themConcerning (ii) the parameter 120578sapl which denotes the

power transmitted by the gear between satelliteplanetarypinions using a yield term has a value ranging between 092and 098 Since Fanchon [19] has reported that the yield inthe case of two meshed pinions is around 98 it can beconcluded that in a differential mechanism where there arefour pinions in contact the lowest value of 120578sapl asymp (098)

4cong

092 and the highest value is likely to be 120578sapl cong 098 Ittherefore seems reasonable to assume that the real yield 120578saplof this complex mechanism is approximately 096

The parameters that need to be investigated more closelyin order to determine their influence on themechanical lossesare the various coefficients of friction (iii) 120583axsa and 120583shThe sliding contacts in the differential gearbox mechanismcan be of very different kinds [20 21] since both thesatelliteplanetary bevel pinion and satellite-pinionsatellite-carrier axis in contact are in the mixed elastohydrodynamiclubrication (MEHL) mode [22ndash24] which is a mixed regimebetween (1) elastohydrodynamic lubrication (EHL) [25 26]where the heavy loads exerted at the surface of the two gears

(due to the presence of both high contact pressures and strongtransmitted torques) induce elastic strains in the two solidsin contact with thin lubricant films and (2) boundary lubri-cation (BL) [27] which may tend to constitute a completelydry frictional contact [28] leading to the development ofscuffing effects [29 30] The contacts between the bevel headpinion (satellites and planetaries) and the plastic shell arein the elastohydrodynamic lubrication mode (EHL) whichcan lead to the development of the hydrodynamic lubricationregime (HL) [31 32] where the lubricant film thickness islarge enough to completely separate the directly apposedsurfaces so that the loads applied between the surfaces arerestored and balanced under specific operating conditionsUnder the above conditions the contacts existing betweenthe satellite pinions and the satellite-carrier axis are of thesteelsteel contact with lubrication type 120583axsa isin [005 007]and those existing between the head pinions and the plasticshell are rather of the steelTeflon contact with lubricationtype 120583sh isin [003 005]

32 Sensitivity Analysis Assessment of the Modelrsquos Responseto a Given Load Parameter and Prediction of the MechanicalLosses In what follows it is proposed to test the modelrsquosresponse to a given load parameter and to determinethe influence of some parameters (120578sapl 120583axsa 120583sh) on themechanical losses occurring in a differential gearbox mecha-nism The expression for the load parameter Δ120596120596rg isin [0 2]means that (i) when Δ120596120596rg = 0 (in cases where 120596lw = 0 and120596rw = 0) the vehicle is travelling on a straight line 120578di = 1 and119897119894= 119875119894119875rg = 0 forall119894 (ii) when 0 lt Δ120596120596rg lt 2 the vehicle is

taking a curvilinear course 120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894(iii) when Δ120596120596rg = 2 one of the driven wheels undergoes

10 The Scientific World Journal

slipping (the extreme situation) the vehicle is stationary120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894

The values adopted for the other parameters are 119889ax =18 times 10

minus3m 119877119904= 22 times 10

minus3m 119877119901= 27 times 10

minus3m 119877sa =49 times 10

minus3m 1205750= 10∘ 12057510158400= 18∘ 1205751= 32∘ 1205752= 40∘ 120572 = 20∘

120575 = 39∘

Figure 10(a) shows the yield of a differential gearbox 120578didepending on the load parameter Δ120596120596rg with various setsof friction coefficients of friction (120583axsa 120583sh) and a lineardecrease in 120578sapl from 098 to 096 In the most gentle oper-ating range that is 0 le Δ120596120596rg le 1 120578di decreased from 098to around 0963 (120583axsa 120583sh) = (003 005) 120578di isin [098 0959](resp 120578di isin [098 0954]) when (120583axsa 120583sh) = (003 005)(resp (120583axsa 120583sh) = (005 007)) and 120578di decrease from 098to around 0949 when (120583axsa 120583sh) showed a concomitantlinear increase from (003 005) to (005 007) In the heavyoperating range that is 1 lt Δ120596120596rg le 2 the maximumdifferential gearbox yield 120578di isin ]0963 0946] was obtainedwith (120583axsa 120583sh) = (003 005) and the minimum yield(120578di isin ]0949 0909]) occurred when (120583axsa 120583sh) and 120578saplincreased linearly from (003 005) to (005 007) concomi-tantly with the load parameter Δ120596120596rg 120578di isin ]0959 0938](resp 120578di isin ]0954 0929]) when (120583axsa 120583sh) = (003 005)(resp (120583axsa 120583sh) = (005 007)) The results presentedin Figure 10(b) show the loss ratio 119897

119894= 119875119894119875rg associated

with 119894-mechanism (where 119894 = (sapl axsa sash plsh)) indecreasing order of influence 119897axsa (red line) 119897sapl (blackline) 119897plsh (green line) 119897sash (blue line) when 120578sapl had a con-stant value of 098 and in the case where 120578sapl decreased from098 to 096 the effects of the loss ratio 119897sapl became greaterthan those of the other parameters (119897axsa 119897axsa 119897sash 119897plsh)It should be noted that when 0 le Δ120596120596rg le 1 then119897axsa isin [0 00143] 119897plsh isin [0 00057] and 119897sash isin [0 00052]whereas when 1 lt Δ120596120596rg le 2 then 119897axsa isin [0 00286]119897plsh isin [0 00115] and 119897sash isin [0 00104] Figure 11 showsthe friction torque ratio 119862

119894119862rg as a function of the friction

coefficient 120583119894with 119894 = (axsa sash plsh) when the other

parameters were taken to be fixed constants The range ofvariation of the friction coefficient 120583

119894was taken to be between

0 and 02 (this extreme value may correspond to steelsteelcontact without any lubrification) In view of these resultsthe maximum 119862

119894119862rg ratio was obtained with 119862axsa119862rg and

the minimum one with 119862sash119862rg The friction torque ratios119862119894119862rg progressed linearly with the friction coefficient 120583

119894

(where 120583sh = 120583sash = 120583plsh) as predicted by the constitutiveequations in Section 2 Note that the results obtained in thisfirst approximation are likely to differ from those obtained bysimulating nonlinear behaviour between 119862

119894119862rg and 120583119894

A complex path of the load parameter Δ120596120596rg wasthen studied in order to test the ability of the model todescribe the differential gearbox yield 120578di and the loss ratios119897119894= 119875119894119875rg associated with the 119894-mechanism (where 119894 =

(sapl axsa sash plsh)) under more realistic conditionsinvolving a straight line situation (Δ120596120596rg = 0 with 120596lw = 0

and 120596rw = 0) a curved line situation (0 lt Δ120596120596rg lt 2)a constant curved line situation (Δ120596120596rg = constant) orwhere one of the drivenwheels undergoes slipping (Δ120596120596rg =2) Figure 12 shows the evolution of the load parameter

Δ120596120596rg as a function of the time 119905 isin [0 1000 s] Theresults obtained with a complex path in terms of the yieldof the differential gearbox (120578di) and the loss ratios (119897

119894) are

shown in Figure 13 Specifically Figure 13(a) gives the yieldof the differential gearbox 120578di as a function of the time119905 isin [0 1000 s] in the case of the complex path plotted inFigure 12 with various sets of friction coefficients (120583axsa 120583sh)(003005) in dashed line (005007) in dash-dot line and(003007) in dotted line During the whole complex path ofΔ120596120596rg the maximum (resp minimum) yield of differentialgearbox was reached with 120583axsa = 003 and 120583sh = 005

(resp 120583axsa = 005 and 120583sh = 007) On the 119896-plate (with119896 = 1 2 3 4) defined by the 119896-time interval [119905119894 119905119891]

119896 it can be

noted that for example (1) in [400 s 500 s]2 120578di asymp 09774with

(005007) 120578di asymp 09591 with (003007) and 120578di asymp 09547with (005007) (2) in [900 s 950 s]

4 120578di asymp 09464 with

(005007) 120578di asymp 09382 with (003007) and 120578di asymp 09294with (005007) The loss ratios due to friction 119897

119894= 119875119894119875rg are

shown in Figure 13(b) The maximum loss ratios obtained indecreasing order of magnitude (along the loading path) were119897sapl (black line) 119897axsa (red line) 119897plsh (green line) and 119897sash(blue line) except for the last loadwhere 119897axsa could be greaterthan 119897sapl with the pair of friction coefficients (005007)and (003007) It should be noted that (1) on the secondplate ([400 s 500 s]

2) 119897max

axsa asymp 00143 with (005007) 119897maxplsh asymp

00057 with (003007) and 120578di asymp 00052 with (005007)(2) on the fourth plate ([900 s 950 s]

4) 119897max

axsa asymp 00286 with(005007) 119897max

plsh asymp 00115 with (003007) and 120578di asymp 00104with (005007)

Although only a sensitivity analysis was performed on themodel the results obtained show quite clearly that this modelcan be used to assess and predict the mechanical frictionlosses occurring in a differential gearbox An experimentalstudy shall be conducted in order to obtain more realisticvalues for some of the parameters such as the frictioncoefficients (120583axsa and 120583sh) which significantly influenceboth the friction torques and the differential gearbox yield

4 Conclusion

In this paper an analytical model is presented for assessingand predicting the mechanical friction losses occurring indifferential gearboxes Some of the parameters involvedin this model can be determined quite easily (geometricparameters) while others which are more delicate dependdirectly on the type of friction occurring in the mechanism(and therefore on the friction coefficients) which affectsthe mechanical losses to a variable extent In order totest the influence of these parameters and determine theability of the model to predict any mechanical losses asensitivity analysis was conducted After this initial numericalapproach an experimental study shall be performed in orderto obtainmore realistic values for some of the parameters andconfirm some of the assumptionsmade here in themodellingprocedure

The Scientific World Journal 11

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is greatly indebted to Dr J-L Ligier for helpfuldiscussions comments and advices The author would liketo thank also Dr Jessica Blanc for her help with this paper

References

[1] S C Lee and H S Cheng ldquoScuffing theory modeling andexperimental correlationsrdquo Journal of Tribology vol 113 no 2pp 327ndash334 1991

[2] D Dowson G Dalmaz Childs M T H C Godet and C MTaylorThin Films in Tribology Elsevier Science 1993

[3] J C Enthoven P M Cann and H A Spikes ldquoTemperatureand scuffingrdquoTribology Transactions vol 36 no 2 pp 258ndash2661993

[4] D Dowson G Dalmaz T H C Childs and C M TaylorLubricants and Lubrication Elsevier Science 1995

[5] C Taylor L Flamand G Dalmaz et al Elastohydrodynamicsmdashrsquo96 Fundamentals and Applications in Lubrication and TractionElsevier Science 1997

[6] M Priest P Ehret L Flamand et al Lubrication at the Frontierthe Role of the Interface and Surface Layers in the Thin Film andBoundary Regime Elsevier Science 1999

[7] A Cornet and J P Deville Physique et Ingenierie des SurfacesEDP Sciences 1998

[8] W Piekoszewski M Szczerek and W Tuszynski ldquoThe actionof lubricants under extreme pressure conditions in a modifiedrdquoWear vol 249 no 3-4 pp 188ndash193 2001

[9] M Wisniewski M Szczerek and W Tuszynski ldquoThe tempera-tures at scuffing and seizure in a four-ball contactrdquo LubricationScience vol 16 no 3 pp 215ndash227 2004

[10] Q Wang ldquoSeizure failure of journal-bearing conformal con-tactsrdquoWear vol 210 no 1-2 pp 8ndash16 1997

