École Sigle

Doctorale

Renée El

Melhem Bât

Pascal Blaise

R Gourdon

M Stéphane

ÉLECTROTECHNIQUE E E A Électronique

M Gérard Scorletti
email: gerard.scorletti@ec-lyon.fr

M Fabrice

Mme Emmanuelle
email: emmanuelle.canet@univ-lyon1.fr

Étage

M Luca

Jean-Yves Buffière
email: jean-yves.buffiere@insa-lyon.fr

M Philippe Boisse

Viviane Polsinelli
email: viviane.polsinelli@univ-lyon2.fr

Brigitte Dubois

J Y Toussaint

M Christian Montes

Keywords: power losses, thermal network, thermal exchanges, axle, hypoid gear, gear friction

le laboratoire d'Énergétique et des Matériaux de l'ECAM de Lyon (LabECAM), et la société Renault Trucks du Groupe Volvo.

Re sume

Pour répondre au besoin des clients ainsi qu'aux réglementations gouvernementales, les constructeurs de camions doivent diminuer la consommation et les émissions de leurs véhicules. Une solution-clé est d'améliorer le rendement de la transmission du camion, dont le pont fait partie. Leur design n'a longtemps été optimisé qu'en fonction de critères de durabilité et de bruit.

L'objectif de ce travail est donc de caractériser le rendement des ponts de camion. La dissipation de puissance au sein du pont est causée par l'engrènement, les roulements, les joints et le barbotage. Des méthodes permettent d'estimer globalement ces pertes de puissance, mais elles ne sont pas forcément adaptées aux ponts. En effet, l'élément principal du pont est un engrenage spiro-conique ou hypoïde et son importance est étudiée : sa forme influe sur le barbotage, tandis que sa géométrie de denture et sa cinématique gouvernent le frottement à l'engrènement. Il semble ainsi important d'évaluer le frottement de ces couples coniques par une approche locale et d'étudier l'influence des paramètres de denture.

Cependant, les pertes de puissance dépendent de la température, via les propriétés de l'huile. Des expériences montrent un important écart de température entre les composants. Il faut donc considérer des températures locales plutôt qu'une température d'huile globale. Le rendement et la durabilité peuvent être impactés par des points chauds. La méthode des réseaux thermiques permet de modéliser les échanges thermiques du pont ainsi que la distribution de températures. Les tests classiques de rendement mesurent uniquement la perte globale et la température d'huile : rien ne permet de confirmer la répartition des pertes entre sources. Une campagne d'essais avec mesures de température est donc réalisée et valide le modèle pour le calcul des températures locales et pour l'estimation des pertes de chaque composant. Ce modèle peut alors être utilisé lors du design de futurs ponts.

Mots-clés : pertes de puissance, réseaux thermiques, échanges thermiques, pont, engrenage hypoïde, frottement à l'engrènement

Re sume e tendu

Introduction

Alors que plus en plus de biens sont transportés à travers le monde, les poids-lourds jouent un rôle important dans le fret terrestre. Ils assurent notamment l'approvisionnement local en nourriture, médicaments, biens de consommation, carburant… En Europe, les poids-lourds doivent respecter la réglementation sur les émissions. Euro 0 remonte à 1990 et Euro VI, sa dernière version, date de fin 2013. Les constructeurs de camions ont ainsi peu à peu amélioré leur technologie en ce qui concerne les émissions de particules, de NOx, d'Hydrocarbures et de Monoxydes de carbone. Cependant, selon la Commission Européenne [START_REF]Reducing CO2 emissions from Heavy-Duty Vehicles[END_REF], les camions, les bus et les cars contribuent pour un quart aux émissions de CO2 dues au transport routier en Europe. Une réglementation spécifique au CO2 doit entrer en vigueur dans un futur proche.

Un moyen de réduire les émissions de particules et de CO2 est de diminuer la consommation de carburant. Au-delà des questions de réglementation, les clients demandent aussi des solutions de transport plus efficaces, principalement pour réduire les dépenses de carburant. Pour ce qui est d'améliorer le rendement, la transmission du véhicule est toujours un élément-clé de développement. Ainsi, il est important de déterminer exactement les pertes de puissance de la transmission pour pouvoir ensuite les réduire.

Cette étude s'intéresse au pont, qui est le dernier élément de la transmission du camion. Son principal composant est un engrenage spiro-conique ou hypoïde. Ces engrenages spéciaux permettent de transmettre la puissance entre des arbres perpendiculaires, et ce avec une durée de vie et un bruit acceptables. La plupart des développements menés sur les ponts se sont justement focalisés sur les aspects de durabilité et de bruyance. Le rendement a longtemps été étudié de manière empirique uniquement. Le premier modèle de rendement d'un pont date de 2013 [START_REF] Kolekar | The efficiency of a hypoid axle -a thermally coupled lubrication model[END_REF] et s'intéresse aux ponts de voiture. S'ils ont une architecture et un fonctionnement similaires, ils ont en revanche des conditions opératoires bien différentes. Cela peut être à l'origine de nouvelles problématiques.

Cependant, les modèles développés pour estimer le rendement des ponts se basent uniquement sur des approches isothermes : le pont est considéré à température d'huile. Cette hypothèse est pourtant remise en question par des essais [START_REF] Xu | Thermal Mapping of an Automotive Rear Drive Axle[END_REF] qui soulignent d'importants écarts de température entre les composants. Cela peut modifier les propriétés de l'huile et ainsi le rendement. De plus, les transmissions sont aujourd'hui plus compactes tandis que plus de chaleur est générée. Les questions de refroidissement et de durabilité doivent être gardées à l'esprit, car des défauts peuvent alors survenir sur les engrenages et les roulements à cause de l'apparition de points chauds. Le comportement thermique des composants et de l'huile doit donc être modélisé et analysé.

Finalement, l'objectif de cette étude est de développer un modèle thermomécanique d'un pont de camion. Une approche globale est utilisée pour considérer le système complet. Les pertes de puissance et les échanges thermiques sont estimés à l'aide de méthodes simples pour permettre un cycle de développement rapide et donc un usage lors de la phase de design. Le rendement du pont mais aussi la distribution de température sont obtenus. ; ii) réaliser une dernière réduction de la vitesse ; iii) permettre au véhicule de tourner en évitant que les roues ne patinent grâce au différentiel. La réduction de vitesse est réalisée grâce à un engrenage spiro-conique, ou hypoïde s'il existe un décalage (offset) entre l'axe du pignon et celui de la couronne. Cet offset induit un pignon hypoïde plus gros que celui du spiroconique pour une couronne de même taille : le pignon est donc plus résistant pour une garde au sol identique. L'hypoïde a aussi un meilleur rapport de conduite et est ainsi plus silencieux. Cependant, le glissement et la perte au contact sont plus importants pour l'hypoïde à cause de l'offset.

En plus du couple conique, la tête du pont comprend plusieurs roulements et le carter est partiellement rempli d'huile afin d'assurer la lubrification des éléments. Les ponts de camion et de voiture ont la même architecture, mais ceux de camion sont plus gros : la couronne a un plus grand diamètre, le volume d'huile est plus important ; mais la puissance transmise est aussi plus grande. Cela peut mener à des rendements similaires.

Rendement d'un pont

Les pertes de puissance du pont sont générées par certains composants. La principale source de frottement est le couple conique. La lubrification par bain d'huile de l'engrenage cause des pertes par barbotage. Les roulements produisent des pertes par frottement mais aussi par traînée. Enfin, les joints à lèvre génèrent du frottement par rotation.

Le rendement des ponts a d'abord été étudié grâce à des moyens expérimentaux. Une première procédure de laboratoire est proposée en 1979 par Hobson [START_REF] Hobson | Axle Efficiency -Test Procedures and Results[END_REF], qui évalue le rendement simplement, en mesurant couples et vitesses. En 1988, Winter et Wech [START_REF] Winter | Measurements and Optimization of the Efficiency of Hypoid Axle Drives of Vehicles[END_REF] utilisent un banc d'essai pour engrenages hypoïdes en boucle fermée et testent divers paramètres. L'offset de l'hypoïde et les lubrifiants influent beaucoup le rendement sous charge, tandis que la précharge des roulements et le barbotage prédominent à haute vitesse. Plus récemment, Xu et al. [START_REF] Xu | Effects of Bearing Preload, Oil Volume, and Operating Temperature on Axle Power Losses[END_REF] soulignent l'importance de la température d'huile. En ce qui concerne les approches théoriques, Kolekar et al. [START_REF] Kolekar | The efficiency of a hypoid axle -a thermally coupled lubrication model[END_REF] établissent un modèle de rendement en 2013. Ils considèrent les différentes sources de pertes et évaluent le rendement pour différentes conditions. Kakavas et al. [START_REF] Kakavas | Hypoid gear vehicle axle efficiency[END_REF] ont récemment développé parallèlement les approches expérimentale et théorique. Chaque source de pertes est estimée sur un cycle routier et la perte globale est comparée avec succès à des mesures.

Les pertes par frottement au joint sont évaluées par des approches simples, comme celle de Simrit [START_REF] Simrit | Radialwellendichtringe[END_REF] qui est utilisée dans le rapport technique ISO/TR 14179-2 [START_REF]Gears -Thermal capacity Part 2 : Thermal load-carrying capacity[END_REF], celle de Linke [START_REF] Linke | Stirnradverzahnung[END_REF], ou celle de Kettler [START_REF] Kettler | Planetengetriebe-Sumpftemperatur[END_REF]. Une comparaison en fonction de la température permet de choisir la formule de Simrit. Les pertes des roulements sont estimées selon les méthodes de Harris [START_REF] Harris | Rolling bearing analysis[END_REF] ou de SKF [START_REF] Skf Group | [END_REF]. Les résultats obtenus divergent cependant selon les conditions d'application et les deux méthodes seront testées dans le second chapitre sur le cas du pont. Les pertes par barbotage ont été beaucoup étudiées pour les engrenages cylindriques. Concernant les spiro-coniques et les hypoïdes, des formules génériques proposées dans les rapports techniques ISO/TR 14179-1 [START_REF]Gears -Thermal capacity Part 1 : Rating gear drives with thermal equilibrium at 95°C sump temperature[END_REF] et ISO/TR 14179-2 [START_REF]Gears -Thermal capacity Part 2 : Thermal load-carrying capacity[END_REF] peuvent être appliquées. Une formule spécifique a aussi été établie par Jeon [START_REF] Jeon | Improving Efficiency in Drive Lines: an Experimental Study on Churning Losses in Hypoid Axle[END_REF] dans une étude sur les ponts automobiles. Ces trois formules donnent toutes des résultats différents et seront aussi testées dans le second chapitre. Enfin, le rendement des engrenages spiro-coniques ou hypoïdes peut être estimé d'après les formules de Buckingham [START_REF] Buckingham | Analytical Mechanics of Gears[END_REF], de Coleman [START_REF] Coleman | Effect of mounting displacements on bevel and hypoid gear tooth strength[END_REF], ou de Niemann et Winter dans l'ISO/TR 14179-2 [START_REF]Gears -Thermal capacity Part 2 : Thermal load-carrying capacity[END_REF]. Comme pour les cylindriques, le rendement des hypoïdes peut être exprimé en fonction du coefficient de frottement et d'un facteur géométrique. Une comparaison en fonction du couple révèle un comportement incohérent de la formule de Coleman. Les deux autres formules sont conservées pour questionner la pertinence d'une approche globale par la suite.