[11] G Spinnler Conception Des MachinesmdashPrincipes Et Applica-tions vol 1 Presses Polytechniques et Universitaires Romandes2002

[12] L Flamand Fatigue Des Surface Techniques De LrsquoIngenieur1993

[13] J L Ligier Materiaux Pour Palier Lisses Techniques DeLrsquoIngenieur 1995

[14] J L Ligier Avaries En Lubrification Ecole Nationales SuperieureDu Petrole Et Des Moteurs Editions Technip 2007

[15] M Aublin R Boncompain M Boulaton et al SystemesMecaniquesmdashTheorie Et Dimensionnement Dunod 2005

[16] J L Ligier and R Gojon ldquoPrediction de lrsquousure et du grippagedrsquoun palier de moteur dieselrdquo Tech Rep SIA 93065 1993

[17] J L Ligier Lubrification des Paliers Moteurs Ecole NationalesSuperieure du Petrole et des Moteurs Editions Technip 1997

[18] S Picard Fonctionnement et Maintenance Du VehiculemdashTransmission amp Freinage Delta Press 1994

[19] J L Fanchon Guide des Sciences et Technologies IndustriellesNathan 2001

[20] J HMathieu E Bergmann and R GrasAnalyse et Technologiedes Surfaces Couches Minces et Tribologie Presses Polytech-niques et Universitaires Romandes Traite desMateriaux Tome2003

[21] V N Constantinescu Laminar Viscous Flow Mechanical Engi-neering Series Springer 1995

[22] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part IWithout consideration of contact adhering layer in the inletzonerdquo Journal of Molecular Liquids vol 130 no 1ndash3 pp 88ndash942007

[23] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part II-Considering the contact adhering layer effect in the inlet zonerdquoJournal of Molecular Liquids vol 130 no 1ndash3 pp 95ndash103 2007

[24] S Li and A Kahraman ldquoA fatigue model for contacts undermixed elastohydrodynamic lubrication conditionrdquo Interna-tional Journal of Fatigue vol 33 no 3 pp 427ndash436 2011

[25] K L Johnson J A Greenwood and S Y Poon ldquoA simple theoryof asperity contact in elastohydro-dynamic lubricationrdquo Wearvol 19 no 1 pp 91ndash108 1972

[26] G Ramsey Elastohydrodynamics Imperial College Press 2001[27] B J Hamrock S R Schmid and B O Jacobson Fundamentals

of Fluid Film Lubrication CRC Press 2004[28] K L Johnson Contact Mechanics Cambridge University Press

1987[29] R Brun Manuel Du Mecanicien et Du Thermicien Editions

Technip 2000[30] P Arques Diagnostic Predictif et Defaillances Des Machines

Theorie Traitement Analyse Reconnaissance Prediction Edi-tions Technip 2007

[31] J Frene D Nicolas B Degueurce D Berthe and M GodetLubrication Hydrodynamique Paliers Et Butees Eyrolles 1990

[32] J Frene ldquoButees et paliers hydrodynamiquerdquo Tech Rep B5055Techniques de lrsquoIngenieur 1995

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Chemical EngineeringInternational Journal of Antennas and

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DistributedSensor Networks

International Journal of

The Scientific World Journal 7

means that the resulting force applied to the head-planetarypinion by the plastic shell119883pl is

119883pl = 119865ax tan120572 sin 120575 (26)

where 120575 is the half-angle pitch radius of the planetary pinion(see Figure 8(a)) Note that given the particular geometry ofthe mechanism the sum of the half-angle pitch radii of theplanetary and satellite pinions satisfies 120575 + 120575 = 1205872 and wetherefore obtain the following relationship sin 120575 = cos 120575

The expressions for both the resulting force 119883pl and thefriction torque119862119891plsh applied to the head-satellite pinion are

119883pl = int2120587

0

int

1205752

1205751015840

0

119875sphpl (120579 120601) 119877

2

sa cos 120579 sin 120579 119889120579 119889120601

119862119891

plsh = int2120587

0

int

1205752

1205751015840

0

120591sphpl (120579 120601) 119877

3

sa sin 120579 119889120579 119889120601

(27)

where 120591sphpl (120579 120601) = 120583plsh119875sphpl (120579 120601) denotes the shear stress

exerted on the outer surface of the planetary pinion with aCoulomb-type friction law (Figure 9) 120583plsh is the coefficientof friction (which is constant) between the satellite pinionand the plastic shell 119875sphpl is the pressure applied to the headsurface of the planetary pinion and 120575

2and 1205751015840

0are the angles

defining the spherical portion of the head-planetary pinionIt is again assumed here that in a first approximation the

pressure field 119875sphpl exerted on the head-planetary pinion isuniform (ie (120579 120601)-independent) and (27) reduce to

119883pl = 21205871198772

sa119875sphpl (

cos (212057510158400) minus cos (2120575

2)

4)

119862119891

plsh = 120583plsh119877sa119862rg

2119877119901

tan120572 cos 120575

times (cos (120575

2+ 1205751015840

0) sin (120575

2minus 1205751015840

0) minus (120575

2minus 1205751015840

0)

sin (12057510158400+ 1205752) sin (1205751015840

0minus 1205752)

)

(28)

235 Total Power Dissipated in the Differential GearboxBased on the above constitutive equations the total powerdissipated in the differential gearbox can bewritten as follows

119875ab = 119875sapl + 119875axsa + 119875sash + 119875plsh

= (1 minus 120578sapl) 119862rg120596rg119867+(Δ120596) + 120583axsa

119862rg

4

119889ax119877119904

Δ120596

+ 120583sash119877sa119862rg

2119877119904

tan120572 sin 120575

times (cos (120575

1+ 1205750) sin (120575

1minus 1205750) minus (120575

1minus 1205750)

sin (1205750+ 1205751) sin (120575

0minus 1205751)

)Δ120596

+ 120583plsh119877sa119862rg

2119877119901

tan120572 cos 120575

times (cos (120575

2+ 1205751015840

0) sin (120575

2minus 1205751015840

0) minus (120575

2minus 1205751015840

0)

sin (12057510158400+ 1205752) sin (1205751015840

0minus 1205752)

)Δ120596

(29)

Using (10) and (29) and assuming that in a first approx-imation 120583sash = 120583plsh = 120583sh (where 120583sh denotes thecoefficient of friction between each of the pinions in contactwith the plastic shell) the total differential gearbox yield canbe written as follows

120578di = 1 minussum119894

119897119894

= 1 minus

[[[[

[

119875sapl

119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897sapl

+119875axsa119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897axsa

+119875sash119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897sash

+119875plsh

119875rg⏟⏟⏟⏟⏟⏟⏟⏟⏟

=119897plsh

]]]]

]

(30)

with

119897sapl = (1 minus 120578sapl)119867+(Δ120596)

119897axsa = 120583axsa119889ax4119877119904

Δ120596

120596rg

119897sash = 120583sh119877sa2119877119904

tan120572 sin 120575

times (cos (120575

1+ 1205750) sin (120575

1minus 1205750) minus (120575

1minus 1205750)

sin (1205750+ 1205751) sin (120575

0minus 1205751)

)Δ120596

120596rg

119897plsh = 120583sh119877sa2119877119901

tan120572 cos 120575

times (cos (120575

2+ 1205751015840

0) sin (120575

2minus 1205751015840

0) minus (120575

2minus 1205751015840

0)

sin (12057510158400+ 1205752) sin (1205751015840

0minus 1205752)

)Δ120596

120596rg

(31)

where 119897119894= 119875119894119875rg denotes the loss ratio of the 119894-mechanism

(with 119894 = (sapl axsa sash plsh)) It is worth noting thatthe differential yield 120578di shows mechanical friction losseswhenever the rotational-speed differs between two of theplanetary pinions associated with the driven wheels that iswhen Δ120596 gt 0

3 Discussions

In the first part of this section we discuss the order of magni-tude of the constitutive parameters of the model presented inSection 2 In the second part we present a sensitivity analysis

8 The Scientific World Journal

0 05 1 15 2085

09

095

1D

iffer

entia

l gea

rbox

yie

ld120578

di

Δ120596120596rg

(a)

0 05 1 15 20

0005

001

0015

002

0025

003

0035

004

PiP

rg

Δ120596120596rg

(b)

Figure 10 (a) Yield of a differential gearbox 120578di as a function of the load parameterΔ120596120596rg for various sets of coefficients of friction (120583axsa 120583sh)(003 005) in a dashed line (005 007) in a dash-dot line (003 007) in a dotted line and in a solid line for (120583axsa 120583sh) and 120578sapl undergoinga concomitant linear increase from (003 005) to (005 007) and 098 to 096 respectively with Δ120596120596rg the thick solid line gives the yieldof the differential gearbox without any losses (which is constant) with the load parameter Δ120596120596rg that is 119897119894 = 119875119894119875rg = 0 forall119894 (b) Loss ratio119897119894= 119875119894119875rg of the 119894-mechanism (with 119894 = (sapl axsa sash plsh)) as a function of the load parameter Δ120596120596rg and with the same sets of

coefficients of friction as in (a) 119897sapl without (black solid line) or with (black dashed line) a linear decrease in 120578sapl from 098 to 096 119897axsa(red line) 119897sash (blue line) and 119897plsh (green line)

0 005 01 015 020

001

002

003

004

005

006

Cf iC

rg

Coefficient of friction 120583i

Figure 11 Friction torque ratio 119862119894119862rg as a function of coefficient of

friction 120583119894with 119894 = (axsa sash plsh) (while the other parameters

were taken to be constant) 119862axsa119862rg increased linearly with 120583axsa(red solid line) 119862sash119862rg increased linearly with 120583sash = 120583sh (bluesolid line)119862plsh119862rg increased linearly with 120583plsh = 120583sh (green solidline) in a range of variation 120583

119894isin [0 02]

in which the response of the model to a given load parameterwas examined in order to assess the effect of this responseon themechanical losses occurring in the differential gearboxmechanism

31 Order of Magnitude of the Constitutive Parameters Thisanalytical model involves thirteen parameters (120578sapl 120583axsa

0 200 400 600 800 10000

02

04

06

08

1

12

14

16

18

2

Time (s)

Δ120596120596

rg

Figure 12 Example of a load parameterΔ120596120596rg with a complex pathdepending on time 119905 isin [0 1000 s] when Δ120596120596rg = 0 (in the casewhere 120596lw = 0 and 120596rw = 0) then 120578di = 1 and 119897119894 = 119875119894119875rg = 0 forall119894 (whenthe vehiclewas travelling along a straight line) when 0 lt Δ120596120596rg lt 2

then 120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894 (when the vehicle was taking acurvilinear course) when Δ120596120596rg = 2 then 120578di lt 1 and 119897119894 = 119875119894119875rg gt0 forall119894 (when one of the driven wheels underwent slipping the vehiclewas stationary) on the plateau Δ120596120596rg = constant and |120596lw minus120596rw| =

constant (the vehicle was taking a constant curvilinear trajectory)

120583sh 119889ax 119877119904 119877119901 119877sa 120572 120575 1205750 1205751015840

0 1205751 1205752) consisting of (i) ten

geometrical parameters (119889ax 119877119904 119877119901 119877sa 120572 120575 1205750 1205751015840

0 1205751 1205752)

depending on the differential gearbox under consideration

The Scientific World Journal 9

0 200 400 600 800 1000091

092

093

094

095

096

097

098

099

1

Time (s)

Diff

eren

tial g

earb

ox y

ield

120578di

(a)

0 200 400 600 800 10000

0005

001

0015

002

0025

003

0035

Time (s)

PiP

rg

(b)

Figure 13 (a) Yield of a differential gearbox 120578119889119894with time 119905 isin [0 1000 s] in the case of the complex path presented in Figure 12 with various

sets of friction coefficients (120583axsa 120583sh) (003005) in a dashed line (005007) in a dash-dot line and (003007) in a dotted line (b) Lossratio 119897

119894= 119875119894119875rg of the 119894-mechanism (with 119894 = (sapl axsa sash plsh)) with time 119905 isin [0 1000 s] in the case of the complex path presented

in Figure 12 with the same sets of friction coefficients as in (a) 119897sapl = 096 which remained constant with time 119905 (black solid line) 119897axsa (redline) 119897sash (blue line) and 119897plsh (green line)

which can be identified fairly easily (ii) one yield param-eter (120578sapl) corresponding to the power transmitted by thegear between the satelliteplanetary pinions (iii) two fric-tion parameters (120583axsa 120583sh) which depend considerably onthe types of materials in contact and the surface conditionstheir state of lubrication and their temperature

Concerning (i) although these specific parameters de-pend on the differential gearbox under consideration theirorder of magnitude can be said to be 119889ax asymp 10ndash20mm 119877

119904asymp

20ndash30mm 119877119901asymp 20ndash30mm 119877sa asymp 30ndash50mm and for the

angle parameters 1205750asymp 10∘ndash15∘ 1205751015840

0asymp 15∘ndash25∘ 120575

1asymp 30∘ndash40∘

1205752asymp 35∘ndash45∘ 120572 asymp 20ndash25∘ 120575 asymp 30∘ndash40∘ readers can refer to

Fanchon [19] for further details about some of themConcerning (ii) the parameter 120578sapl which denotes the

power transmitted by the gear between satelliteplanetarypinions using a yield term has a value ranging between 092and 098 Since Fanchon [19] has reported that the yield inthe case of two meshed pinions is around 98 it can beconcluded that in a differential mechanism where there arefour pinions in contact the lowest value of 120578sapl asymp (098)

4cong

092 and the highest value is likely to be 120578sapl cong 098 Ittherefore seems reasonable to assume that the real yield 120578saplof this complex mechanism is approximately 096

The parameters that need to be investigated more closelyin order to determine their influence on themechanical lossesare the various coefficients of friction (iii) 120583axsa and 120583shThe sliding contacts in the differential gearbox mechanismcan be of very different kinds [20 21] since both thesatelliteplanetary bevel pinion and satellite-pinionsatellite-carrier axis in contact are in the mixed elastohydrodynamiclubrication (MEHL) mode [22ndash24] which is a mixed regimebetween (1) elastohydrodynamic lubrication (EHL) [25 26]where the heavy loads exerted at the surface of the two gears

(due to the presence of both high contact pressures and strongtransmitted torques) induce elastic strains in the two solidsin contact with thin lubricant films and (2) boundary lubri-cation (BL) [27] which may tend to constitute a completelydry frictional contact [28] leading to the development ofscuffing effects [29 30] The contacts between the bevel headpinion (satellites and planetaries) and the plastic shell arein the elastohydrodynamic lubrication mode (EHL) whichcan lead to the development of the hydrodynamic lubricationregime (HL) [31 32] where the lubricant film thickness islarge enough to completely separate the directly apposedsurfaces so that the loads applied between the surfaces arerestored and balanced under specific operating conditionsUnder the above conditions the contacts existing betweenthe satellite pinions and the satellite-carrier axis are of thesteelsteel contact with lubrication type 120583axsa isin [005 007]and those existing between the head pinions and the plasticshell are rather of the steelTeflon contact with lubricationtype 120583sh isin [003 005]

32 Sensitivity Analysis Assessment of the Modelrsquos Responseto a Given Load Parameter and Prediction of the MechanicalLosses In what follows it is proposed to test the modelrsquosresponse to a given load parameter and to determinethe influence of some parameters (120578sapl 120583axsa 120583sh) on themechanical losses occurring in a differential gearbox mecha-nism The expression for the load parameter Δ120596120596rg isin [0 2]means that (i) when Δ120596120596rg = 0 (in cases where 120596lw = 0 and120596rw = 0) the vehicle is travelling on a straight line 120578di = 1 and119897119894= 119875119894119875rg = 0 forall119894 (ii) when 0 lt Δ120596120596rg lt 2 the vehicle is

taking a curvilinear course 120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894(iii) when Δ120596120596rg = 2 one of the driven wheels undergoes

10 The Scientific World Journal

slipping (the extreme situation) the vehicle is stationary120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894

The values adopted for the other parameters are 119889ax =18 times 10

minus3m 119877119904= 22 times 10

minus3m 119877119901= 27 times 10

minus3m 119877sa =49 times 10

minus3m 1205750= 10∘ 12057510158400= 18∘ 1205751= 32∘ 1205752= 40∘ 120572 = 20∘

120575 = 39∘

Figure 10(a) shows the yield of a differential gearbox 120578didepending on the load parameter Δ120596120596rg with various setsof friction coefficients of friction (120583axsa 120583sh) and a lineardecrease in 120578sapl from 098 to 096 In the most gentle oper-ating range that is 0 le Δ120596120596rg le 1 120578di decreased from 098to around 0963 (120583axsa 120583sh) = (003 005) 120578di isin [098 0959](resp 120578di isin [098 0954]) when (120583axsa 120583sh) = (003 005)(resp (120583axsa 120583sh) = (005 007)) and 120578di decrease from 098to around 0949 when (120583axsa 120583sh) showed a concomitantlinear increase from (003 005) to (005 007) In the heavyoperating range that is 1 lt Δ120596120596rg le 2 the maximumdifferential gearbox yield 120578di isin ]0963 0946] was obtainedwith (120583axsa 120583sh) = (003 005) and the minimum yield(120578di isin ]0949 0909]) occurred when (120583axsa 120583sh) and 120578saplincreased linearly from (003 005) to (005 007) concomi-tantly with the load parameter Δ120596120596rg 120578di isin ]0959 0938](resp 120578di isin ]0954 0929]) when (120583axsa 120583sh) = (003 005)(resp (120583axsa 120583sh) = (005 007)) The results presentedin Figure 10(b) show the loss ratio 119897

119894= 119875119894119875rg associated

with 119894-mechanism (where 119894 = (sapl axsa sash plsh)) indecreasing order of influence 119897axsa (red line) 119897sapl (blackline) 119897plsh (green line) 119897sash (blue line) when 120578sapl had a con-stant value of 098 and in the case where 120578sapl decreased from098 to 096 the effects of the loss ratio 119897sapl became greaterthan those of the other parameters (119897axsa 119897axsa 119897sash 119897plsh)It should be noted that when 0 le Δ120596120596rg le 1 then119897axsa isin [0 00143] 119897plsh isin [0 00057] and 119897sash isin [0 00052]whereas when 1 lt Δ120596120596rg le 2 then 119897axsa isin [0 00286]119897plsh isin [0 00115] and 119897sash isin [0 00104] Figure 11 showsthe friction torque ratio 119862

119894119862rg as a function of the friction

coefficient 120583119894with 119894 = (axsa sash plsh) when the other

parameters were taken to be fixed constants The range ofvariation of the friction coefficient 120583

119894was taken to be between

0 and 02 (this extreme value may correspond to steelsteelcontact without any lubrification) In view of these resultsthe maximum 119862

119894119862rg ratio was obtained with 119862axsa119862rg and

the minimum one with 119862sash119862rg The friction torque ratios119862119894119862rg progressed linearly with the friction coefficient 120583

119894

(where 120583sh = 120583sash = 120583plsh) as predicted by the constitutiveequations in Section 2 Note that the results obtained in thisfirst approximation are likely to differ from those obtained bysimulating nonlinear behaviour between 119862

119894119862rg and 120583119894

A complex path of the load parameter Δ120596120596rg wasthen studied in order to test the ability of the model todescribe the differential gearbox yield 120578di and the loss ratios119897119894= 119875119894119875rg associated with the 119894-mechanism (where 119894 =

(sapl axsa sash plsh)) under more realistic conditionsinvolving a straight line situation (Δ120596120596rg = 0 with 120596lw = 0

and 120596rw = 0) a curved line situation (0 lt Δ120596120596rg lt 2)a constant curved line situation (Δ120596120596rg = constant) orwhere one of the drivenwheels undergoes slipping (Δ120596120596rg =2) Figure 12 shows the evolution of the load parameter

Δ120596120596rg as a function of the time 119905 isin [0 1000 s] Theresults obtained with a complex path in terms of the yieldof the differential gearbox (120578di) and the loss ratios (119897

119894) are

shown in Figure 13 Specifically Figure 13(a) gives the yieldof the differential gearbox 120578di as a function of the time119905 isin [0 1000 s] in the case of the complex path plotted inFigure 12 with various sets of friction coefficients (120583axsa 120583sh)(003005) in dashed line (005007) in dash-dot line and(003007) in dotted line During the whole complex path ofΔ120596120596rg the maximum (resp minimum) yield of differentialgearbox was reached with 120583axsa = 003 and 120583sh = 005

(resp 120583axsa = 005 and 120583sh = 007) On the 119896-plate (with119896 = 1 2 3 4) defined by the 119896-time interval [119905119894 119905119891]

119896 it can be

noted that for example (1) in [400 s 500 s]2 120578di asymp 09774with

(005007) 120578di asymp 09591 with (003007) and 120578di asymp 09547with (005007) (2) in [900 s 950 s]

4 120578di asymp 09464 with

(005007) 120578di asymp 09382 with (003007) and 120578di asymp 09294with (005007) The loss ratios due to friction 119897

119894= 119875119894119875rg are

shown in Figure 13(b) The maximum loss ratios obtained indecreasing order of magnitude (along the loading path) were119897sapl (black line) 119897axsa (red line) 119897plsh (green line) and 119897sash(blue line) except for the last loadwhere 119897axsa could be greaterthan 119897sapl with the pair of friction coefficients (005007)and (003007) It should be noted that (1) on the secondplate ([400 s 500 s]

2) 119897max

axsa asymp 00143 with (005007) 119897maxplsh asymp

00057 with (003007) and 120578di asymp 00052 with (005007)(2) on the fourth plate ([900 s 950 s]

4) 119897max

axsa asymp 00286 with(005007) 119897max

plsh asymp 00115 with (003007) and 120578di asymp 00104with (005007)

Although only a sensitivity analysis was performed on themodel the results obtained show quite clearly that this modelcan be used to assess and predict the mechanical frictionlosses occurring in a differential gearbox An experimentalstudy shall be conducted in order to obtain more realisticvalues for some of the parameters such as the frictioncoefficients (120583axsa and 120583sh) which significantly influenceboth the friction torques and the differential gearbox yield

4 Conclusion

In this paper an analytical model is presented for assessingand predicting the mechanical friction losses occurring indifferential gearboxes Some of the parameters involvedin this model can be determined quite easily (geometricparameters) while others which are more delicate dependdirectly on the type of friction occurring in the mechanism(and therefore on the friction coefficients) which affectsthe mechanical losses to a variable extent In order totest the influence of these parameters and determine theability of the model to predict any mechanical losses asensitivity analysis was conducted After this initial numericalapproach an experimental study shall be performed in orderto obtainmore realistic values for some of the parameters andconfirm some of the assumptionsmade here in themodellingprocedure

The Scientific World Journal 11

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is greatly indebted to Dr J-L Ligier for helpfuldiscussions comments and advices The author would liketo thank also Dr Jessica Blanc for her help with this paper