Comportement thermique d'un pont

La plupart des études sur les ponts s'intéressent uniquement aux pertes ([4]- [START_REF] Xu | Effects of Bearing Preload, Oil Volume, and Operating Temperature on Axle Power Losses[END_REF], [START_REF] Mohammadpour | Transmission efficiency and noise, vibration and harshness refinement of differential hypoid gear pairs[END_REF]) et travaillent toujours à température d'huile fixe. Les quelques études thermiques considèrent généralement le pont comme isotherme, à température de la cuve d'huile. C'est le cas de Kolekar et al. [START_REF] Kolekar | The efficiency of a hypoid axle -a thermally coupled lubrication model[END_REF], mais aussi de Kakavas et al. [START_REF] Kakavas | Hypoid gear vehicle axle efficiency[END_REF]. Ils calculent ainsi une température d'huile globale et évaluent le rendement en fonction des conditions ambiantes sur un cycle routier. Ce type d'approche permet d'évaluer le rendement global du pont, mais ne peut pas confirmer la répartition des pertes entre les sources. Seule une mesure de la température des composants pourrait indiquer la prédominance d'une source.

Le comportement thermique du pont est étudié expérimentalement par Hurley [START_REF] Hurley | An Experimental Investigation of Thermal Behavior of an Automotive Rear Axle[END_REF] et Xu et al. [START_REF] Xu | Thermal Mapping of an Automotive Rear Drive Axle[END_REF] : la température est enregistrée en de nombreux points du pont. Pour différentes conditions opératoires, ils observent des écarts de température entre les composants, par exemple entre pignon et roulements. Il apparaît aussi que le pignon est plus chaud que l'huile, ce qui est important pour des questions de durabilité mais aussi pour le calcul du coefficient de frottement. Quelques-uns se sont intéressés spécifiquement au couple conique, comme Handschuh [START_REF] Handschuh | Thermal behaviour of spiral bevel gears[END_REF] ou Wang et al. [START_REF] Wang | Investigation into the meshing friction heat generation and transient thermal characteristics of spiral bevel gears[END_REF]. Le premier observe expérimentalement l'influence de la charge et de la lubrification par jet sur la température de masse du pignon. Il constate également que le pied et la tête sont les parties les plus chaudes de la dent. Wang et al. montrent que la forme du contact influe sur la génération de chaleur par frottement. Un contact étendu produit ainsi moins de pertes localement.

Finalement, aucune étude n'a établi de modèle thermique d'un pont complet. Cela peut pourtant avoir une influence sur la température de masse des engrenages ou d'autres éléments. Cela permettrait également de connaître la répartition des pertes entre sources.

Chapitre 2 -Pertes de puissance du pont

Rendement du pont de camion

Un pont à simple réduction pour une application longue distance est étudié. Des essais de rendement sont menés sur ce pont à l'aide d'une méthode similaire à celle de Hobson [START_REF] Hobson | Axle Efficiency -Test Procedures and Results[END_REF]. Deux bancs sont utilisés : l'un permet de mesurer la perte sous charge et l'autre la perte à vide. Les couples et vitesses sont mesurés, ainsi que la température de la cuve d'huile, qui est maintenue à 80°C grâce à des ventilateurs.

Trois paliers de puissance sont testés sur le premier banc et six paliers de vitesse sur le second banc. Les résultats obtenus, de l'ordre de quelques kilowatts de pertes, sont ensuite comparés aux calculs.

Les pertes indépendantes de la charge sont d'abord étudiées. Le frottement au joint est évalué selon la formule de Simrit et ne génère pas plus de 200 W de pertes. Les pertes dans les roulements sont estimées selon Harris et SKF. Pour les conditions sans charge, les deux méthodes donnent des résultats similaires car certains roulements sont préchargés. En revanche, les résultats divergent sous charge. La méthode SKF prédit une perte plus importante que celle de Harris, or cela induit un très bon rendement du couple conique, ce qui n'est pas réaliste. Cette méthode est donc écartée. Les pertes par barbotage sont calculées selon les formules de Mauz et de Jeon et comparées pour les conditions sans charge. La formule de Mauz surestime complètement le barbotage tandis que la formule de Jeon le sous-estime. Une étude plu poussée sur ces pertes est donc menée juste après.

Etude du barbotage des engrenages hypoïdes

La formule de Jeon [START_REF] Jeon | Improving Efficiency in Drive Lines: an Experimental Study on Churning Losses in Hypoid Axle[END_REF]

Amélioration de la formule de barbotage pour les ponts de camion

La formule de Jeon n'est pas cohérente avec les paramètres géométriques identifiés grâce aux essais, car elle utilise notamment la largeur : deux engrenages de même diamètre extérieur génèrent ainsi des pertes différentes. Une nouvelle formulation de la perte par barbotage d'un engrenage spiroconique est proposée : Afin de prendre en compte ces particularités géométriques, un logiciel de calcul de l'engrènement a été développé et validé par Barday [START_REF] Barday | Meshing of Bevel and Hypoid Gear Sets[END_REF]. La géométrie de la denture est établie en simulant le processus de fabrication, mais des mesures topographiques peuvent aussi être utilisées. Les modèles en Eléments Finis d'une dent du pignon et d'une de la couronne sont créés et une analyse tridimensionnelle est réalisée pour déterminer la flexion et le cisaillement sur ces deux dents.

𝑃 𝑐ℎ𝑢𝑟𝑛𝑖𝑛𝑔 = 1 2 𝜌 Ω 3 𝑅 𝑜 3 𝑆 𝑚 𝐶 𝑚 (0.
L'engrènement des hypoïdes est théoriquement un contact linéaire. Cependant, la charge et les déformations en fonctionnement induisent plutôt un contact elliptique. Cette ellipse est généralement discrétisée en plusieurs contacts linéaires ( [START_REF] Xu | A Model to Predict Friction Losses of Hypoid Gears[END_REF], [START_REF] Kolivand | Prediction of mechanical gear mesh efficiency of hypoid gear pairs[END_REF]). C'est aussi le cas du modèle de Barday qui détermine les lignes de contact potentiel et les discrétise en segments de longueur variable. Les conditions de contact varient d'une ligne de contact à une autre, mais également entre les segments au sein d'une même ligne : les vitesses de surface, la charge et le rayon de courbure évoluent et modifient la largeur de contact, l'épaisseur de film d'huile et la cinématique du contact. Cette dernière est étudiée à partir des vitesses de surface du pignon 𝑉 1 et de la couronne 𝑉 2 . La vitesse de glissement 𝑉 𝑠 , la somme des vitesses de surface 𝑉 Σ et la vitesse d'entraînement 𝑉 𝑒 sont alors calculées : le long de l'axe mineur de l'ellipse, comme pour les engrenages cylindriques. En revanche, il y a aussi du glissement selon l'axe majeur de l'ellipse sur les engrenages hypoïdes, à cause des angles de spirale différents. Cela induit également un glissement tout le long de l'engrènement.

𝑉 𝑠 = 𝑉 1 -𝑉 2 (0.4) 𝑉 ∑ = 𝑉 1 + 𝑉 2 (0.5) 𝑉 𝑒 = 𝑉 ∑ 2 (0.

Estimation des pertes

2 V 1 V e δe V s δ 𝑥 𝑦 V 2 V 1 V e δe V s δ 𝑥 𝑦 V 2 V 1 V e δe V s δ 𝑥 𝑦 (a) (b) (c)

Coefficient de frottement

Le coefficient de frottement est un paramètre de premier ordre pour les pertes à l'engrènement. Luimême est principalement influencé par la charge, les vitesses d'entraînement et de glissement, les propriétés de l'huile et la rugosité de surface. Ces paramètres définissent le contact élastohydrodynamique entre les dentures et les différents régimes de lubrification : lubrification limite, en film complet, ou mixte. Ce dernier régime, qui mêle frottements dus aux aspérités et au cisaillement fluide, est le plus courant dans les engrenages. Des formules évaluent le coefficient de frottement de manière à considérer les conditions locales de contact. La plupart se basent sur des mesures obtenues à l'aide de machines qui reproduisent les conditions de contact à l'engrènement. La formule de Benedict et Kelley [START_REF] Benedict | Instantaneous Coefficients of Gear Tooth Friction[END_REF] est fréquemment utilisée, bien qu'elle ne considère pas la rugosité de surface. Ce paramètre est bien intégré dans les formules proposées par Diab et al. [START_REF] Diab | Prediction of Power Losses Due to Tooth Friction in Gears[END_REF] et par Xu [START_REF] Xu | Development of a Generalized Mechanical Efficiency Prediction Methodology for Gear Pairs[END_REF]. Ainsi, la température de chaque élément est utilisée plutôt qu'une température globale. Ce calcul transitoire est réalisé grâce à un algorithme. L'intégration numérique est faite avec la méthode de prédiction-correction d'Adams [START_REF] Changenet | Modélisation du comportement thermique des transmissions par engrenages[END_REF]. Les seules conditions limites du réseau sont la vitesse et la température de l'air environnant.

Exploitation du modèle

Le modèle thermomécanique est maintenant utilisé pour analyser l'influence de certains paramètres sur les pertes et les températures de masse. Ainsi, un lubrifiant plus fluide induit une température de masse des engrenages plus basse mais pas forcément un meilleur rendement de pont. Une surface de denture plus lisse est un avantage à la fois en termes de pertes à l'engrènement et de température de

Conclusion

La consommation de carburant et les émissions des camions doivent être diminuées, notamment en améliorant le rendement de la transmission et donc du pont. Ainsi, cette étude s'attache à caractériser le rendement des ponts de camion grâce à un modèle thermomécanique. Celui-ci estime à la fois les pertes de puissance et les échanges thermiques par une approche globale du système. Une approche locale est aussi utilisée pour analyser le frottement à l'engrènement des engrenages hypoïdes. Le premier chapitre décrit le pont de camion et son couple conique : hypoïde ou spiro-conique. La plupart des études concernant les ponts de voiture, les différences avec ceux des camions sont présentées, car elles influencent les pertes même si le rendement reste comparable. La littérature s'intéressa d'abord au rendement du pont par le biais d'expériences, puis par des études théoriques, et identifia ainsi des paramètres influents (conditions opératoires, macro-et micro-géométrie des engrenages). Des modèles existent pour chaque source de perte mais ils ne sont pas toujours cohérents entre eux. Le comportement thermique du pont fut principalement étudié par des approches isothermes. Or, des essais montrent des écarts de température entre composants [START_REF] Xu | Thermal Mapping of an Automotive Rear Drive Axle[END_REF]. Cela peut engendrer des problèmes de refroidissement et de durabilité, mais aussi modifier les pertes via la viscosité d'huile. Connaître les températures des composants permet aussi de valider la répartition des pertes entre les différentes sources. Le second chapitre s'attache à estimer le rendement du pont de camion. Des essais sont réalisés pour évaluer les pertes dépendantes et indépendantes de la charge. Les pertes sont ensuite calculées d'après les formules issues de la littérature et une méthode est sélectionnée pour chaque source de pertes. Pour le frottement du joint, la formule de Simrit est choisie. Pour les roulements, il apparaît que la formule de SKF surestime la perte sous charge ; des résultats plus cohérents sont obtenus avec la formule de Harris qui est conservé pour la suite. Pour le frottement à l'engrènement, l'estimation fournie par l'ISO/TR 14179-2 donne une bonne tendance, mais une étude plus poussée est envisagée. Pour le barbotage, les solutions de Mauz et Jeon donnent des résultats irréalistes. Le barbotage des spiro-coniques et des hypoïdes est donc étudié plus en détails grâce à un banc spécifique. Des essais sont réalisés sur plusieurs cônes et engrenages spiro-coniques pour différentes huiles et conditions opératoires. Les résultats montrent l'influence de conditions opératoires (vitesse, ratio d'immersion, viscosité d'huile), mais aussi d'un paramètre géométrique (diamètre extérieur). En revanche, les dents ont peu d'impact sur le barbotage. Un nouveau modèle basé sur celui de Jeon est proposé pour prendre en compte les paramètres identifiés, ainsi que les deux régimes d'écoulement. Grâce à cette formule, l'estimation des pertes du pont est plus cohérente avec les mesures. Le troisième chapitre s'intéresse à la principale source de pertes du pont de camion : le frottement à l'engrènement des hypoïdes. Les paramètres géométriques de ces engrenages particuliers sont déterminés par les choix de conception et de fabrication. Ils sont pris en compte dans le modèle développé par Barday [START_REF] Barday | Meshing of Bevel and Hypoid Gear Sets[END_REF], qui discrétise le contact elliptique de l'engrènement en plusieurs contacts linéaires. La cinématique des engrenages hypoïdes est différente de celle des autres engrenages : les vitesses et le glissement ne suivent pas les axes de l'ellipse de contact et ils peuvent varier en son sein. Le frottement à l'engrènement est étudié sur deux couples coniques (un spiro-conique et un hypoïde), suivant une approche locale basée sur les données du logiciel de Barday. Les conditions de contact et la perte sont analysées le long de l'engrènement. Une comparaison est ensuite réalisée avec l'approche globale de l'ISO pour en discuter la validité. Cette dernière surestime toujours les pertes à cause de l'estimation de la vitesse de glissement. L'approche locale utilisée avec un coefficient de frottement global obtient ainsi des résultats plus proches des mesures. Un coefficient de frottement local pourrait cependant donner des résultats plus précis. Les formules pour le calculer se basent généralement sur les mêmes paramètres : la charge, les vitesses d'entraînement et de glissement, la viscosité d'huile et la rugosité de surface. Deux paramètres sont analysés : la rugosité de surface et les vitesses définies par les angles d'orientation. Leur influence sur le coefficient de frottement est importante et ils doivent donc être considérés pour les questions de rendement.