References

[1] S C Lee and H S Cheng ldquoScuffing theory modeling andexperimental correlationsrdquo Journal of Tribology vol 113 no 2pp 327ndash334 1991

[2] D Dowson G Dalmaz Childs M T H C Godet and C MTaylorThin Films in Tribology Elsevier Science 1993

[3] J C Enthoven P M Cann and H A Spikes ldquoTemperatureand scuffingrdquoTribology Transactions vol 36 no 2 pp 258ndash2661993

[4] D Dowson G Dalmaz T H C Childs and C M TaylorLubricants and Lubrication Elsevier Science 1995

[5] C Taylor L Flamand G Dalmaz et al Elastohydrodynamicsmdashrsquo96 Fundamentals and Applications in Lubrication and TractionElsevier Science 1997

[6] M Priest P Ehret L Flamand et al Lubrication at the Frontierthe Role of the Interface and Surface Layers in the Thin Film andBoundary Regime Elsevier Science 1999

[7] A Cornet and J P Deville Physique et Ingenierie des SurfacesEDP Sciences 1998

[8] W Piekoszewski M Szczerek and W Tuszynski ldquoThe actionof lubricants under extreme pressure conditions in a modifiedrdquoWear vol 249 no 3-4 pp 188ndash193 2001

[9] M Wisniewski M Szczerek and W Tuszynski ldquoThe tempera-tures at scuffing and seizure in a four-ball contactrdquo LubricationScience vol 16 no 3 pp 215ndash227 2004

[10] Q Wang ldquoSeizure failure of journal-bearing conformal con-tactsrdquoWear vol 210 no 1-2 pp 8ndash16 1997

[11] G Spinnler Conception Des MachinesmdashPrincipes Et Applica-tions vol 1 Presses Polytechniques et Universitaires Romandes2002

[12] L Flamand Fatigue Des Surface Techniques De LrsquoIngenieur1993

[13] J L Ligier Materiaux Pour Palier Lisses Techniques DeLrsquoIngenieur 1995

[14] J L Ligier Avaries En Lubrification Ecole Nationales SuperieureDu Petrole Et Des Moteurs Editions Technip 2007

[15] M Aublin R Boncompain M Boulaton et al SystemesMecaniquesmdashTheorie Et Dimensionnement Dunod 2005

[16] J L Ligier and R Gojon ldquoPrediction de lrsquousure et du grippagedrsquoun palier de moteur dieselrdquo Tech Rep SIA 93065 1993

[17] J L Ligier Lubrification des Paliers Moteurs Ecole NationalesSuperieure du Petrole et des Moteurs Editions Technip 1997

[18] S Picard Fonctionnement et Maintenance Du VehiculemdashTransmission amp Freinage Delta Press 1994

[19] J L Fanchon Guide des Sciences et Technologies IndustriellesNathan 2001

[20] J HMathieu E Bergmann and R GrasAnalyse et Technologiedes Surfaces Couches Minces et Tribologie Presses Polytech-niques et Universitaires Romandes Traite desMateriaux Tome2003

[21] V N Constantinescu Laminar Viscous Flow Mechanical Engi-neering Series Springer 1995

[22] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part IWithout consideration of contact adhering layer in the inletzonerdquo Journal of Molecular Liquids vol 130 no 1ndash3 pp 88ndash942007

[23] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part II-Considering the contact adhering layer effect in the inlet zonerdquoJournal of Molecular Liquids vol 130 no 1ndash3 pp 95ndash103 2007

[24] S Li and A Kahraman ldquoA fatigue model for contacts undermixed elastohydrodynamic lubrication conditionrdquo Interna-tional Journal of Fatigue vol 33 no 3 pp 427ndash436 2011

[25] K L Johnson J A Greenwood and S Y Poon ldquoA simple theoryof asperity contact in elastohydro-dynamic lubricationrdquo Wearvol 19 no 1 pp 91ndash108 1972

[26] G Ramsey Elastohydrodynamics Imperial College Press 2001[27] B J Hamrock S R Schmid and B O Jacobson Fundamentals

of Fluid Film Lubrication CRC Press 2004[28] K L Johnson Contact Mechanics Cambridge University Press

1987[29] R Brun Manuel Du Mecanicien et Du Thermicien Editions

Technip 2000[30] P Arques Diagnostic Predictif et Defaillances Des Machines

Theorie Traitement Analyse Reconnaissance Prediction Edi-tions Technip 2007

[31] J Frene D Nicolas B Degueurce D Berthe and M GodetLubrication Hydrodynamique Paliers Et Butees Eyrolles 1990

[32] J Frene ldquoButees et paliers hydrodynamiquerdquo Tech Rep B5055Techniques de lrsquoIngenieur 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 The Scientific World Journal

0 05 1 15 2085

09

095

1D

iffer

entia

l gea

rbox

yie

ld120578

di

Δ120596120596rg

(a)

0 05 1 15 20

0005

001

0015

002

0025

003

0035

004

PiP

rg

Δ120596120596rg

(b)

Figure 10 (a) Yield of a differential gearbox 120578di as a function of the load parameterΔ120596120596rg for various sets of coefficients of friction (120583axsa 120583sh)(003 005) in a dashed line (005 007) in a dash-dot line (003 007) in a dotted line and in a solid line for (120583axsa 120583sh) and 120578sapl undergoinga concomitant linear increase from (003 005) to (005 007) and 098 to 096 respectively with Δ120596120596rg the thick solid line gives the yieldof the differential gearbox without any losses (which is constant) with the load parameter Δ120596120596rg that is 119897119894 = 119875119894119875rg = 0 forall119894 (b) Loss ratio119897119894= 119875119894119875rg of the 119894-mechanism (with 119894 = (sapl axsa sash plsh)) as a function of the load parameter Δ120596120596rg and with the same sets of

coefficients of friction as in (a) 119897sapl without (black solid line) or with (black dashed line) a linear decrease in 120578sapl from 098 to 096 119897axsa(red line) 119897sash (blue line) and 119897plsh (green line)

0 005 01 015 020

001

002

003

004

005

006

Cf iC

rg

Coefficient of friction 120583i

Figure 11 Friction torque ratio 119862119894119862rg as a function of coefficient of

friction 120583119894with 119894 = (axsa sash plsh) (while the other parameters

were taken to be constant) 119862axsa119862rg increased linearly with 120583axsa(red solid line) 119862sash119862rg increased linearly with 120583sash = 120583sh (bluesolid line)119862plsh119862rg increased linearly with 120583plsh = 120583sh (green solidline) in a range of variation 120583

119894isin [0 02]

in which the response of the model to a given load parameterwas examined in order to assess the effect of this responseon themechanical losses occurring in the differential gearboxmechanism

31 Order of Magnitude of the Constitutive Parameters Thisanalytical model involves thirteen parameters (120578sapl 120583axsa

0 200 400 600 800 10000

02

04

06

08

1

12

14

16

18

2

Time (s)

Δ120596120596

rg

Figure 12 Example of a load parameterΔ120596120596rg with a complex pathdepending on time 119905 isin [0 1000 s] when Δ120596120596rg = 0 (in the casewhere 120596lw = 0 and 120596rw = 0) then 120578di = 1 and 119897119894 = 119875119894119875rg = 0 forall119894 (whenthe vehiclewas travelling along a straight line) when 0 lt Δ120596120596rg lt 2

then 120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894 (when the vehicle was taking acurvilinear course) when Δ120596120596rg = 2 then 120578di lt 1 and 119897119894 = 119875119894119875rg gt0 forall119894 (when one of the driven wheels underwent slipping the vehiclewas stationary) on the plateau Δ120596120596rg = constant and |120596lw minus120596rw| =

constant (the vehicle was taking a constant curvilinear trajectory)

120583sh 119889ax 119877119904 119877119901 119877sa 120572 120575 1205750 1205751015840

0 1205751 1205752) consisting of (i) ten

geometrical parameters (119889ax 119877119904 119877119901 119877sa 120572 120575 1205750 1205751015840

0 1205751 1205752)

depending on the differential gearbox under consideration

The Scientific World Journal 9

0 200 400 600 800 1000091

092

093

094

095

096

097

098

099

1

Time (s)

Diff

eren

tial g

earb

ox y

ield

120578di

(a)

0 200 400 600 800 10000

0005

001

0015

002

0025

003

0035

Time (s)

PiP

rg

(b)

Figure 13 (a) Yield of a differential gearbox 120578119889119894with time 119905 isin [0 1000 s] in the case of the complex path presented in Figure 12 with various

sets of friction coefficients (120583axsa 120583sh) (003005) in a dashed line (005007) in a dash-dot line and (003007) in a dotted line (b) Lossratio 119897

119894= 119875119894119875rg of the 119894-mechanism (with 119894 = (sapl axsa sash plsh)) with time 119905 isin [0 1000 s] in the case of the complex path presented

in Figure 12 with the same sets of friction coefficients as in (a) 119897sapl = 096 which remained constant with time 119905 (black solid line) 119897axsa (redline) 119897sash (blue line) and 119897plsh (green line)

which can be identified fairly easily (ii) one yield param-eter (120578sapl) corresponding to the power transmitted by thegear between the satelliteplanetary pinions (iii) two fric-tion parameters (120583axsa 120583sh) which depend considerably onthe types of materials in contact and the surface conditionstheir state of lubrication and their temperature

Concerning (i) although these specific parameters de-pend on the differential gearbox under consideration theirorder of magnitude can be said to be 119889ax asymp 10ndash20mm 119877

119904asymp

20ndash30mm 119877119901asymp 20ndash30mm 119877sa asymp 30ndash50mm and for the

angle parameters 1205750asymp 10∘ndash15∘ 1205751015840

0asymp 15∘ndash25∘ 120575

1asymp 30∘ndash40∘

1205752asymp 35∘ndash45∘ 120572 asymp 20ndash25∘ 120575 asymp 30∘ndash40∘ readers can refer to

Fanchon [19] for further details about some of themConcerning (ii) the parameter 120578sapl which denotes the

power transmitted by the gear between satelliteplanetarypinions using a yield term has a value ranging between 092and 098 Since Fanchon [19] has reported that the yield inthe case of two meshed pinions is around 98 it can beconcluded that in a differential mechanism where there arefour pinions in contact the lowest value of 120578sapl asymp (098)

4cong

092 and the highest value is likely to be 120578sapl cong 098 Ittherefore seems reasonable to assume that the real yield 120578saplof this complex mechanism is approximately 096

The parameters that need to be investigated more closelyin order to determine their influence on themechanical lossesare the various coefficients of friction (iii) 120583axsa and 120583shThe sliding contacts in the differential gearbox mechanismcan be of very different kinds [20 21] since both thesatelliteplanetary bevel pinion and satellite-pinionsatellite-carrier axis in contact are in the mixed elastohydrodynamiclubrication (MEHL) mode [22ndash24] which is a mixed regimebetween (1) elastohydrodynamic lubrication (EHL) [25 26]where the heavy loads exerted at the surface of the two gears

(due to the presence of both high contact pressures and strongtransmitted torques) induce elastic strains in the two solidsin contact with thin lubricant films and (2) boundary lubri-cation (BL) [27] which may tend to constitute a completelydry frictional contact [28] leading to the development ofscuffing effects [29 30] The contacts between the bevel headpinion (satellites and planetaries) and the plastic shell arein the elastohydrodynamic lubrication mode (EHL) whichcan lead to the development of the hydrodynamic lubricationregime (HL) [31 32] where the lubricant film thickness islarge enough to completely separate the directly apposedsurfaces so that the loads applied between the surfaces arerestored and balanced under specific operating conditionsUnder the above conditions the contacts existing betweenthe satellite pinions and the satellite-carrier axis are of thesteelsteel contact with lubrication type 120583axsa isin [005 007]and those existing between the head pinions and the plasticshell are rather of the steelTeflon contact with lubricationtype 120583sh isin [003 005]

32 Sensitivity Analysis Assessment of the Modelrsquos Responseto a Given Load Parameter and Prediction of the MechanicalLosses In what follows it is proposed to test the modelrsquosresponse to a given load parameter and to determinethe influence of some parameters (120578sapl 120583axsa 120583sh) on themechanical losses occurring in a differential gearbox mecha-nism The expression for the load parameter Δ120596120596rg isin [0 2]means that (i) when Δ120596120596rg = 0 (in cases where 120596lw = 0 and120596rw = 0) the vehicle is travelling on a straight line 120578di = 1 and119897119894= 119875119894119875rg = 0 forall119894 (ii) when 0 lt Δ120596120596rg lt 2 the vehicle is

taking a curvilinear course 120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894(iii) when Δ120596120596rg = 2 one of the driven wheels undergoes

10 The Scientific World Journal

slipping (the extreme situation) the vehicle is stationary120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894

The values adopted for the other parameters are 119889ax =18 times 10

minus3m 119877119904= 22 times 10

minus3m 119877119901= 27 times 10

minus3m 119877sa =49 times 10

minus3m 1205750= 10∘ 12057510158400= 18∘ 1205751= 32∘ 1205752= 40∘ 120572 = 20∘

120575 = 39∘

Figure 10(a) shows the yield of a differential gearbox 120578didepending on the load parameter Δ120596120596rg with various setsof friction coefficients of friction (120583axsa 120583sh) and a lineardecrease in 120578sapl from 098 to 096 In the most gentle oper-ating range that is 0 le Δ120596120596rg le 1 120578di decreased from 098to around 0963 (120583axsa 120583sh) = (003 005) 120578di isin [098 0959](resp 120578di isin [098 0954]) when (120583axsa 120583sh) = (003 005)(resp (120583axsa 120583sh) = (005 007)) and 120578di decrease from 098to around 0949 when (120583axsa 120583sh) showed a concomitantlinear increase from (003 005) to (005 007) In the heavyoperating range that is 1 lt Δ120596120596rg le 2 the maximumdifferential gearbox yield 120578di isin ]0963 0946] was obtainedwith (120583axsa 120583sh) = (003 005) and the minimum yield(120578di isin ]0949 0909]) occurred when (120583axsa 120583sh) and 120578saplincreased linearly from (003 005) to (005 007) concomi-tantly with the load parameter Δ120596120596rg 120578di isin ]0959 0938](resp 120578di isin ]0954 0929]) when (120583axsa 120583sh) = (003 005)(resp (120583axsa 120583sh) = (005 007)) The results presentedin Figure 10(b) show the loss ratio 119897

119894= 119875119894119875rg associated

with 119894-mechanism (where 119894 = (sapl axsa sash plsh)) indecreasing order of influence 119897axsa (red line) 119897sapl (blackline) 119897plsh (green line) 119897sash (blue line) when 120578sapl had a con-stant value of 098 and in the case where 120578sapl decreased from098 to 096 the effects of the loss ratio 119897sapl became greaterthan those of the other parameters (119897axsa 119897axsa 119897sash 119897plsh)It should be noted that when 0 le Δ120596120596rg le 1 then119897axsa isin [0 00143] 119897plsh isin [0 00057] and 119897sash isin [0 00052]whereas when 1 lt Δ120596120596rg le 2 then 119897axsa isin [0 00286]119897plsh isin [0 00115] and 119897sash isin [0 00104] Figure 11 showsthe friction torque ratio 119862

119894119862rg as a function of the friction

coefficient 120583119894with 119894 = (axsa sash plsh) when the other

parameters were taken to be fixed constants The range ofvariation of the friction coefficient 120583

119894was taken to be between

0 and 02 (this extreme value may correspond to steelsteelcontact without any lubrification) In view of these resultsthe maximum 119862

119894119862rg ratio was obtained with 119862axsa119862rg and

the minimum one with 119862sash119862rg The friction torque ratios119862119894119862rg progressed linearly with the friction coefficient 120583

119894

(where 120583sh = 120583sash = 120583plsh) as predicted by the constitutiveequations in Section 2 Note that the results obtained in thisfirst approximation are likely to differ from those obtained bysimulating nonlinear behaviour between 119862

119894119862rg and 120583119894

A complex path of the load parameter Δ120596120596rg wasthen studied in order to test the ability of the model todescribe the differential gearbox yield 120578di and the loss ratios119897119894= 119875119894119875rg associated with the 119894-mechanism (where 119894 =

(sapl axsa sash plsh)) under more realistic conditionsinvolving a straight line situation (Δ120596120596rg = 0 with 120596lw = 0

and 120596rw = 0) a curved line situation (0 lt Δ120596120596rg lt 2)a constant curved line situation (Δ120596120596rg = constant) orwhere one of the drivenwheels undergoes slipping (Δ120596120596rg =2) Figure 12 shows the evolution of the load parameter

Δ120596120596rg as a function of the time 119905 isin [0 1000 s] Theresults obtained with a complex path in terms of the yieldof the differential gearbox (120578di) and the loss ratios (119897

119894) are

shown in Figure 13 Specifically Figure 13(a) gives the yieldof the differential gearbox 120578di as a function of the time119905 isin [0 1000 s] in the case of the complex path plotted inFigure 12 with various sets of friction coefficients (120583axsa 120583sh)(003005) in dashed line (005007) in dash-dot line and(003007) in dotted line During the whole complex path ofΔ120596120596rg the maximum (resp minimum) yield of differentialgearbox was reached with 120583axsa = 003 and 120583sh = 005

(resp 120583axsa = 005 and 120583sh = 007) On the 119896-plate (with119896 = 1 2 3 4) defined by the 119896-time interval [119905119894 119905119891]

119896 it can be

noted that for example (1) in [400 s 500 s]2 120578di asymp 09774with

(005007) 120578di asymp 09591 with (003007) and 120578di asymp 09547with (005007) (2) in [900 s 950 s]

4 120578di asymp 09464 with

(005007) 120578di asymp 09382 with (003007) and 120578di asymp 09294with (005007) The loss ratios due to friction 119897

119894= 119875119894119875rg are

shown in Figure 13(b) The maximum loss ratios obtained indecreasing order of magnitude (along the loading path) were119897sapl (black line) 119897axsa (red line) 119897plsh (green line) and 119897sash(blue line) except for the last loadwhere 119897axsa could be greaterthan 119897sapl with the pair of friction coefficients (005007)and (003007) It should be noted that (1) on the secondplate ([400 s 500 s]

2) 119897max

axsa asymp 00143 with (005007) 119897maxplsh asymp

00057 with (003007) and 120578di asymp 00052 with (005007)(2) on the fourth plate ([900 s 950 s]

4) 119897max

axsa asymp 00286 with(005007) 119897max

plsh asymp 00115 with (003007) and 120578di asymp 00104with (005007)

Although only a sensitivity analysis was performed on themodel the results obtained show quite clearly that this modelcan be used to assess and predict the mechanical frictionlosses occurring in a differential gearbox An experimentalstudy shall be conducted in order to obtain more realisticvalues for some of the parameters such as the frictioncoefficients (120583axsa and 120583sh) which significantly influenceboth the friction torques and the differential gearbox yield

4 Conclusion

In this paper an analytical model is presented for assessingand predicting the mechanical friction losses occurring indifferential gearboxes Some of the parameters involvedin this model can be determined quite easily (geometricparameters) while others which are more delicate dependdirectly on the type of friction occurring in the mechanism(and therefore on the friction coefficients) which affectsthe mechanical losses to a variable extent In order totest the influence of these parameters and determine theability of the model to predict any mechanical losses asensitivity analysis was conducted After this initial numericalapproach an experimental study shall be performed in orderto obtainmore realistic values for some of the parameters andconfirm some of the assumptionsmade here in themodellingprocedure

The Scientific World Journal 11

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is greatly indebted to Dr J-L Ligier for helpfuldiscussions comments and advices The author would liketo thank also Dr Jessica Blanc for her help with this paper

References

[1] S C Lee and H S Cheng ldquoScuffing theory modeling andexperimental correlationsrdquo Journal of Tribology vol 113 no 2pp 327ndash334 1991

[2] D Dowson G Dalmaz Childs M T H C Godet and C MTaylorThin Films in Tribology Elsevier Science 1993

[3] J C Enthoven P M Cann and H A Spikes ldquoTemperatureand scuffingrdquoTribology Transactions vol 36 no 2 pp 258ndash2661993

[4] D Dowson G Dalmaz T H C Childs and C M TaylorLubricants and Lubrication Elsevier Science 1995

[5] C Taylor L Flamand G Dalmaz et al Elastohydrodynamicsmdashrsquo96 Fundamentals and Applications in Lubrication and TractionElsevier Science 1997

[6] M Priest P Ehret L Flamand et al Lubrication at the Frontierthe Role of the Interface and Surface Layers in the Thin Film andBoundary Regime Elsevier Science 1999

[7] A Cornet and J P Deville Physique et Ingenierie des SurfacesEDP Sciences 1998

[8] W Piekoszewski M Szczerek and W Tuszynski ldquoThe actionof lubricants under extreme pressure conditions in a modifiedrdquoWear vol 249 no 3-4 pp 188ndash193 2001

[9] M Wisniewski M Szczerek and W Tuszynski ldquoThe tempera-tures at scuffing and seizure in a four-ball contactrdquo LubricationScience vol 16 no 3 pp 215ndash227 2004

[10] Q Wang ldquoSeizure failure of journal-bearing conformal con-tactsrdquoWear vol 210 no 1-2 pp 8ndash16 1997

[11] G Spinnler Conception Des MachinesmdashPrincipes Et Applica-tions vol 1 Presses Polytechniques et Universitaires Romandes2002

[12] L Flamand Fatigue Des Surface Techniques De LrsquoIngenieur1993

[13] J L Ligier Materiaux Pour Palier Lisses Techniques DeLrsquoIngenieur 1995

[14] J L Ligier Avaries En Lubrification Ecole Nationales SuperieureDu Petrole Et Des Moteurs Editions Technip 2007

[15] M Aublin R Boncompain M Boulaton et al SystemesMecaniquesmdashTheorie Et Dimensionnement Dunod 2005

[16] J L Ligier and R Gojon ldquoPrediction de lrsquousure et du grippagedrsquoun palier de moteur dieselrdquo Tech Rep SIA 93065 1993

[17] J L Ligier Lubrification des Paliers Moteurs Ecole NationalesSuperieure du Petrole et des Moteurs Editions Technip 1997

[18] S Picard Fonctionnement et Maintenance Du VehiculemdashTransmission amp Freinage Delta Press 1994

[19] J L Fanchon Guide des Sciences et Technologies IndustriellesNathan 2001

[20] J HMathieu E Bergmann and R GrasAnalyse et Technologiedes Surfaces Couches Minces et Tribologie Presses Polytech-niques et Universitaires Romandes Traite desMateriaux Tome2003

[21] V N Constantinescu Laminar Viscous Flow Mechanical Engi-neering Series Springer 1995

[22] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part IWithout consideration of contact adhering layer in the inletzonerdquo Journal of Molecular Liquids vol 130 no 1ndash3 pp 88ndash942007