Le quatrième chapitre présente le modèle thermique du pont, basé sur la méthode des réseaux thermiques. Les échanges thermiques entre les éléments du pont sont contrôlés par des résistances thermiques (conduction, convection, rayonnement). Les pertes sont estimées d'après les formules vues précédemment et la chaleur générée est injectée au noeud-source correspondant. Les pertes et les températures sont calculées grâce à un algorithme permettant d'obtenir des résultats en régime permanent ou transitoire. Le réseau thermique apparaît comme plus fiable, que ce soit par rapport à l'ISO/TR 14179-2 [START_REF]Gears -Thermal capacity Part 2 : Thermal load-carrying capacity[END_REF] pour les échanges thermiques avec l'environnement ou par rapport à l'ISO/TR 13989-2 [START_REF]Calculation of scuffing load capacity of cylindrical, bevel and hypoid gears -Part 2: Integral temperature method[END_REF] pour le calcul des températures de masse du couple conique. Une campagne d'essais sert à valider le modèle thermomécanique du pont en mesurant la température en de nombreux points de celui-ci. Les résultats montrent des écarts de température entre les éléments. La comparaison avec le modèle est cohérente pour les différentes conditions opératoires : les températures et les pertes calculées sont représentatives. Un autre pont est testé et modélisé et des résultats probants sont obtenus : le modèle thermomécanique peut servir d'outil de design. L'influence de paramètres comme le lubrifiant, la rugosité de surface des dentures et le flux d'air autour du carter est étudiée.

Finalement, le modèle thermomécanique du pont développé dans cette étude permet d'obtenir son rendement et la distribution de températures. En ce qui concerne les applications industrielles pour le Groupe Volvo, cet outil peut être utilisé lors du design pour d'une part optimiser le rendement du pont, la durabilité et la bruyance pouvant être aussi prises en compte grâce un logiciel préexistant. D'autre part, les défauts dus aux points chauds sur les roulements ou les engrenages peuvent être étudiés. En parallèle de cette étude, un modèle a également été établi sur les trains planétaires [START_REF] Boni | Experimental investigations on churning power losses generated in a planetary gear set[END_REF]. En le couplant avec le modèle de pont, il serait possible d'étudier les ponts à double réduction.

Des résultats expérimentaux doivent encore être analysés. En effet, deux ponts supplémentaires ont été testés lors des essais thermiques pour étudier d'autres paramètres : la méthode de taillage (facehobbing et face-milling) et le ratio (4,625 et 2,643). Ces résultats d'essais sont importants afin de valider la cohérence du modèle pour des paramètres spécifiques.

La complexité du frottement dans les engrenages spiro-coniques et hypoïdes a été relevée. Les stries d'usinage sont obliques par rapport à la direction de roulement, or cela a un impact sur le coefficient de frottement [START_REF] Diab | Prediction of Power Losses Due to Tooth Friction in Gears[END_REF]. De plus, le pignon et la couronne présentent des rugosités de surface différentes. Des essais sur machine bi-disque permettraient de caractériser leur coefficient de frottement. Cependant, sur cette machine, les directions de roulement des deux disques sont identiques, tandis que ce n'est pas le cas sur les spiro-coniques et les hypoïdes. Cette configuration particulière est reproduite sur certains bancs ( [START_REF] Omasta | Effect of surface velocity directions on elastohydrodynamic film shape[END_REF], [START_REF] Stahl | Theoretical and Experimental investigations on EHL point contacts with different entrainment velocity directions[END_REF]), mais seule l'épaisseur de film est étudiée. Les essais préliminaires du Chapitre 3 montrent l'influence de l'angle d'orientation sur le coefficient de frottement. Il serait intéressant d'aller plus loin, en caractérisant plus précisément l'impact de cet angle et en testant également des éprouvettes rugueuses, comme sur la machine bi-disque.

Enfin, les roulements peuvent aussi être pointés. Ils sont actuellement considérés comme des éléments isothermes. Le modèle pourrait être affiné en décomposant les roulements en plusieurs éléments, afin de prendre en compte l'écart de température entre les bagues extérieure et intérieure. Cela permettrait une étude plus précise des défauts de roulement ou de l'expansion des bagues. In Europe, heavy duty vehicles have to fulfil a regulation concerning emissions: Euro 0 dates from 1990 and the last version (Euro VI) was implemented end 2013. Truck manufacturers have then gradually improved their technology regarding emissions of particulates, NOx, HydroCarbons and Carbon Monoxide. Nevertheless, according to the European Commission, "trucks, buses and coaches produce about a quarter of CO2 emissions from road transport in the EU and around 5% of the EU's total greenhouse gas emissions -a greater share than international aviation or shipping" [START_REF]Reducing CO2 emissions from Heavy-Duty Vehicles[END_REF]. A specific CO2 regulation should be enforced in a near future.

Nomenclature

One way to decrease emissions of particulates and CO2 is to improve fuel consumption: less fuel used means less oxides produced. Besides regulation perspectives, customers also ask for more efficient solutions of transport, mainly to decrease operating costs. Among efficiency improvements, the vehicle powertrain is still a key development for the truck industry: the engine is the source of power and the driveline must transmit it to the wheels. It is therefore important to determine precisely the power losses of the driveline in order to decrease them.

The present study focuses on the axle, which is the last part of the truck driveline. Its main component is a gear set: a spiral bevel or a hypoid one. This type of gear is special as it transmits power between perpendicular shafts. It is used in power transmissions not only for automotive but also for aerospace applications. Most of the developments done on axles have centred on durability and noise aspects. For a long time, efficiency has been studied only through empirical approach. The first model for axle efficiency dates from 2013 [START_REF] Kolekar | The efficiency of a hypoid axle -a thermally coupled lubrication model[END_REF] and studies automotive axles. These are similar in structure and working to truck axles, but operate in quite different conditions. This may bring in new issues.

Nevertheless, models developed for axle efficiency use only isothermal approaches: axle is considered at oil sump temperature. This hypothesis is however refuted by the experiments of Xu et al. [START_REF] Xu | Thermal Mapping of an Automotive Rear Drive Axle[END_REF] that highlight noticeable temperature differences between components. This can influence lubricant viscosity and so efficiency. Moreover, heat is still generated while systems tend toward higher compactness. Cooling and durability issues must then be kept in mind, as local hot spots can be created: gears and rolling element bearings can be subject to failure. The thermal behaviour of components and oil must be modelled and analysed.

Finally, the present study aims to develop a thermo-mechanical model for truck axles. A global approach is used to take into account the whole system. Power losses and thermal exchanges are then estimated thanks to simple models in order to allow a fast increment and to be used at design stage. Axle efficiency and temperature distribution are thus obtained.

The first chapter describes the truck drive axle and defines its power losses. Literature is reviewed regarding axle efficiency and thermal behaviour. As most of the studies concern car and medium duty axles, differences with truck axles are highlighted. Different methods to calculate power losses for mechanical transmissions in general are summarized there.

The second chapter attempts to estimate power losses of truck axles. Experiments are conducted to measure the efficiency of a specific axle. Power losses are then calculated according to the different methods and compared to each other's and to test results. The aim is to determine if a simplified analytical approach suffices to estimate axle efficiency. Churning losses are then studied more thoroughly. As experimental data concern only oil temperature and global power loss, estimation of global power loss is validated but not the power loss breakdown.

The third chapter focuses on the major power loss source of truck axles: gear mesh friction. The gear set of an axle is a spiral bevel or a hypoid one, mainly for durability reasons. However, there is more sliding at the meshing of hypoid gears and so more friction, due to the offset between pinion and crown gear. As kinematic and friction aspects of hypoid gears are not well understood, they are investigated in this part. Manufacturing process is also considered from cutting to finishing, as this induces gear characteristics, like macro-and micro-geometry. Regarding power loss estimation, global and local approaches are compared. Finally, formulae for friction coefficient calculation are reviewed and some influential parameters are studied.

The last chapter refers to the model of axle thermal behaviour. Due to power losses, heat is produced at different point of the system. This can induce local hot spots and modify oil properties. Efficiency and durability can be influenced. A thermo-mechanical model is then developed thanks to the thermal network method. This model and an isothermal one are compared regarding thermal exchanges and temperature evaluation. The selected model is validated against specific measurements of global power loss and temperature of several components. Influential parameters are finally investigated.

Literature review

Introduction

This first chapter is a review of previous works on axles regarding both efficiency and thermal behaviour. The aim is to have a basis of theoretical and experimental studies in order to drive the development of a global thermo-mechanical approach of the axle. Indeed, the use of simple models is a key to address efficiency issue at design stage, as this allows for a fast increment.

The chapter begins with a description of the truck drive axle and its specificities. As most of the studies concern car and medium duty axles, it is important to underline the differences. Axle efficiency is then presented thanks to a definition of power loss sources. Previous efficiency tests and models on axles are reviewed and each power loss contribution is detailed with general calculation methods. The thermal behaviour is finally investigated through testing results and global models from the literature.

Drive axle architecture

The mechanical system studied here is a truck drive axle, which is part of the truck driveline. This component, presented on Figure 1.1, has three main roles: i) to transmit power from the drive train to the wheel shafts (both parts are usually perpendicular); ii) to make a final speed reduction; iii) to allow the vehicle to turn without wheel slippage, thanks to the differential.

Chapter 1: Literature review On a truck, there could be one to four drive axles depending on vehicle use. An example of a truck with three drive axles is presented on Figure 1.2: as this vehicle is for severe construction application, it needs to have all wheels driven. Subsequently, only the drive axle is considered in this study and is simply called an axle. Single reduction axles ensure speed reduction with the gear set only. This gear set is a spiral bevel or a hypoid one, as presented on Figure 1.3. The main difference is the offset between pinion and crown gear axis, which is null for the spiral bevel gear. Thanks to this offset, it is possible to position the propeller shaft lower, which is helpful in terms of architecture. Moreover, this induces a bigger hypoid pinion gear than the spiral bevel one for a crown gear of the same size; pinion is so more durable, notably regarding tooth bending resistance, and ground clearance is kept. The hypoid gear has also a better contact ratio and is then quieter. However, there is more sliding at hypoid gear mesh due to pinion offset.