[23] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part II-Considering the contact adhering layer effect in the inlet zonerdquoJournal of Molecular Liquids vol 130 no 1ndash3 pp 95ndash103 2007

[24] S Li and A Kahraman ldquoA fatigue model for contacts undermixed elastohydrodynamic lubrication conditionrdquo Interna-tional Journal of Fatigue vol 33 no 3 pp 427ndash436 2011

[25] K L Johnson J A Greenwood and S Y Poon ldquoA simple theoryof asperity contact in elastohydro-dynamic lubricationrdquo Wearvol 19 no 1 pp 91ndash108 1972

[26] G Ramsey Elastohydrodynamics Imperial College Press 2001[27] B J Hamrock S R Schmid and B O Jacobson Fundamentals

of Fluid Film Lubrication CRC Press 2004[28] K L Johnson Contact Mechanics Cambridge University Press

1987[29] R Brun Manuel Du Mecanicien et Du Thermicien Editions

Technip 2000[30] P Arques Diagnostic Predictif et Defaillances Des Machines

Theorie Traitement Analyse Reconnaissance Prediction Edi-tions Technip 2007

[31] J Frene D Nicolas B Degueurce D Berthe and M GodetLubrication Hydrodynamique Paliers Et Butees Eyrolles 1990

[32] J Frene ldquoButees et paliers hydrodynamiquerdquo Tech Rep B5055Techniques de lrsquoIngenieur 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 9

0 200 400 600 800 1000091

092

093

094

095

096

097

098

099

1

Time (s)

Diff

eren

tial g

earb

ox y

ield

120578di

(a)

0 200 400 600 800 10000

0005

001

0015

002

0025

003

0035

Time (s)

PiP

rg

(b)

Figure 13 (a) Yield of a differential gearbox 120578119889119894with time 119905 isin [0 1000 s] in the case of the complex path presented in Figure 12 with various

sets of friction coefficients (120583axsa 120583sh) (003005) in a dashed line (005007) in a dash-dot line and (003007) in a dotted line (b) Lossratio 119897

119894= 119875119894119875rg of the 119894-mechanism (with 119894 = (sapl axsa sash plsh)) with time 119905 isin [0 1000 s] in the case of the complex path presented

in Figure 12 with the same sets of friction coefficients as in (a) 119897sapl = 096 which remained constant with time 119905 (black solid line) 119897axsa (redline) 119897sash (blue line) and 119897plsh (green line)

which can be identified fairly easily (ii) one yield param-eter (120578sapl) corresponding to the power transmitted by thegear between the satelliteplanetary pinions (iii) two fric-tion parameters (120583axsa 120583sh) which depend considerably onthe types of materials in contact and the surface conditionstheir state of lubrication and their temperature

Concerning (i) although these specific parameters de-pend on the differential gearbox under consideration theirorder of magnitude can be said to be 119889ax asymp 10ndash20mm 119877

119904asymp

20ndash30mm 119877119901asymp 20ndash30mm 119877sa asymp 30ndash50mm and for the

angle parameters 1205750asymp 10∘ndash15∘ 1205751015840

0asymp 15∘ndash25∘ 120575

1asymp 30∘ndash40∘

1205752asymp 35∘ndash45∘ 120572 asymp 20ndash25∘ 120575 asymp 30∘ndash40∘ readers can refer to

Fanchon [19] for further details about some of themConcerning (ii) the parameter 120578sapl which denotes the

power transmitted by the gear between satelliteplanetarypinions using a yield term has a value ranging between 092and 098 Since Fanchon [19] has reported that the yield inthe case of two meshed pinions is around 98 it can beconcluded that in a differential mechanism where there arefour pinions in contact the lowest value of 120578sapl asymp (098)

4cong

092 and the highest value is likely to be 120578sapl cong 098 Ittherefore seems reasonable to assume that the real yield 120578saplof this complex mechanism is approximately 096

The parameters that need to be investigated more closelyin order to determine their influence on themechanical lossesare the various coefficients of friction (iii) 120583axsa and 120583shThe sliding contacts in the differential gearbox mechanismcan be of very different kinds [20 21] since both thesatelliteplanetary bevel pinion and satellite-pinionsatellite-carrier axis in contact are in the mixed elastohydrodynamiclubrication (MEHL) mode [22ndash24] which is a mixed regimebetween (1) elastohydrodynamic lubrication (EHL) [25 26]where the heavy loads exerted at the surface of the two gears

(due to the presence of both high contact pressures and strongtransmitted torques) induce elastic strains in the two solidsin contact with thin lubricant films and (2) boundary lubri-cation (BL) [27] which may tend to constitute a completelydry frictional contact [28] leading to the development ofscuffing effects [29 30] The contacts between the bevel headpinion (satellites and planetaries) and the plastic shell arein the elastohydrodynamic lubrication mode (EHL) whichcan lead to the development of the hydrodynamic lubricationregime (HL) [31 32] where the lubricant film thickness islarge enough to completely separate the directly apposedsurfaces so that the loads applied between the surfaces arerestored and balanced under specific operating conditionsUnder the above conditions the contacts existing betweenthe satellite pinions and the satellite-carrier axis are of thesteelsteel contact with lubrication type 120583axsa isin [005 007]and those existing between the head pinions and the plasticshell are rather of the steelTeflon contact with lubricationtype 120583sh isin [003 005]

32 Sensitivity Analysis Assessment of the Modelrsquos Responseto a Given Load Parameter and Prediction of the MechanicalLosses In what follows it is proposed to test the modelrsquosresponse to a given load parameter and to determinethe influence of some parameters (120578sapl 120583axsa 120583sh) on themechanical losses occurring in a differential gearbox mecha-nism The expression for the load parameter Δ120596120596rg isin [0 2]means that (i) when Δ120596120596rg = 0 (in cases where 120596lw = 0 and120596rw = 0) the vehicle is travelling on a straight line 120578di = 1 and119897119894= 119875119894119875rg = 0 forall119894 (ii) when 0 lt Δ120596120596rg lt 2 the vehicle is

taking a curvilinear course 120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894(iii) when Δ120596120596rg = 2 one of the driven wheels undergoes

10 The Scientific World Journal

slipping (the extreme situation) the vehicle is stationary120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894

The values adopted for the other parameters are 119889ax =18 times 10

minus3m 119877119904= 22 times 10

minus3m 119877119901= 27 times 10

minus3m 119877sa =49 times 10

minus3m 1205750= 10∘ 12057510158400= 18∘ 1205751= 32∘ 1205752= 40∘ 120572 = 20∘

120575 = 39∘

Figure 10(a) shows the yield of a differential gearbox 120578didepending on the load parameter Δ120596120596rg with various setsof friction coefficients of friction (120583axsa 120583sh) and a lineardecrease in 120578sapl from 098 to 096 In the most gentle oper-ating range that is 0 le Δ120596120596rg le 1 120578di decreased from 098to around 0963 (120583axsa 120583sh) = (003 005) 120578di isin [098 0959](resp 120578di isin [098 0954]) when (120583axsa 120583sh) = (003 005)(resp (120583axsa 120583sh) = (005 007)) and 120578di decrease from 098to around 0949 when (120583axsa 120583sh) showed a concomitantlinear increase from (003 005) to (005 007) In the heavyoperating range that is 1 lt Δ120596120596rg le 2 the maximumdifferential gearbox yield 120578di isin ]0963 0946] was obtainedwith (120583axsa 120583sh) = (003 005) and the minimum yield(120578di isin ]0949 0909]) occurred when (120583axsa 120583sh) and 120578saplincreased linearly from (003 005) to (005 007) concomi-tantly with the load parameter Δ120596120596rg 120578di isin ]0959 0938](resp 120578di isin ]0954 0929]) when (120583axsa 120583sh) = (003 005)(resp (120583axsa 120583sh) = (005 007)) The results presentedin Figure 10(b) show the loss ratio 119897

119894= 119875119894119875rg associated

with 119894-mechanism (where 119894 = (sapl axsa sash plsh)) indecreasing order of influence 119897axsa (red line) 119897sapl (blackline) 119897plsh (green line) 119897sash (blue line) when 120578sapl had a con-stant value of 098 and in the case where 120578sapl decreased from098 to 096 the effects of the loss ratio 119897sapl became greaterthan those of the other parameters (119897axsa 119897axsa 119897sash 119897plsh)It should be noted that when 0 le Δ120596120596rg le 1 then119897axsa isin [0 00143] 119897plsh isin [0 00057] and 119897sash isin [0 00052]whereas when 1 lt Δ120596120596rg le 2 then 119897axsa isin [0 00286]119897plsh isin [0 00115] and 119897sash isin [0 00104] Figure 11 showsthe friction torque ratio 119862

119894119862rg as a function of the friction

coefficient 120583119894with 119894 = (axsa sash plsh) when the other

parameters were taken to be fixed constants The range ofvariation of the friction coefficient 120583

119894was taken to be between

0 and 02 (this extreme value may correspond to steelsteelcontact without any lubrification) In view of these resultsthe maximum 119862

119894119862rg ratio was obtained with 119862axsa119862rg and

the minimum one with 119862sash119862rg The friction torque ratios119862119894119862rg progressed linearly with the friction coefficient 120583

119894

(where 120583sh = 120583sash = 120583plsh) as predicted by the constitutiveequations in Section 2 Note that the results obtained in thisfirst approximation are likely to differ from those obtained bysimulating nonlinear behaviour between 119862

119894119862rg and 120583119894

A complex path of the load parameter Δ120596120596rg wasthen studied in order to test the ability of the model todescribe the differential gearbox yield 120578di and the loss ratios119897119894= 119875119894119875rg associated with the 119894-mechanism (where 119894 =

(sapl axsa sash plsh)) under more realistic conditionsinvolving a straight line situation (Δ120596120596rg = 0 with 120596lw = 0

and 120596rw = 0) a curved line situation (0 lt Δ120596120596rg lt 2)a constant curved line situation (Δ120596120596rg = constant) orwhere one of the drivenwheels undergoes slipping (Δ120596120596rg =2) Figure 12 shows the evolution of the load parameter

Δ120596120596rg as a function of the time 119905 isin [0 1000 s] Theresults obtained with a complex path in terms of the yieldof the differential gearbox (120578di) and the loss ratios (119897

119894) are

shown in Figure 13 Specifically Figure 13(a) gives the yieldof the differential gearbox 120578di as a function of the time119905 isin [0 1000 s] in the case of the complex path plotted inFigure 12 with various sets of friction coefficients (120583axsa 120583sh)(003005) in dashed line (005007) in dash-dot line and(003007) in dotted line During the whole complex path ofΔ120596120596rg the maximum (resp minimum) yield of differentialgearbox was reached with 120583axsa = 003 and 120583sh = 005

(resp 120583axsa = 005 and 120583sh = 007) On the 119896-plate (with119896 = 1 2 3 4) defined by the 119896-time interval [119905119894 119905119891]

119896 it can be

noted that for example (1) in [400 s 500 s]2 120578di asymp 09774with

(005007) 120578di asymp 09591 with (003007) and 120578di asymp 09547with (005007) (2) in [900 s 950 s]

4 120578di asymp 09464 with

(005007) 120578di asymp 09382 with (003007) and 120578di asymp 09294with (005007) The loss ratios due to friction 119897

119894= 119875119894119875rg are

shown in Figure 13(b) The maximum loss ratios obtained indecreasing order of magnitude (along the loading path) were119897sapl (black line) 119897axsa (red line) 119897plsh (green line) and 119897sash(blue line) except for the last loadwhere 119897axsa could be greaterthan 119897sapl with the pair of friction coefficients (005007)and (003007) It should be noted that (1) on the secondplate ([400 s 500 s]