An amboid gear has inversely an offset above pinion axis (or negative offset). This is an unusual case for axles but is interesting for architecture aspects. It is not studied here.

Figure 1.3: Architecture of spiral bevel and hypoid gear set

For hub reduction axles, a second reduction occurs in wheel hubs. This allows to transmit higher torques and is used for severe applications.

In addition to the gear set, the drive head is composed of several rolling element bearings, which are highlighted on Figure 1.4 by dashed frames in green. There are two tapered roller bearings supporting the differential and at least two others on the input shaft. An additional cylindrical roller bearing can reinforce the input shaft on some axles. The axle housing is partially filled with oil. This ensures the dip lubrication of gear set and differential bearings. Input shaft bearings are splash lubricated thanks to the crown gear or dip lubricated depending on oil level and axle installation angle.

Automobile and truck axles have the same architecture, but truck ones are larger: crown wheel has a larger diameter, oil sump volume is more important; but the power transmitted is also higher. However, this may lead to comparable efficiency. For information only, some figures are displayed in Table 1.1 regarding light, medium and heavy duty axles. Light duty axles are for car application. Medium duty axles are for distribution trucks while heavy duty ones are for long haul and construction trucks.

Table 1.1: Representative figures for axles depending on application and duty

Car application 1

Truck application

Axle efficiency

Drive axle power losses

The axle generates power losses: some of its components are source of dissipation. The main source of friction is the gear set, especially for a hypoid one, as sliding is more important. Dip lubrication of the gear set generates non-negligible oil churning losses. Rolling element bearings induce losses due to friction between rolling elements and races and also due to dip lubrication, or at least splash lubrication. The seals generate friction between lip and rotating shaft.

The efficiency is simply defined as the ratio between output and input power:

Efficiency = 𝑃 𝑜𝑢𝑡 𝑃 𝑖𝑛 (1.1)
If the gear set is considered as having no transmission error, this is equivalent to:

Efficiency = 1 𝑟𝑎𝑡𝑖𝑜 𝑀 𝑜𝑢𝑡 𝑀 𝑖𝑛 (1.2)
where 𝑀 𝑜𝑢𝑡 and 𝑀 𝑖𝑛 are the output and input torques, and 𝑟𝑎𝑡𝑖𝑜 is the axle ratio.

Experimental and theoretical investigations

Axle efficiency study started through experimental means. A first laboratory procedure was proposed in 1979 by Hobson [START_REF] Hobson | Axle Efficiency -Test Procedures and Results[END_REF]. He considered different testing methods to evaluate efficiency: measurement of heat rejection, axle temperature rise, balance beam method, and torque and speed measurements. The two thermal approaches were eliminated as they require a long time for stabilisation. Finally, the torque and speed measurements solution was selected to quantify power losses due to its reduced complexity and its advantages over the balance beam method. A complete test map is then defined, but also a standard test to allow a first approach, as represented on Figure 1.5. This standard test is based on US conditions with an oil temperature at 66°C. The rated torque is calculated according to the Gross Combination Weight (GCW) of the vehicle.

Figure 1.5: Matrix of potential data points (standard test indicated by dashed grid) [4]

In 1988, Winter and Wech [START_REF] Winter | Measurements and Optimization of the Efficiency of Hypoid Axle Drives of Vehicles[END_REF] worked with a back-to-back test rig for hypoid gears, combining four axles and operating under car conditions. They investigated various factors as gear geometry, operating conditions, lubricant… Hypoid offset and lubricant appear to have a big influence on axle efficiency, power losses increasing with higher offset. This is mostly true under heavy load as gear friction is the major source, whereas bearing preload, dip lubrication and sliding seals predominate at high speed. More recently in 2012, Xu et al. [START_REF] Xu | Effects of Bearing Preload, Oil Volume, and Operating Temperature on Axle Power Losses[END_REF] performed tests on a car axle to study the influence of bearing preload, oil volume and temperature. Oil temperature is considered as the first influencing parameter. Bearing preload comes in second position and last oil volume. Nevertheless, it is important to note that oil quantity inside a car axle is limited: here, volume varies between 1.5L and 3.5L. Moreover, operating temperature depends mostly on the heat generated by power losses: if a lubricant generates less losses at a same temperature level, then temperature will decrease slightly and so the efficiency. There could finally be no gain.

Concerning theoretical studies on axles, Kolekar et al. [START_REF] Kolekar | The efficiency of a hypoid axle -a thermally coupled lubrication model[END_REF] established an efficiency model in 2013. Their quasi-static formula considers the different power loss sources and is coupled to a transient thermal analysis for calculating oil temperature. They estimated the total power loss for two lubricants on various simulated cycles and showed that the lubricant choice depending on drive application is a key in terms of efficiency. Formulae used in this study are analysed in detail in Part 3.3. Mohammadpour et al. [START_REF] Mohammadpour | Transmission efficiency and noise, vibration and harshness refinement of differential hypoid gear pairs[END_REF] focused on hypoid gears in axles with a multi-body dynamics model using an EHL contact. They investigated the gear pair efficiency and its link with Noise Vibration Harshness refinement. They underlined that this NVH refinement is a better indicator of the transmission efficiency than an efficiency measurement or calculation on a drive cycle. Indeed, a drive cycle is generally established for specific road conditions. Kakavas et al. [START_REF] Kakavas | Hypoid gear vehicle axle efficiency[END_REF] recently developed both experimental and model approaches on axle efficiency. Each source of power losses is evaluated on a complete drive cycle for cars, the NEDC (New European Drive Cycle). Then, the global power loss is calculated and compared to measurements, obtaining good agreement. Figure 1.6 presents the simulated power loss for each source of dissipation. When the vehicle accelerates, high torque is applied: gear friction and bearing losses are predominant; when speed is stabilised and so torque is lower, gear friction drops. For this type of drive cycle and according to these calculations, the major contribution is apparently due to bearings, up to almost the total power loss. They proceeded to an analysis on gear parameters such as spiral angle, offset, face width, ease off… and defined a solution to increase efficiency without losing on durability and noise characteristics. As Winter and Wech [START_REF] Winter | Measurements and Optimization of the Efficiency of Hypoid Axle Drives of Vehicles[END_REF], they concluded that a smaller offset but also a smaller spiral angle generates lower friction; for durability aspects, this can be balanced with a larger face width. On the opposite, ease off and surface modifications appear to have almost no influence on hypoid efficiency.

General power loss calculations

The power losses generated in the axle and described previously are separated in load-dependent and load-independent losses [START_REF] Höhn | Thermal rating of gear drives: balance between power loss and heat dissipation[END_REF]:

𝑃 𝑙𝑜𝑠𝑠 = 𝑃 𝑙𝑜𝑎𝑑 + 𝑃 𝑛𝑜 𝑙𝑜𝑎𝑑 (1.3)
With:

𝑃 𝑙𝑜𝑎𝑑 = 𝑃 𝑔𝑒𝑎𝑟 + 𝑃 𝑏𝑒𝑎𝑟𝑖𝑛𝑔 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 (1.4)
𝑃 𝑛𝑜 𝑙𝑜𝑎𝑑 = 𝑃 𝑏𝑒𝑎𝑟𝑖𝑛𝑔 𝑑𝑟𝑎𝑔 + 𝑃 𝑐ℎ𝑢𝑟𝑛𝑖𝑛𝑔 + 𝑃 𝑠𝑒𝑎𝑙 (1.5)

These power loss sources have been thoroughly studied for various systems, and standards notably exist, but not specifically for axles. They are detailed hereafter. Since the axle works at low speeds, up to an order of 2000 rpm on input side, load-dependent losses have a major impact [START_REF] Diab | Prediction of Power Losses Due to Tooth Friction in Gears[END_REF]. Thus, great attention is paid to friction losses at gear mesh but also in rolling element bearings.

Shaft seal friction

Seal power loss results of the friction between seal lip and rotating shaft. This is no major contribution for power losses in axles as shown by the experimental results of Takaki et al. [START_REF] Takaki | Optimization of Hypoid Gear Design for High Efficiency Drives[END_REF] on Figure 1.7. This is explained by the low rotational speeds of the system. Simple approaches are thus summarized, as the one proposed by Simrit [START_REF] Simrit | Radialwellendichtringe[END_REF] which is used in the ISO/TR 14179-2 [START_REF]Gears -Thermal capacity Part 2 : Thermal load-carrying capacity[END_REF], or by Linke [START_REF] Linke | Stirnradverzahnung[END_REF] and by Kettler [START_REF] Kettler | Planetengetriebe-Sumpftemperatur[END_REF] who are taking the temperature into account.

Freudenberg-Simrit formula defines shaft seal friction losses as:

𝑃 𝑠𝑒𝑎𝑙 = 7.69 × 10 -6 𝑑 𝑠ℎ 2 𝑛 (1.6)
where 𝑑 𝑠ℎ is the shaft diameter (mm) and 𝑛 the rotational speed (rpm).

Kettler formula is close to Simrit one, but it takes the temperature influence on oil viscosity by means of a coefficient 𝐹 𝐷 :

𝑃 𝑠𝑒𝑎𝑙 = 7,9163 × 10 -6 𝐹 𝐷 d 𝑠ℎ 2 𝑛 (1.7)
Linke proposed another type of formula:

𝑃 𝑠𝑒𝑎𝑙 = [145 -1.6 × 𝜃 + 350 × log log(𝜈 40 + 0.8)] × 10 -7 𝑑 𝑠ℎ 2 𝑛 (1.8)
where 𝜃 is the temperature (°C), and 𝜈 40 is the oil kinematic viscosity at 40°C (mm²/s).

A comparison between Simrit and Linke models 3 is performed for a 75W-90 lubricant at 80°C (ν 40 =120mm²/s), which is typical for a European long haul truck application on highway. Results are represented on Figure 1.8: losses increase with speed for both solutions but Linke gives a higher estimation than Simrit. The amount of seal losses is still negligible regarding the global power losses of the axle that are up to several kilowatts. As a matter of fact, they are often neglected ([2], [START_REF] Kakavas | Hypoid gear vehicle axle efficiency[END_REF]). However, when considering load-independent losses only, seal losses can contribute. No discrimination between formulae can be done based on speed comparison. Results according to temperature are then plotted on Figure 1.9. A less viscous oil is also tested in this case (ν 40 =40mm²/s).

When temperature increases, the estimation according to Linke formula comes close to zero. This is 3 As the work from Kettler [START_REF] Kettler | Planetengetriebe-Sumpftemperatur[END_REF] is not accessible, the definition of coefficient 𝐹 𝐷 is not known and no comparison with Kettler formula can be held.

Rolling Element Bearing drag and friction

Rolling Element Bearings (REB) generate friction losses as their rolling elements are rolling and sliding on raceways and also drag losses due to the lubricant. These power losses are defined as:

𝑃 𝑏𝑒𝑎𝑟𝑖𝑛𝑔 𝑑𝑟𝑎𝑔 + 𝑃 𝑏𝑒𝑎𝑟𝑖𝑛𝑔 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 = (𝑀 𝑏0 + 𝑀 𝑏1 ) 𝜋 𝑛 30
(1.9)
where 𝑀 𝑏0 is the no-load bearing loss torque or drag torque (N.m), 𝑀 𝑏1 the load bearing loss torque or friction torque (N.m), and 𝑛 is the rotational speed (rpm).

Harris [START_REF] Harris | Rolling bearing analysis[END_REF] established well-known formulae for these torque losses. Typically, for tapered or cylindrical roller bearings, loss torques are:

𝑀 𝑏0 = 1.6 𝑓 0 𝑑 𝑀 3 × 10 -8
for 𝜈 𝑙𝑢𝑏 𝑛 < 2000 mm 2 /s. min (1.10)

𝑀 𝑏0 = 𝑓 0 (𝜈 𝑙𝑢𝑏 𝑛) 2 3
⁄ 𝑑 𝑀 3 × 10 -10 for 𝜈 𝑙𝑢𝑏 𝑛 ≥ 2000 mm 2 /s. min (1.11)

𝑀 𝑏1 = 𝑓 1 𝑃 1 𝑑 𝑀 × 10 -3 (1.12)
where 𝑓 0 and 𝑓 1 are factors depending respectively on bearing and lubrication types, and on bearing design and load, 𝜈 𝑙𝑢𝑏 is the lubricant kinematic viscosity (mm²/s), 𝑑 𝑀 is the bearing mean diameter (mm) and 𝑃 1 is the equivalent bearing load (N). For the ISO technical report 14179-2 [START_REF]Gears -Thermal capacity Part 2 : Thermal load-carrying capacity[END_REF], Eschmann based his formulae on Harris work to calculate friction and drag forces. SKF Group [START_REF] Skf Group | [END_REF] developed his own numerical formula, using a total frictional moment:

𝑀 𝑆𝐾𝐹 = 𝜙 𝑖𝑠ℎ 𝜙 𝑟𝑠 M rr + 𝑀 𝑠𝑙 + 𝑀 𝑠𝑒𝑎𝑙 + 𝑀 𝑑𝑟𝑎𝑔 (1.13)
𝑀 𝑟𝑟 is the rolling frictional moment (N.mm) and 𝑀 𝑠𝑙 is the sliding frictional moment (N.mm), defined as:

M rr = 𝐺 𝑟𝑟 (𝜈 𝑙𝑢𝑏 𝑛) 0.6 (1.14) 𝑀 𝑠𝑙 = 𝐺 𝑠𝑙 𝜇 𝑠𝑙 (1.15)
where 𝐺 𝑟𝑟 and 𝐺 𝑠𝑙 are variables depending on bearing type, dimension and load, and 𝜇 𝑠𝑙 is a sliding friction coefficient. 𝑀 𝑟𝑟 and 𝑀 𝑠𝑙 are thus load dependent moments.

𝑀 𝑠𝑒𝑎𝑙 is the seal friction moment (N.mm) and 𝑀 𝑑𝑟𝑎𝑔 is the frictional moment of drag torque, churning, splashing… (N.mm) defined as:

𝑀 𝑠𝑒𝑎𝑙 = 𝐾 𝑆1 𝑑 𝑠 𝛽 𝑠 + 𝐾 𝑆2 (1.16
)

𝑀 𝑑𝑟𝑎𝑔 = 10 × 𝑉 𝑀 𝐾 𝑟𝑜𝑙𝑙 𝐵 𝑑 𝑀 4 𝑛 2 (1.17)
where 𝐾 𝑆1 and 𝐾 𝑆2 are constant depending on seal type and bearing type and size, 𝑑 𝑠 is the seal counter diameter (mm), 𝛽 𝑠 is an exponent depending on seal and bearing type, 𝑉 𝑀 is a variable function of oil level, 𝐾 𝑟𝑜𝑙𝑙 is a constant related to roller bearing, and 𝐵 is the bearing inner ring width (mm).

Finally, these moments can be categorised in the same manner as for Harris formula, depending on load or on rotational speed:

𝑀 𝑏0 = (𝑀 𝑠𝑒𝑎𝑙 + 𝑀 𝑑𝑟𝑎𝑔 ) × 10 -3 (1.18) 𝑀 𝑏1 = (𝜙 𝑖𝑠ℎ 𝜙 𝑟𝑠 M rr + 𝑀 𝑠𝑙 ) × 10 -3 (1.19)
𝜙 𝑖𝑠ℎ and 𝜙 𝑟𝑠 are factors to take into account the effects of respectively shear heating and starvation. Kakavas et al. [START_REF] Kakavas | Hypoid gear vehicle axle efficiency[END_REF] and Kolekar et al. [START_REF] Kolekar | The efficiency of a hypoid axle -a thermally coupled lubrication model[END_REF], refer to SKF approach in their work.

An analysis is conducted for both Harris and SKF formulae, considering a tapered roller bearing with a mean diameter of 110 mm. This bearing is located on the input shaft of the axle. Losses are calculated under load (𝐹 𝑟 = 7000 N and 𝐹 𝑎 =15000 N) for speed increasing from 200 to 2000 rpm and lubricant at 80°C. Figure 1.10 displays the results. A great discrepancy between Harris and SKF is observed: with increasing speed, power losses according to SKF are doubled compared to Harris estimations.

An important fact in this configuration is that axial load is predominant on radial load, conducting to a particular equivalent load on the bearing. If axial load is virtually decreased, the ratio is inversed: the results obtained in this configuration are plotted on Figure 1.11. The two models are then more coherent. In order to understand the difference between both formulae, losses are decomposed into their different contributions: no-load losses and load losses. Breakdown of the results is plotted on Figure 1.12. Concerning SKF, Figure 1.12 (a) highlights the big contribution of load losses in comparison to noload losses when speed is over 1000 rpm. On Figure 1.12 (b), Harris solution presents a relative equilibrium between friction and drag; the drag becomes predominant with speed, which seems coherent.

The discrepancy observed between Harris and SKF models under specific conditions raises the question of the application domain validity. Indeed, it seems that SKF formula overestimates load losses when the rolling element bearing is loaded mainly radially. In tapered roller bearings, contacts occur between roller ends and ring flanges, due to axial load. The shape and size of this contact depend on bearing design and operating conditions. But it is generally a large contact, which generates friction. As Harris formulae are empirically determined, this phenomena is taken into account. SKF model is a numerical approach and it does not explicitly consider this loss.

Nevertheless, further investigations on the complete bearing set of the axle are held in the second chapter.

Oil churning

The aim of gear lubrication is to ensure both contact durability and heat evacuation. As gear peripheral speeds are rather low in truck applications, dip lubrication is used for the gear set. This induces oil churning: oil is moved by the gears, splashed inside the housing, and trapped between the mating gear teeth.

Investigations on churning started with experiments on smooth plane discs by Daily and Nece [START_REF] Daily | Chamber Dimension Effects on Induced Flow and Frictional Resistance of Enclosed Rotating Disks[END_REF]. Luke and Olver [START_REF] Luke | A study of churning losses in dip-lubricated spur gear[END_REF] proceeded to a wide experimental study on dip-lubricated spur gears in order to understand the discrepancy between the formulae proposed previously by Boness (1989), Terekhov (1991) and [START_REF] Höhn | Thermal rating of gear drives: balance between power loss and heat dissipation[END_REF]. Changenet et al. [START_REF] Changenet | A Note on Flow Regimes and Churning Loss Modeling[END_REF] described the existence of distinct flow regimes: at low speeds, diameter is the major influential parameter on churning losses; at higher speeds, face width has more importance. Daily and Nece [START_REF] Daily | Chamber Dimension Effects on Induced Flow and Frictional Resistance of Enclosed Rotating Disks[END_REF], and more recently Changenet and Velex [START_REF] Changenet | Housing Influence on Churning Losses in Geared Transmissions[END_REF], investigated the influence of axial and radial clearances thanks to deflectors and flanges. According to these various studies on spur gears, it appears that the main factors influencing churning losses are: rotational speed, gear diameter and casing. Few studies have been conducted on churning losses of spiral bevel and hypoid gears. However, two ISO technical reports (14179-1 [START_REF]Gears -Thermal capacity Part 1 : Rating gear drives with thermal equilibrium at 95°C sump temperature[END_REF] and 14179-2 [START_REF]Gears -Thermal capacity Part 2 : Thermal load-carrying capacity[END_REF]) propose a formula to calculate these losses, whatever the gear.

ISO/TR 14179-1 decomposes churning losses as the sum of the ones associated with gear faces 𝑃 𝑠𝑖𝑑𝑒𝑠 (kW) and the ones of tooth surfaces 𝑃 𝑡𝑜𝑜𝑡ℎ (kW). For gear faces, it is defined as:

𝑃 𝑠𝑖𝑑𝑒𝑠 = 1.474 × 𝑓 𝑔 𝜈 𝑙𝑢𝑏 𝑛 3 𝐷 5.7 𝐴 𝑔 × 10 26 (1.20)
where 𝑓 𝑔 is a gear dip factor (0 ≤ 𝑓 𝑔 ≤ 1), 𝜈 𝑙𝑢𝑏 is the kinematic viscosity of oil at operating temperature(mm²/s), 𝑛 is the rotational speed (rpm), 𝐷 is the gear outside diameter (mm), and 𝐴 𝑔 is the arrangement constant (=0,2). For tooth surfaces, it is:

𝑃 𝑡𝑜𝑜𝑡ℎ = 7.37 × 𝑓 𝑔 𝜈 𝑙𝑢𝑏 𝑛 3 𝐷 4.7 𝐹 𝑤 ( 𝑅 𝑓 √𝑡𝑎𝑛 𝛽 ) 𝐴 𝑔 × 10 26 (1.21)
where 𝐹 𝑤 is the total face width (mm), and 𝛽 is the generated helix angle (°) (if 𝛽 < 10° then consider 𝛽 = 10°). 𝑅 𝑓 is a roughness factor which is defined as:

𝑅 𝑓 = 7.93 - 4.648 𝑚 𝑡 (1.22)
where 𝑚 𝑡 is the transverse tooth module.

The ISO technical report 14179-2 calculates the churning losses according to Mauz formula:

𝑃 𝑐ℎ𝑢𝑟𝑛𝑖𝑛𝑔 = 𝐶 𝑠𝑝 𝐶 1 exp (𝐶 2 ( 𝑣 𝑡 𝑣 𝑡0 )) Ω (1.23)
where 𝐶 𝑠𝑝 is a splash oil factor, 𝐶 1 and 𝐶 2 are factors depending on tooth width and immersion depth, 𝜈 𝑡 is the tangential speed (m/s), 𝜈 𝑡0 is the reference tangential speed (=10 m/s), and Ω is the gear angular velocity (rad/s). The various factors are defined as:

𝐶 𝑠𝑝 = ( 4 ℎ 𝑒,𝑚𝑎𝑥 3 ℎ 𝑐 ) 1.5 2 ℎ 𝑐 𝑙 ℎ (1.24) 𝐶 1 = 0.063 ( ℎ 𝑒1 + ℎ 𝑒2 ℎ 𝑒0 ) + 0.0128 ( 𝑏 𝑡𝑜𝑜𝑡ℎ 𝑏 0 ) 3 (1.25) 𝐶 2 = ℎ 𝑒1 + ℎ 𝑒2 80 ℎ 𝑒0 + 0.2 (1.26)
where ℎ 𝑒1,𝑒2 are the tip circle immersion depths of both gear and ℎ 𝑒,𝑚𝑎𝑥 the maximum one (mm), ℎ 𝑒0 is the reference value of immersion depth (mm), ℎ 𝑐 is the height of point of contact (mm), 𝑙 ℎ is the hydraulic length (mm), 𝑏 𝑡𝑜𝑜𝑡ℎ is the tooth width (m), and 𝑏 0 its reference value (=0.01 m).