2) 119897max

axsa asymp 00143 with (005007) 119897maxplsh asymp

00057 with (003007) and 120578di asymp 00052 with (005007)(2) on the fourth plate ([900 s 950 s]

4) 119897max

axsa asymp 00286 with(005007) 119897max

plsh asymp 00115 with (003007) and 120578di asymp 00104with (005007)

Although only a sensitivity analysis was performed on themodel the results obtained show quite clearly that this modelcan be used to assess and predict the mechanical frictionlosses occurring in a differential gearbox An experimentalstudy shall be conducted in order to obtain more realisticvalues for some of the parameters such as the frictioncoefficients (120583axsa and 120583sh) which significantly influenceboth the friction torques and the differential gearbox yield

4 Conclusion

In this paper an analytical model is presented for assessingand predicting the mechanical friction losses occurring indifferential gearboxes Some of the parameters involvedin this model can be determined quite easily (geometricparameters) while others which are more delicate dependdirectly on the type of friction occurring in the mechanism(and therefore on the friction coefficients) which affectsthe mechanical losses to a variable extent In order totest the influence of these parameters and determine theability of the model to predict any mechanical losses asensitivity analysis was conducted After this initial numericalapproach an experimental study shall be performed in orderto obtainmore realistic values for some of the parameters andconfirm some of the assumptionsmade here in themodellingprocedure

The Scientific World Journal 11

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is greatly indebted to Dr J-L Ligier for helpfuldiscussions comments and advices The author would liketo thank also Dr Jessica Blanc for her help with this paper

References

[1] S C Lee and H S Cheng ldquoScuffing theory modeling andexperimental correlationsrdquo Journal of Tribology vol 113 no 2pp 327ndash334 1991

[2] D Dowson G Dalmaz Childs M T H C Godet and C MTaylorThin Films in Tribology Elsevier Science 1993

[3] J C Enthoven P M Cann and H A Spikes ldquoTemperatureand scuffingrdquoTribology Transactions vol 36 no 2 pp 258ndash2661993

[4] D Dowson G Dalmaz T H C Childs and C M TaylorLubricants and Lubrication Elsevier Science 1995

[5] C Taylor L Flamand G Dalmaz et al Elastohydrodynamicsmdashrsquo96 Fundamentals and Applications in Lubrication and TractionElsevier Science 1997

[6] M Priest P Ehret L Flamand et al Lubrication at the Frontierthe Role of the Interface and Surface Layers in the Thin Film andBoundary Regime Elsevier Science 1999

[7] A Cornet and J P Deville Physique et Ingenierie des SurfacesEDP Sciences 1998

[8] W Piekoszewski M Szczerek and W Tuszynski ldquoThe actionof lubricants under extreme pressure conditions in a modifiedrdquoWear vol 249 no 3-4 pp 188ndash193 2001

[9] M Wisniewski M Szczerek and W Tuszynski ldquoThe tempera-tures at scuffing and seizure in a four-ball contactrdquo LubricationScience vol 16 no 3 pp 215ndash227 2004

[10] Q Wang ldquoSeizure failure of journal-bearing conformal con-tactsrdquoWear vol 210 no 1-2 pp 8ndash16 1997

[11] G Spinnler Conception Des MachinesmdashPrincipes Et Applica-tions vol 1 Presses Polytechniques et Universitaires Romandes2002

[12] L Flamand Fatigue Des Surface Techniques De LrsquoIngenieur1993

[13] J L Ligier Materiaux Pour Palier Lisses Techniques DeLrsquoIngenieur 1995

[14] J L Ligier Avaries En Lubrification Ecole Nationales SuperieureDu Petrole Et Des Moteurs Editions Technip 2007

[15] M Aublin R Boncompain M Boulaton et al SystemesMecaniquesmdashTheorie Et Dimensionnement Dunod 2005

[16] J L Ligier and R Gojon ldquoPrediction de lrsquousure et du grippagedrsquoun palier de moteur dieselrdquo Tech Rep SIA 93065 1993

[17] J L Ligier Lubrification des Paliers Moteurs Ecole NationalesSuperieure du Petrole et des Moteurs Editions Technip 1997

[18] S Picard Fonctionnement et Maintenance Du VehiculemdashTransmission amp Freinage Delta Press 1994

[19] J L Fanchon Guide des Sciences et Technologies IndustriellesNathan 2001

[20] J HMathieu E Bergmann and R GrasAnalyse et Technologiedes Surfaces Couches Minces et Tribologie Presses Polytech-niques et Universitaires Romandes Traite desMateriaux Tome2003

[21] V N Constantinescu Laminar Viscous Flow Mechanical Engi-neering Series Springer 1995

[22] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part IWithout consideration of contact adhering layer in the inletzonerdquo Journal of Molecular Liquids vol 130 no 1ndash3 pp 88ndash942007

[23] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part II-Considering the contact adhering layer effect in the inlet zonerdquoJournal of Molecular Liquids vol 130 no 1ndash3 pp 95ndash103 2007

[24] S Li and A Kahraman ldquoA fatigue model for contacts undermixed elastohydrodynamic lubrication conditionrdquo Interna-tional Journal of Fatigue vol 33 no 3 pp 427ndash436 2011

[25] K L Johnson J A Greenwood and S Y Poon ldquoA simple theoryof asperity contact in elastohydro-dynamic lubricationrdquo Wearvol 19 no 1 pp 91ndash108 1972

[26] G Ramsey Elastohydrodynamics Imperial College Press 2001[27] B J Hamrock S R Schmid and B O Jacobson Fundamentals

of Fluid Film Lubrication CRC Press 2004[28] K L Johnson Contact Mechanics Cambridge University Press

1987[29] R Brun Manuel Du Mecanicien et Du Thermicien Editions

Technip 2000[30] P Arques Diagnostic Predictif et Defaillances Des Machines

Theorie Traitement Analyse Reconnaissance Prediction Edi-tions Technip 2007

[31] J Frene D Nicolas B Degueurce D Berthe and M GodetLubrication Hydrodynamique Paliers Et Butees Eyrolles 1990

[32] J Frene ldquoButees et paliers hydrodynamiquerdquo Tech Rep B5055Techniques de lrsquoIngenieur 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 The Scientific World Journal

slipping (the extreme situation) the vehicle is stationary120578di lt 1 and 119897119894 = 119875119894119875rg gt 0 forall119894

The values adopted for the other parameters are 119889ax =18 times 10

minus3m 119877119904= 22 times 10

minus3m 119877119901= 27 times 10

minus3m 119877sa =49 times 10

minus3m 1205750= 10∘ 12057510158400= 18∘ 1205751= 32∘ 1205752= 40∘ 120572 = 20∘

120575 = 39∘

Figure 10(a) shows the yield of a differential gearbox 120578didepending on the load parameter Δ120596120596rg with various setsof friction coefficients of friction (120583axsa 120583sh) and a lineardecrease in 120578sapl from 098 to 096 In the most gentle oper-ating range that is 0 le Δ120596120596rg le 1 120578di decreased from 098to around 0963 (120583axsa 120583sh) = (003 005) 120578di isin [098 0959](resp 120578di isin [098 0954]) when (120583axsa 120583sh) = (003 005)(resp (120583axsa 120583sh) = (005 007)) and 120578di decrease from 098to around 0949 when (120583axsa 120583sh) showed a concomitantlinear increase from (003 005) to (005 007) In the heavyoperating range that is 1 lt Δ120596120596rg le 2 the maximumdifferential gearbox yield 120578di isin ]0963 0946] was obtainedwith (120583axsa 120583sh) = (003 005) and the minimum yield(120578di isin ]0949 0909]) occurred when (120583axsa 120583sh) and 120578saplincreased linearly from (003 005) to (005 007) concomi-tantly with the load parameter Δ120596120596rg 120578di isin ]0959 0938](resp 120578di isin ]0954 0929]) when (120583axsa 120583sh) = (003 005)(resp (120583axsa 120583sh) = (005 007)) The results presentedin Figure 10(b) show the loss ratio 119897

119894= 119875119894119875rg associated

with 119894-mechanism (where 119894 = (sapl axsa sash plsh)) indecreasing order of influence 119897axsa (red line) 119897sapl (blackline) 119897plsh (green line) 119897sash (blue line) when 120578sapl had a con-stant value of 098 and in the case where 120578sapl decreased from098 to 096 the effects of the loss ratio 119897sapl became greaterthan those of the other parameters (119897axsa 119897axsa 119897sash 119897plsh)It should be noted that when 0 le Δ120596120596rg le 1 then119897axsa isin [0 00143] 119897plsh isin [0 00057] and 119897sash isin [0 00052]whereas when 1 lt Δ120596120596rg le 2 then 119897axsa isin [0 00286]119897plsh isin [0 00115] and 119897sash isin [0 00104] Figure 11 showsthe friction torque ratio 119862

119894119862rg as a function of the friction

coefficient 120583119894with 119894 = (axsa sash plsh) when the other

parameters were taken to be fixed constants The range ofvariation of the friction coefficient 120583

119894was taken to be between

0 and 02 (this extreme value may correspond to steelsteelcontact without any lubrification) In view of these resultsthe maximum 119862

119894119862rg ratio was obtained with 119862axsa119862rg and

the minimum one with 119862sash119862rg The friction torque ratios119862119894119862rg progressed linearly with the friction coefficient 120583

119894

(where 120583sh = 120583sash = 120583plsh) as predicted by the constitutiveequations in Section 2 Note that the results obtained in thisfirst approximation are likely to differ from those obtained bysimulating nonlinear behaviour between 119862

119894119862rg and 120583119894

A complex path of the load parameter Δ120596120596rg wasthen studied in order to test the ability of the model todescribe the differential gearbox yield 120578di and the loss ratios119897119894= 119875119894119875rg associated with the 119894-mechanism (where 119894 =

(sapl axsa sash plsh)) under more realistic conditionsinvolving a straight line situation (Δ120596120596rg = 0 with 120596lw = 0

and 120596rw = 0) a curved line situation (0 lt Δ120596120596rg lt 2)a constant curved line situation (Δ120596120596rg = constant) orwhere one of the drivenwheels undergoes slipping (Δ120596120596rg =2) Figure 12 shows the evolution of the load parameter

Δ120596120596rg as a function of the time 119905 isin [0 1000 s] Theresults obtained with a complex path in terms of the yieldof the differential gearbox (120578di) and the loss ratios (119897

119894) are

shown in Figure 13 Specifically Figure 13(a) gives the yieldof the differential gearbox 120578di as a function of the time119905 isin [0 1000 s] in the case of the complex path plotted inFigure 12 with various sets of friction coefficients (120583axsa 120583sh)(003005) in dashed line (005007) in dash-dot line and(003007) in dotted line During the whole complex path ofΔ120596120596rg the maximum (resp minimum) yield of differentialgearbox was reached with 120583axsa = 003 and 120583sh = 005

(resp 120583axsa = 005 and 120583sh = 007) On the 119896-plate (with119896 = 1 2 3 4) defined by the 119896-time interval [119905119894 119905119891]

119896 it can be

noted that for example (1) in [400 s 500 s]2 120578di asymp 09774with

(005007) 120578di asymp 09591 with (003007) and 120578di asymp 09547with (005007) (2) in [900 s 950 s]