Figure 1.13 illustrates the various immersions considered. An interesting choice of this formula is not to take into account the lubricant viscosity. In his work on an automotive axle, Jeon [START_REF] Jeon | Improving Efficiency in Drive Lines: an Experimental Study on Churning Losses in Hypoid Axle[END_REF] studied churning losses for hypoid gears and established a specific relationship:

𝑃 𝑐ℎ𝑢𝑟𝑛𝑖𝑛𝑔 = 1 2 𝜌 Ω 3 𝑏 𝑅 𝑝 2 𝑆 𝑚 𝐶 𝑚 (1.27)
where 𝜌 is the lubricant density, Ω is the gear angular velocity (rad/s), 𝑏 is the gear width (m), 𝑅 𝑝 is the gear pitch radius (m), 𝑆 𝑚 is the gear immersed surface area (m²). 𝐶 𝑚 is the dimensionless drag torque, defined as:

𝐶 𝑚 = 2.19 ( ℎ 𝑙𝑢𝑏 𝑅 𝑝 ) 0.15 ( 𝑉 𝑙𝑢𝑏 𝑅 𝑝 3 ) -0.20 𝑅𝑒 -0.25 𝐹𝑟 -0.53 (1.28)
where ℎ 𝑙𝑢𝑏 is the gear immersion depth (m), 𝑉 𝑙𝑢𝑏 is the oil sump volume (m 3 ), 𝑅𝑒 is the Reynolds number, and 𝐹𝑟 is the Froud number.

Jeon proposed a second formula of the dimensionless drag torque to consider an air-oil mix around gears:

𝐶 𝑚 = 10 ( ℎ 𝑙𝑢𝑏 𝑅 𝑝 ) 0.18 ( 𝑉 𝑙𝑢𝑏 𝑅 𝑝 3 ) -0.22 𝑅𝑒 -0.42 𝐹𝑟 -0.45 ( 𝜈 𝑙𝑢𝑏 𝜈 𝑎𝑖𝑟 ) -0.18 (1.29)
where 𝜈 𝑙𝑢𝑏 and 𝜈 𝑎𝑖𝑟 are respectively lubricant and air kinematic viscosity (mm²/s). Kakavas et al. [START_REF] Kakavas | Hypoid gear vehicle axle efficiency[END_REF] used this last formula.

Calculations are made according to ISO technical reports and Jeon formulae for a spiral bevel gear pair. The diameter of the crown wheel is 157 mm. Equation (1.28) is used for calculations according to Jeon.

The results for speed increasing from 200 to 2000 rpm are presented on Figure 1.14. The three methods give different estimations of churning losses, which can be more than doubled. Each solution is then tested on the axle case in the second chapter.

ℎ 𝑒1 ℎ 𝑒,𝑚𝑎𝑥 ℎ 𝑒2 ℎ 𝑐

Figure 1.14: Comparison between Jeon and ISO technical report formulae for churning losses

Kolekar et al. [START_REF] Kolekar | The efficiency of a hypoid axle -a thermally coupled lubrication model[END_REF] used Equations (1.27) and (1.28) proposed by Jeon, and combined it with Changenet and Velex approach for a gear pair [START_REF] Changenet | A Model for the Prediction of Churning Losses in Geared Transmissions-Preliminary Results[END_REF]. Indeed, contrary to Mauz formula, Equation (1.27) is valid for a single gear only; the gear pair is considered by adding both contributions. Changenet and Velex [START_REF] Changenet | A Model for the Prediction of Churning Losses in Geared Transmissions-Preliminary Results[END_REF] investigated the churning losses of a meshing spur gear, especially by testing different rotation directions as represented on Figure 1.15. They showed that the addition of the two individual contributions is realistic for a clockwise rotation, but not for a counter-clockwise rotation, which generates higher losses. [START_REF] Changenet | A Model for the Prediction of Churning Losses in Geared Transmissions-Preliminary Results[END_REF] The gear set of a rear axle rotates clockwise when the vehicle is going forward. Thus, according to the solution of Changenet and Velex, the contribution of pinion and crown gear should just be added. However, for a meshed gear pair, Jeon [START_REF] Jeon | Improving Efficiency in Drive Lines: an Experimental Study on Churning Losses in Hypoid Axle[END_REF] measured a greater increase of churning losses: this can be due to oil trapping between gear teeth or to a modification of the oil flow around the crown gear, as observed by Jeon. The power loss of the pair is then bigger than the addition of each gear contribution.

Figure 1.15: Definition of rotation directions

Hypoid gear mesh friction

The hypoid gear set is the main component of the drive axle and it can be also the main source of friction. Due to pinion offset, sliding occurs all along the contact line. This induces an efficiency of

Mauz

Jeon roughly 96% to 98% [START_REF] Kolivand | Prediction of mechanical gear mesh efficiency of hypoid gear pairs[END_REF]. The gear mesh friction loss is commonly described as the product of friction coefficient, load or applied pressure, and relative velocity between gear teeth surfaces.

The efficiency of the hypoid gear set was studied by Buckingham [START_REF] Buckingham | Analytical Mechanics of Gears[END_REF], who considers the hypoid as a combination between a spiral bevel gear and a worm gear. In 1949, he expressed its efficiency as:

𝐸 ℎ𝑦𝑝𝑜𝑖𝑑 = 1 1 + 𝜙 𝑠𝑝𝑖𝑟𝑎𝑙 𝑏𝑒𝑣𝑒𝑙 + 𝜙 𝑤𝑜𝑟𝑚 (1.30)
where 𝜙 𝑠𝑝𝑖𝑟𝑎𝑙 𝑏𝑒𝑣𝑒𝑙 is the friction loss factor of spiral bevel gear taking into account the conjugate gear tooth action, and 𝜙 𝑤𝑜𝑟𝑚 is the friction loss factor of worm gear considering the sliding velocities. Details of the formula are given in Annex 1.A. However, Buckingham does not use directly the sliding velocity in his formula; he determines a friction coefficient depending on sliding velocity value.

In 1975, Coleman [START_REF] Coleman | Effect of mounting displacements on bevel and hypoid gear tooth strength[END_REF] proposed another formula of the hypoid efficiency:

𝐸 ℎ𝑦𝑝𝑜𝑖𝑑 = 100 1 + √𝑀 𝑚𝑎𝑥 /𝑀 (0.01 + 𝜇 sec 𝜙 √(tan 𝜓 𝑃 -tan 𝜓 𝐺 ) 2 + 0.0225 ) (1.31)
where 𝑀 and 𝑀 𝑚𝑎𝑥 are respectively the applied and maximum gear torque (N.m), 𝜙 is the pressure angle (°), and 𝜓 𝑃 and 𝜓 𝐺 are pinion and gear mean spiral angle (°). The friction coefficient 𝜇 ranges from 0.01 to 0.12 depending on lubricant and tooth surface, but is generally equal to 0.05. According to this formula, the efficiency can be improved by reducing pressure and spiral angles or by increasing operating load.

ISO/TR 14179-2 [START_REF]Gears -Thermal capacity Part 2 : Thermal load-carrying capacity[END_REF] uses the work of Niemann and Winter (1983) to determine hypoid friction loss. An equivalent helical gear substitutes for the hypoid one for gear mesh calculation. Hypoid friction power loss is defined as:

𝑃 ℎ𝑦𝑝𝑜𝑖𝑑 = 𝐹 𝑛 𝜈 𝑔𝑚 𝜇 𝑚 (1.32)
where 𝐹 𝑛 is the tooth normal load (N), 𝜈 𝑔𝑚 is the mean sliding speed (m/s), and 𝜇 𝑚 is the mean friction coefficient. Details of the formula are also given in Annex 1.A.

These three analytical formulae (Buckingham, Coleman and ISO/TR 14179-2) present different structures. To hold a comparison, an analogy to spur and helical gears efficiency is proposed: according to Velex and Ville [START_REF] Velex | An Analytical Approach to Tooth Friction Losses in Spur and Helical Gears-Influence of Profile Modifications[END_REF], classical efficiency formulae for spur and helical gears given by [START_REF] Buckingham | Analytical Mechanics of Gears[END_REF] or Niemann and Winter (1989) can be expressed as:

𝐸 𝑠𝑝𝑢𝑟 = 1 -𝜇 𝐻 𝑣 (1.33)
where 𝐻 𝑣 is a factor depending on spur or helical gear geometry. This means that load and speed can have an influence only through friction coefficient and that efficiency depends only on gear geometry if friction coefficient is kept constant. Concerning hypoid efficiency, the ISO technical report formula can be expressed with a similar structure:

𝐸 ℎ𝑦𝑝𝑜𝑖𝑑 = 1 - 𝑃 ℎ𝑦𝑝𝑜𝑖𝑑 𝑃 𝑖𝑛 = 1 -𝜇 𝑚 𝐻 𝑣 ℎ𝑦𝑝𝑜𝑖𝑑 (1.34)
where 𝐻 𝑣 ℎ𝑦𝑝𝑜𝑖𝑑 is a factor depending on hypoid gear geometry. This is less evident for the two other formulae.

A Besides these analytical formulae of the gear efficiency, some discretisation approaches of the contact friction are researched. Xu and Kahraman [START_REF] Xu | Prediction of friction-related power losses of hypoid gear pairs[END_REF] developed a method using a hypoid gear contact analysis model and a gear pair mechanical efficiency formula. They also proposed a friction coefficient calculation based on an elastohydrodynamic lubrication model. They identified surface roughness amplitude and lubricant temperature as influential parameters of efficiency. Kolivand et al. [START_REF] Kolivand | Prediction of mechanical gear mesh efficiency of hypoid gear pairs[END_REF] improved this previous work, notably by taking into account surface asperities for the friction coefficient calculations.

Takaki et al. [START_REF] Takaki | Optimization of Hypoid Gear Design for High Efficiency Drives[END_REF] evaluates the hypoid efficiency by calculating friction loss at each meshing point along the contact path based on real tooth geometry.

In a first time, analytical formulae (Buckingham and ISO/TR 14179-2) are used for hypoid gear loss estimation in the second chapter. The appropriateness of this global approach is then questioned.

Axle thermal behaviour

The power loss sources reviewed in the previous part generate heat at different points of the system. In order to establish a thermo-mechanical model of the axle, it is necessary to know more precisely the thermal exchanges with the environment but also inside the axle. This allows to determine not only the oil temperature, but also the operating temperature of each component. Indeed, this information is important to predict bearing and gear failures, like scoring [START_REF] Handschuh | Thermal behaviour of spiral bevel gears[END_REF], pitting, or even scuffing. It seems also rather imprecise to calculate the gear set bulk temperature considering only oil temperature and gears operating condition [START_REF]Calculation of scuffing load capacity of cylindrical, bevel and hypoid gears -Part 2: Integral temperature method[END_REF], ignoring the global system around it. Previous work has already shown some limitations for this kind of method: for example, the position of jet lubrication is not taken into account while influencing bulk temperature [START_REF] Handschuh | Effect of Lubricant Jet Location on Spiral Bevel Gear Operating Temperatures[END_REF].

Thermal behaviour of the complete axle

Most of axle studies focus only on power losses ([4]- [START_REF] Xu | Effects of Bearing Preload, Oil Volume, and Operating Temperature on Axle Power Losses[END_REF], [START_REF] Mohammadpour | Transmission efficiency and noise, vibration and harshness refinement of differential hypoid gear pairs[END_REF]). Operating conditions are thus all under control. Among them, oil sump temperature is set up to a target value. For those who deal with thermal aspects, a common hypothesis is to consider the axle as an isothermal system at oil sump temperature. Indeed, Kolekar et al. [START_REF] Kolekar | The efficiency of a hypoid axle -a thermally coupled lubrication model[END_REF] developed a simple thermal model of the axle, where the whole axle is at oil sump temperature. This temperature is calculated, as forced convection between air and axle occurs. On-road cycles are thus simulated and the efficiency is evaluated depending on environmental conditions. Kakavas et al. [START_REF] Kakavas | Hypoid gear vehicle axle efficiency[END_REF] adopted a similar approach in order to work on the NEDC complete drive cycle. The calculated oil temperature is in good agreement with the measured one.

This kind of approach may be sufficient to estimate the global efficiency; but at the same time, component temperatures are an indicator of the power loss share between the different sources. For example, no experimental evidence confirms the breakdown presented on Figure 1.6 by Kakavas et al.

If the real power loss split is unknown, it can be difficult to know which component should be improved.

Some have investigated the thermal behaviour of the axle using experimental means: Hurley [START_REF] Hurley | An Experimental Investigation of Thermal Behavior of an Automotive Rear Axle[END_REF] and Xu et al. [START_REF] Xu | Thermal Mapping of an Automotive Rear Drive Axle[END_REF] recorded the temperature at multiple points on axle parts. For various operating conditions, they observed the temperature difference between components: for example on Figure 1.17, pinion tooth (sensors P1-P2) are hotter than input shaft bearing inner races (sensors P3-P4 and P6-P7). This information helps to estimate that the amount of power losses generated by the gear mesh is bigger than the one of the bearings for the tested conditions. On Figure 1.18, the power loss decreases during each load step, as oil temperature increases and oil becomes less viscous. Another interesting point is to compare oil temperature presented on Figure 1.18 to pinion temperature of Figure 1.17 for the same test. It appears clearly that gear bulk temperature is higher than oil temperature, up to 30°C. It is then important to choose the correct temperature for durability issues, but also for the friction coefficient calculation. A local approach on the gear set can be investigated to address these issues.

Thermal behaviour of the gear set

Some have focused specifically on the gear set, as Handschuh [START_REF] Handschuh | Thermal behaviour of spiral bevel gears[END_REF] or Wang et al. [START_REF] Wang | Investigation into the meshing friction heat generation and transient thermal characteristics of spiral bevel gears[END_REF].

Handschuh conducted both experimental and modelling approaches for aerospace applications. He notably measured the pinion bulk temperature and observed a great influence of load and lubrication jet position. He also developed a thermal model of one tooth, representative of a complete gear: a three-dimensional Finite Element model of the tooth allows to consider the heat flux along active profile; at each time step, a new Hertzian contact is activated. For an identical friction coefficient, the most tilted and extended contact (case 3) has a reduced frictional heat generation. Thus, as this induces fewer losses, they also addressed the efficiency issue. On both study, the heat flux is equally divided between the mating gears.

It is not specified how the global system can influence on local temperature of the gear mesh contact. Nevertheless it seems important to consider the axle as a whole, with components at different temperatures exchanging between them. Finally, no present study has developed a model to obtain these components temperature.

Conclusion

The truck drive axle was first presented with its main components and specificities. The most important part is the gear set, which is a spiral bevel or a hypoid one. The literature mainly presents studies on automotive and light to medium duty axles. These are similar in structure and working to heavy duty axles, but their operating conditions and dimensions are different. This may result in new issues.

The power loss sources of the axle were identified: shaft seal friction, rolling element bearing drag and friction, oil churning and gear mesh friction. Previous studies evaluated globally or separately these power losses by experimental or theoretical means. Influential parameters are highlighted through these investigations: operating conditions, such as lubricant temperature, rotational speed, load…; gear macro-geometry parameters, like hypoid offset, spiral angle, face width, diameter, ease-off… and micro-geometry parameters, like surface roughness.

Nevertheless, these proposed power loss models do not always agree with each other. Depending on application and operating conditions, great discrepancies can be observed between estimations. Some of the models used by Kolekar et al. [START_REF] Kolekar | The efficiency of a hypoid axle -a thermally coupled lubrication model[END_REF] or Kakavas et al. [START_REF] Kakavas | Hypoid gear vehicle axle efficiency[END_REF] are under question. Appropriate formulae should be selected for the truck axle case, after specific comparisons held in the next chapter.

The thermal behaviour of the axle was investigated thanks to isothermal approaches and for efficiency purposes. This has some limitations, as the estimated power loss breakdown cannot be validated with only oil temperature. Some thorough testing campaigns underline the temperature difference between components like gears and bearings, suggesting power loss amount. Oil sump temperature appears not to be the correct parameter for durability and friction aspects. Specific local thermal models were developed for the spiral bevel gear set to address both durability and efficiency issues, and support the importance of contact pattern.

However, no detailed thermal model for axles has been developed yet to compute both efficiency and temperatures. Thus, the aim of this work is to develop a thermo-mechanical model of the axle.

Introduction

The first chapter has underlined that different relationships can be used to estimate the power loss sources of a truck axle. This one aims to analyse if simplified analytical approaches, as the one presented in ISO/TR 14179-2 [START_REF]Gears -Thermal capacity Part 2 : Thermal load-carrying capacity[END_REF], are suitable to quantify the efficiency of such gear unit. As a first approach, the efficiency of a specific axle is estimated through experiments and compared to various theoretical formulae.

Truck axle efficiency

Considered drive axle

This study focuses on a single reduction axle for a long haul application. The major part of interest is the drive head. The technical drawing of Figure 2.1 helps to locate the main studied components:

-Pinion gear (A) and crown gear (B) -Rolling Element Bearings (1) to ( 5) -Seal [START_REF] Xu | Effects of Bearing Preload, Oil Volume, and Operating Temperature on Axle Power Losses[END_REF] Although it is also represented on Figure 2.1, the differential is considered unused, as if the truck was running along a straight line. Then, it does not generate any power loss.

Figure 2.1: Location of main components on drive head technical drawing

The gear set is an hypoid one and its characteristics are given in Table 2.1. The main features of the bearings are listed in Table 2.2. The seal is a one lip seal with a shaft diameter of 80 mm.

The axle housing is partially filled with oil: the normal fill corresponds to 14L.

Efficiency tests

Tests were conducted in order to measure the efficiency of the considered axle, using a similar procedure than Hobson's one [START_REF] Hobson | Axle Efficiency -Test Procedures and Results[END_REF]. Two types of tests were conducted: with and without an applied input torque.

For tests with an applied load, the bench described on For tests without an applied load, the bench described on On both benches, oil temperature in the sump is monitored and maintained at 80°C by the use of fans. Characteristics of the oil used for the tests are presented in Table 2.3.

The brand-new axle is run-in in order to obtain stabilised power losses. For a typical condition under load (120kW), the variation of power losses between two break-in cycles must be lower than 200W in order to consider that the running-in procedure is achieved.

During load-dependent losses tests, the three conditions presented on Table 2.4 are reproduced. Condition B is a typical highway condition for long haul trucks in Europe. For each of these conditions, two different oil levels are considered: a normal and a high one, respectively corresponding to 14L and 20L of oil volume. According to the tests under load, the efficiency of this axle ranges from 94% to 98% depending on conditions. A higher oil level induces more power losses, except under high torque where losses are at the same level.

These experimental results are compared to calculations in the next part.

Calculations of load-independent losses

In the previous chapter, the load-independent sources of dissipation were identified as:  Shaft seal friction  Rolling Element Bearing drag and friction (friction is considered as there is preload on some REBs)  Oil churning These power losses are now evaluated by using theoretical approaches presented in the literature and compared to the experimental results [START_REF] Fossier | Investigations on drive axle thermal behaviour: power loss and heat-transfer estimations[END_REF]. To this end, the whole system is assumed to be isothermal at oil sump temperature, which is equal to 80°C.

Input shaft seal friction

The equation proposed by Simrit [9] is used. This is the reference for ISO/TR 14179-2 [START_REF]Gears -Thermal capacity Part 2 : Thermal load-carrying capacity[END_REF] and appears to be the most robust formula for seal loss calculation. As presented on Figure 2.5, seal power losses are smaller than 200 W. This is not a major source of dissipation when compared to the axle power losses, especially under load conditions (Figure 2.3).

Rolling Element Bearing drag and friction

Concerning the Rolling Element Bearing losses, a comparison between Harris and SKF formulae is performed for every test conditions. Factors used for Harris formula are presented in Table 2.5. Results are plotted on Figure 2.6. As some bearings have a preload, the losses of no-load test conditions are the drag ones but also a part of bearing friction ones. For no-load test conditions ( For the loaded cases, the load losses are added:

-REBs losses increase, depending on load and model (calculated) -Gear mesh friction

The efficiency of the gear mesh is then deduced for both Harris and SKF models and presented on Figure 2.7. According to SKF formula, hypoid gear mesh efficiency is close to or over 99%. This is unlikely, as the hypoid efficiency is more around 96-98% [START_REF] Kolivand | Prediction of mechanical gear mesh efficiency of hypoid gear pairs[END_REF]. Thus, SKF formula gives unrealistic results and is not adapted for this case. For Harris model, the evaluated efficiency ranges approximately from 98% to 99%. Although this is still a good efficiency, Harris model calculates more coherent results. It is kept for the continuation.

Figure 2.8: Comparison between global measurements (no-load tests), deduced churning loss and churning loss calculation results according to Mauz and Jeon formulae

It appears clearly that Mauz formula overestimates the power losses, as calculated churning losses represent on their own more than the measured load-independent losses. On the contrary, the estimation given by Jeon formula is too low when compared to experimental measurements and to the deduced oil churning losses. Indeed, at 80 km/h, the churning losses are around 1 kW whereas Jeon gives 270 W. Both models are not adapted to this case. Thus, a further study is required on oil churning losses and is conducted in the next part of this chapter.

Investigations on oil churning losses of hypoid gears

As underlined in the first chapter, the literature is rather limited concerning oil churning of hypoid and spiral bevel gears in general. Moreover, as shown in the previous part, Mauz and Jeon formulae seem unsuitable for the truck axle application. However, Jeon worked specifically on an automotive axle, which is similar to a truck axle. A closer observation of his work is then made. The established relationship for oil churning of spiral bevel gears is:

𝑃 𝑐ℎ𝑢𝑟𝑛𝑖𝑛𝑔 = 1 2 𝜌 Ω 3 𝑏 𝑅 𝑝 2 𝑆 𝑚 𝐶 𝑚 (2.1)
where 𝜌 is the lubricant density, Ω is the gear angular velocity (rad/s), 𝑏 is the gear width (m), 𝑅 𝑝 is the gear pitch radius (m), 𝑆 𝑚 is the gear immersed surface area (m²), and 𝐶 𝑚 is the dimensionless drag torque. This formula is based on the main geometrical features of the gear: width, pitch radius, immersed area… However, Jeon has tested only two geometries and their pitch diameters are close

Test rig

A specific test rig is used in order to study oil churning losses. The test set-up is fully described by Changenet and Velex [START_REF] Changenet | Housing Influence on Churning Losses in Geared Transmissions[END_REF] and is represented on Figure 2.9. Hereafter are summarised the major information concerning the rig.

Truck axle application

In this study, a general approach of churning losses of spiral bevel and hypoid gears is made. Additionally to this, the truck axle application is also investigated. In order to use the test bench results for this case, some operating conditions must be covered during experiments. A similitude approach is then used: equivalent parameters are peripheral speed, oil viscosity and immersion. An immersion ratio ℎ 𝑙𝑢𝑏 /𝑅 𝑜 is defined as the ratio between the immersion depth ℎ 𝑙𝑢𝑏 and the gear outer radius 𝑅 𝑜 , as illustrated on Figure 2.13. Tested gears are left handed, whereas the truck axle crown gear is right handed. An opposite direction of rotation is then used to obtain the same behaviour.