4 120578di asymp 09464 with

(005007) 120578di asymp 09382 with (003007) and 120578di asymp 09294with (005007) The loss ratios due to friction 119897

119894= 119875119894119875rg are

shown in Figure 13(b) The maximum loss ratios obtained indecreasing order of magnitude (along the loading path) were119897sapl (black line) 119897axsa (red line) 119897plsh (green line) and 119897sash(blue line) except for the last loadwhere 119897axsa could be greaterthan 119897sapl with the pair of friction coefficients (005007)and (003007) It should be noted that (1) on the secondplate ([400 s 500 s]

2) 119897max

axsa asymp 00143 with (005007) 119897maxplsh asymp

00057 with (003007) and 120578di asymp 00052 with (005007)(2) on the fourth plate ([900 s 950 s]

4) 119897max

axsa asymp 00286 with(005007) 119897max

plsh asymp 00115 with (003007) and 120578di asymp 00104with (005007)

Although only a sensitivity analysis was performed on themodel the results obtained show quite clearly that this modelcan be used to assess and predict the mechanical frictionlosses occurring in a differential gearbox An experimentalstudy shall be conducted in order to obtain more realisticvalues for some of the parameters such as the frictioncoefficients (120583axsa and 120583sh) which significantly influenceboth the friction torques and the differential gearbox yield

4 Conclusion

In this paper an analytical model is presented for assessingand predicting the mechanical friction losses occurring indifferential gearboxes Some of the parameters involvedin this model can be determined quite easily (geometricparameters) while others which are more delicate dependdirectly on the type of friction occurring in the mechanism(and therefore on the friction coefficients) which affectsthe mechanical losses to a variable extent In order totest the influence of these parameters and determine theability of the model to predict any mechanical losses asensitivity analysis was conducted After this initial numericalapproach an experimental study shall be performed in orderto obtainmore realistic values for some of the parameters andconfirm some of the assumptionsmade here in themodellingprocedure

The Scientific World Journal 11

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is greatly indebted to Dr J-L Ligier for helpfuldiscussions comments and advices The author would liketo thank also Dr Jessica Blanc for her help with this paper

References

[1] S C Lee and H S Cheng ldquoScuffing theory modeling andexperimental correlationsrdquo Journal of Tribology vol 113 no 2pp 327ndash334 1991

[2] D Dowson G Dalmaz Childs M T H C Godet and C MTaylorThin Films in Tribology Elsevier Science 1993

[3] J C Enthoven P M Cann and H A Spikes ldquoTemperatureand scuffingrdquoTribology Transactions vol 36 no 2 pp 258ndash2661993

[4] D Dowson G Dalmaz T H C Childs and C M TaylorLubricants and Lubrication Elsevier Science 1995

[5] C Taylor L Flamand G Dalmaz et al Elastohydrodynamicsmdashrsquo96 Fundamentals and Applications in Lubrication and TractionElsevier Science 1997

[6] M Priest P Ehret L Flamand et al Lubrication at the Frontierthe Role of the Interface and Surface Layers in the Thin Film andBoundary Regime Elsevier Science 1999

[7] A Cornet and J P Deville Physique et Ingenierie des SurfacesEDP Sciences 1998

[8] W Piekoszewski M Szczerek and W Tuszynski ldquoThe actionof lubricants under extreme pressure conditions in a modifiedrdquoWear vol 249 no 3-4 pp 188ndash193 2001

[9] M Wisniewski M Szczerek and W Tuszynski ldquoThe tempera-tures at scuffing and seizure in a four-ball contactrdquo LubricationScience vol 16 no 3 pp 215ndash227 2004

[10] Q Wang ldquoSeizure failure of journal-bearing conformal con-tactsrdquoWear vol 210 no 1-2 pp 8ndash16 1997

[11] G Spinnler Conception Des MachinesmdashPrincipes Et Applica-tions vol 1 Presses Polytechniques et Universitaires Romandes2002

[12] L Flamand Fatigue Des Surface Techniques De LrsquoIngenieur1993

[13] J L Ligier Materiaux Pour Palier Lisses Techniques DeLrsquoIngenieur 1995

[14] J L Ligier Avaries En Lubrification Ecole Nationales SuperieureDu Petrole Et Des Moteurs Editions Technip 2007

[15] M Aublin R Boncompain M Boulaton et al SystemesMecaniquesmdashTheorie Et Dimensionnement Dunod 2005

[16] J L Ligier and R Gojon ldquoPrediction de lrsquousure et du grippagedrsquoun palier de moteur dieselrdquo Tech Rep SIA 93065 1993

[17] J L Ligier Lubrification des Paliers Moteurs Ecole NationalesSuperieure du Petrole et des Moteurs Editions Technip 1997

[18] S Picard Fonctionnement et Maintenance Du VehiculemdashTransmission amp Freinage Delta Press 1994

[19] J L Fanchon Guide des Sciences et Technologies IndustriellesNathan 2001

[20] J HMathieu E Bergmann and R GrasAnalyse et Technologiedes Surfaces Couches Minces et Tribologie Presses Polytech-niques et Universitaires Romandes Traite desMateriaux Tome2003

[21] V N Constantinescu Laminar Viscous Flow Mechanical Engi-neering Series Springer 1995

[22] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part IWithout consideration of contact adhering layer in the inletzonerdquo Journal of Molecular Liquids vol 130 no 1ndash3 pp 88ndash942007

[23] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part II-Considering the contact adhering layer effect in the inlet zonerdquoJournal of Molecular Liquids vol 130 no 1ndash3 pp 95ndash103 2007

[24] S Li and A Kahraman ldquoA fatigue model for contacts undermixed elastohydrodynamic lubrication conditionrdquo Interna-tional Journal of Fatigue vol 33 no 3 pp 427ndash436 2011

[25] K L Johnson J A Greenwood and S Y Poon ldquoA simple theoryof asperity contact in elastohydro-dynamic lubricationrdquo Wearvol 19 no 1 pp 91ndash108 1972

[26] G Ramsey Elastohydrodynamics Imperial College Press 2001[27] B J Hamrock S R Schmid and B O Jacobson Fundamentals

of Fluid Film Lubrication CRC Press 2004[28] K L Johnson Contact Mechanics Cambridge University Press

1987[29] R Brun Manuel Du Mecanicien et Du Thermicien Editions

Technip 2000[30] P Arques Diagnostic Predictif et Defaillances Des Machines

Theorie Traitement Analyse Reconnaissance Prediction Edi-tions Technip 2007

[31] J Frene D Nicolas B Degueurce D Berthe and M GodetLubrication Hydrodynamique Paliers Et Butees Eyrolles 1990

[32] J Frene ldquoButees et paliers hydrodynamiquerdquo Tech Rep B5055Techniques de lrsquoIngenieur 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 11

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is greatly indebted to Dr J-L Ligier for helpfuldiscussions comments and advices The author would liketo thank also Dr Jessica Blanc for her help with this paper

References

[1] S C Lee and H S Cheng ldquoScuffing theory modeling andexperimental correlationsrdquo Journal of Tribology vol 113 no 2pp 327ndash334 1991

[2] D Dowson G Dalmaz Childs M T H C Godet and C MTaylorThin Films in Tribology Elsevier Science 1993

[3] J C Enthoven P M Cann and H A Spikes ldquoTemperatureand scuffingrdquoTribology Transactions vol 36 no 2 pp 258ndash2661993

[4] D Dowson G Dalmaz T H C Childs and C M TaylorLubricants and Lubrication Elsevier Science 1995

[5] C Taylor L Flamand G Dalmaz et al Elastohydrodynamicsmdashrsquo96 Fundamentals and Applications in Lubrication and TractionElsevier Science 1997

[6] M Priest P Ehret L Flamand et al Lubrication at the Frontierthe Role of the Interface and Surface Layers in the Thin Film andBoundary Regime Elsevier Science 1999

[7] A Cornet and J P Deville Physique et Ingenierie des SurfacesEDP Sciences 1998

[8] W Piekoszewski M Szczerek and W Tuszynski ldquoThe actionof lubricants under extreme pressure conditions in a modifiedrdquoWear vol 249 no 3-4 pp 188ndash193 2001

[9] M Wisniewski M Szczerek and W Tuszynski ldquoThe tempera-tures at scuffing and seizure in a four-ball contactrdquo LubricationScience vol 16 no 3 pp 215ndash227 2004

[10] Q Wang ldquoSeizure failure of journal-bearing conformal con-tactsrdquoWear vol 210 no 1-2 pp 8ndash16 1997

[11] G Spinnler Conception Des MachinesmdashPrincipes Et Applica-tions vol 1 Presses Polytechniques et Universitaires Romandes2002

[12] L Flamand Fatigue Des Surface Techniques De LrsquoIngenieur1993

[13] J L Ligier Materiaux Pour Palier Lisses Techniques DeLrsquoIngenieur 1995

[14] J L Ligier Avaries En Lubrification Ecole Nationales SuperieureDu Petrole Et Des Moteurs Editions Technip 2007

[15] M Aublin R Boncompain M Boulaton et al SystemesMecaniquesmdashTheorie Et Dimensionnement Dunod 2005

[16] J L Ligier and R Gojon ldquoPrediction de lrsquousure et du grippagedrsquoun palier de moteur dieselrdquo Tech Rep SIA 93065 1993

[17] J L Ligier Lubrification des Paliers Moteurs Ecole NationalesSuperieure du Petrole et des Moteurs Editions Technip 1997

[18] S Picard Fonctionnement et Maintenance Du VehiculemdashTransmission amp Freinage Delta Press 1994

[19] J L Fanchon Guide des Sciences et Technologies IndustriellesNathan 2001

[20] J HMathieu E Bergmann and R GrasAnalyse et Technologiedes Surfaces Couches Minces et Tribologie Presses Polytech-niques et Universitaires Romandes Traite desMateriaux Tome2003

[21] V N Constantinescu Laminar Viscous Flow Mechanical Engi-neering Series Springer 1995

[22] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part IWithout consideration of contact adhering layer in the inletzonerdquo Journal of Molecular Liquids vol 130 no 1ndash3 pp 88ndash942007

[23] Y Zhang ldquoAnalytical solution to a mode of mixed elastohy-drodynamic lubrication with mixed contact regimes part II-Considering the contact adhering layer effect in the inlet zonerdquoJournal of Molecular Liquids vol 130 no 1ndash3 pp 95ndash103 2007

[24] S Li and A Kahraman ldquoA fatigue model for contacts undermixed elastohydrodynamic lubrication conditionrdquo Interna-tional Journal of Fatigue vol 33 no 3 pp 427ndash436 2011

[25] K L Johnson J A Greenwood and S Y Poon ldquoA simple theoryof asperity contact in elastohydro-dynamic lubricationrdquo Wearvol 19 no 1 pp 91ndash108 1972

[26] G Ramsey Elastohydrodynamics Imperial College Press 2001[27] B J Hamrock S R Schmid and B O Jacobson Fundamentals

of Fluid Film Lubrication CRC Press 2004[28] K L Johnson Contact Mechanics Cambridge University Press

1987[29] R Brun Manuel Du Mecanicien et Du Thermicien Editions

Technip 2000[30] P Arques Diagnostic Predictif et Defaillances Des Machines

Theorie Traitement Analyse Reconnaissance Prediction Edi-tions Technip 2007

[31] J Frene D Nicolas B Degueurce D Berthe and M GodetLubrication Hydrodynamique Paliers Et Butees Eyrolles 1990

[32] J Frene ldquoButees et paliers hydrodynamiquerdquo Tech Rep B5055Techniques de lrsquoIngenieur 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of