Considering the axle crown gear, peripheral speeds presented in Table 2.8 range from 2.5 to 12.4 m/s. They are quite low as the truck tyres have a big radius (0.48 m in this case). Pinion gear has slightly higher peripheral speeds (from 3.6 to 18 m/s). Indeed, due to offset, hypoid pinion gear has a bigger diameter than the spiral bevel one for the same crown gear size. The tested gears experience the same peripheral speeds (from 6.8 to 19.7 m/s), except for very low speeds as the motor drive is not stable under 800 rpm. It is important to notice that a single reduction axle is considered here; in the case of a double reduction axle, the main speed reduction takes place in the wheel hubs and the gear set thus rotates at higher speeds. Tested oil D is a typical drivetrain lubricant and is the same as the one used for previous axle tests of the study. The oil temperature range considered here evolves from 20°C to 100°C, which corresponds to a viscosity of 352 cSt to 16 cSt.

Figure 2.14: Immersion of crown and pinion gears in an axle (a) with inclination and (b) without inclination

Concerning oil immersion, the rolling element bearings of the differential are small in comparison to the crown gear and their lubrication must be ensured thanks to a sufficient oil level. Thus, the crown gear is highly immersed: the static immersion ratio ranges from 0.7 to 0.9. However, oil level decreases Oil level

(a) (b)
during axle operation due to oil splashing. The maximum dynamic immersion ratio used during experiments is then 0.6.

The immersion of the pinion is harder to reproduce since the axle is generally tilted when mounted on a truck, as illustrated on Figure 2.14. Nevertheless, for practical reasons, most of the tests on axles were done with no inclination. This leads to a relatively important immersion of the pinion. However, rotational speeds in automotive axle are higher and so peripheral speeds, even if the crown gear diameter is smaller. Truck speeds can be insufficient to generate these phenomena. As the test rig housing is a parallelepiped, use of deflectors allows for changing both the distance between tested gear and wall and the wall shape, as presented on Figure 2.10, while keeping the same oil volume.

Experimental results

All the results obtained during the study are sum-up in Annex 2.A.

The influence of operating conditions on churning losses is investigated. Tests are performed on every gear geometry with various rotational speeds, immersion levels and oil properties (density and viscosity).

For each tested gear, churning losses are measured with an increasing speed, from 1000 to 2000 rpm. The other operating conditions (immersion, oil properties) are kept constant. Results for oil C at 40°C and an immersion ratio of 0.5 are plotted on Figure 2.16. Churning losses increases greatly with the rotational speed: the proportionality is to the power 1.7. Another series of tests is done to observe the effect of immersion on losses. Measurements are performed at various immersion levels. Figure 2.17 presents the results at 2000 rpm and for oil C at 40°C. It shows that the immersion ratio is a parameter of first order on churning losses. Churning losses clearly increase with oil viscosity. At low viscosity (under 30 cSt), losses remain constant. This phenomena was not observed by Jeon: in his experiments, viscosity of the lubricant is over 30 cSt. Changenet et al. [START_REF] Changenet | A Note on Flow Regimes and Churning Loss Modeling[END_REF] have already underlined this behaviour for cylindrical gears. Measurement results are similar for the various oils and no influence of oil densities is observed, because this characteristic is close for all tested oils. In a second time, the focus is put on the effect of gear geometrical parameters: outer diameter, width, face width, face angle, and number of teeth. Comparison between the various tested gears is made. Gear 1 and cone 1 have the same geometry, apart from the presence of teeth, as indicated in Table 2.6. Their respective churning losses are plotted on Figure 2.19 for different immersions. The same tendencies are observed, so teeth have limited influence on churning losses. Gear 1 and gear 4 display the same power loss behaviour, according to rotational speed on Figure 2.16 or to oil immersion on Figure 2.17. However, these gears have different geometrical parameters: number of teeth, face angle, width… while they have similar outer diameter. As well, Figure 2.19 presents comparable results for cone 1 and cone 2. These cones also have close outer diameter but dissimilar face width, face angle and width. In conclusion, the gear outer diameter is a geometrical parameter of first order for churning losses compared to other ones. For a bigger outer diameter (gear 3), higher churning losses are experienced; conversely, there are less losses for a smaller outer diameter (gear 2).

Comparison with existing calculation methods

Some comparisons of measured power losses with those predicted from the existing relationships are done. According to ISO/TR 14179-1, churning losses are proportional to Ω 3 which is not in accordance with experiments (proportional to Ω 1.7 ). Because of this divergence, ISO/TR 14179-1 is disregarded in the following analyses. Another comparison, associated with the influence of oil properties, is given in Figure 2.21. As it has been underlined, two flow regimes clearly emerge from measurements whereas calculation methods are based on a single regime: the churning losses always decrease with viscosity according to Jeon's approach and it remains constant by using the relationships of Mauz.

The discrepancy observed between measurements and existing formulae implies modifications.

Improvement of oil churning formula for truck axles

Specifications for spiral bevel and hypoid gears

According to the gear geometry study of the previous part, gear teeth have no influence on churning losses. This induces a similar immersed surface area for both gears and cones. Calculation of this surface is detailed by Laruelle et al. [START_REF] Laruelle | Experimental investigations and analysis on churning losses of splash lubricated spiral bevel gears[END_REF] and reported in Annex 2.B: it considers an envelope geometry of the gear.

Using formula (2.1) and test measurements, an experimental dimensionless churning torque 𝐶 𝑚 is calculated as: where ℎ 𝑙𝑢𝑏 is the gear immersion depth (m), 𝑉 𝑙𝑢𝑏 is the volume of oil sump (m 3 ), 𝑅𝑒 is the Reynolds number, and 𝐹𝑟 is the Froud number. This discrepancy implies a modification, whether of the dimensionless churning torque or of the churning power loss formula, to be in agreement with dimensionless analysis. Experimental results underline that gear width has a limited influence on churning losses, contrary to gear outer diameter. Formula (2.1) is then modified to the following equation:

𝑃 𝑐ℎ𝑢𝑟𝑛𝑖𝑛𝑔 = 1 2 𝜌 Ω 3 𝑅 𝑜 3 𝑆 𝑚 𝐶 𝑚 (2.4)
where 𝑅 𝑜 is the gear outer radius (m). Using this new formula, a better concordance is observed on For dimensional analysis, these seven parameters can be expressed by fundamental unities: length, time and mass. According to Buckingham 𝜋 theorem [START_REF] Misic | Dimensional analysis in physics and the Buckingham theorem[END_REF], three principal parameters 𝑅 𝑜 , Ω, and 𝜌 are selected and four dimensionless parameters can be constituted to define the dimensionless churning torque:

𝐶 𝑚 = 𝑓𝑐𝑡 [𝑅𝑒, 𝐹𝑟, ℎ 𝑙𝑢𝑏 𝑅 𝑜 ⁄ , 𝑉 𝑙𝑢𝑏 𝑅 𝑜 3 ⁄ ] (2.6)
with Reynolds and Froude numbers defined as follows:

𝑅𝑒 = Ω 𝑅 𝑜 2 𝜈 𝑙𝑢𝑏 (2.7) 𝐹𝑟 = Ω 2 𝑅 𝑜 𝑔 (2.8)
where 𝜈 𝑙𝑢𝑏 is the oil kinematic viscosity (mm²/s).

According to the various tests presented in previous part, the dimensionless churning torque is now analysed and compared with Jeon approach:

𝐶 𝑚 = 𝛾 1 ( ℎ 𝑙𝑢𝑏 𝑅 𝑜 ) 𝛾 2 ( 𝑉 𝑙𝑢𝑏 𝑅 𝑜 3 ) 𝛾 3 𝑅𝑒 𝛾 4 𝐹𝑟 𝛾 5 (2.9)
Evolution of 𝐶 𝑚 is first studied regarding Reynolds number. Speed is thus kept constant in order to have no variation on Froude number.

Figure 2.23 shows the existence of two oil flow regimes, with a transition around a Reynolds of 20 000.

For the regime bellow the transition, torque decreases with the Reynolds number as found by Jeon: 𝛾 4 equals -0.25. The second regime is not described by Jeon, as it is out of his scope of tests in terms of oil viscosity. The Reynolds number, and so viscous forces, have no more influence for this regime and 𝛾 4 is null.

The influence of Froude number is isolated through tests where only the rotational speed varies as the ones given in Figure 2.16. The constant 𝛾 5 is then deduced: a value of -0.53 is found which is in agreement with Jeon's formula.

Figure 2.23: Two flow regimes of churning depending on Reynolds number

Concerning oil immersion and volume, the test bench shape is far from the axle one. Tests with and without deflector that modifies the housing shape are performed for different oil levels. In order to characterise the influence of oil immersion on churning, the immersion ratio is isolated from the experimental dimensionless churning torque formula according to Equation (2.9):

𝐶 𝑚 𝑒𝑥𝑝 ( 𝑉 𝑙𝑢𝑏 𝑅 𝑜 3 ) -0.20 𝑅𝑒 -0.25 𝐹𝑟 -0.53 = 𝛾 1 ( ℎ 𝑙𝑢𝑏 𝑅 𝑜 ) 𝛾 2 (2.10)
This expression is used as ordinate on Figure 2.24 for both housing configurations. This highlights that oil immersion influence changes notably with housing shape. This induces different coefficients 𝛾 2 depending on the housing. As Jeon experiments are closer to the truck axle case, coefficients 𝛾 2 and 𝛾 3 are kept as 0.15 and -0.20.

However, Figure 2.25 highlights discrepancy between test results and Jeon theoretical formula. This can be induced by the different approaches adopted. Indeed, no comparison could be performed between gear immersed surface areas defined by Jeon and Laruelle; also, the dimensionless experimental data depend on gear radius but not on gear width as proposed by Jeon.

Figure 2.25: Comparison between measurements and Jeon theoretical formula

Therefore, a modification of the coefficient 𝛾 1 is needed. A comparison is thus made between a test performed by Jeon and the test presented on Figure 2.25 in order to check experimental coherence. An identical immersion ratio of 0.6 and the same rotational speed of 1000 rpm are selected. A good agreement between Jeon data and measurements is shown on Figure 2.26. This allows the definition of a corrected 𝛾 1 .

Finally, two different flow regimes are defined to calculate Cm for the truck application:

for Re ≤ 20 000 : The newly estimated churning losses are higher than the ones calculated according to Jeon formula. However, churning losses are still underestimated. It is important to highlight that calculated churning losses of pinion gear are low compared to crown gear ones. Indeed, its immersion surface is smaller.

𝐶 𝑚 = 1.
As explained in the first chapter, a meshed hypoid gear pair generates more losses than the addition of each gear contribution. According to Jeon [START_REF] Jeon | Improving Efficiency in Drive Lines: an Experimental Study on Churning Losses in Hypoid Axle[END_REF], the churning losses of a complete gear set can be more than doubled when compared to that of crown gear alone. Figure 2.28 illustrates this assumption for the axle case. The new formula shows good agreement with test results.

Figure 2.28: Increase of the churning losses for a complete gear set

Jeon has observed a modification of the oil flow due to pinion presence. To take this phenomenon into account, the immersion level is modified for the churning loss calculation in order to obtain similar power loss amount. Finally, the immersion level is increased by 30% for the meshed gear pair.

Thanks to this new churning loss relationship, more coherent results are obtained with the no-load measurements, as presented on Figure 2.29.

Calculations of load-dependent losses

As load-independent losses are now well estimated, a new focus is made on load losses:  Rolling Element Bearing friction  Hypoid gear mesh friction Bearing friction losses were already studied in Part 2.3.2 with the bearing drag losses. Then, hypoid gear mesh friction is investigated hereafter according to analytical formulae of the literature, still with an isothermal system at 80°C. Figure 2.30 gives an order of hypoid gear mesh friction losses deduced from the previous calculations. This highlights a possible error on measurement B at normal oil level: the trend is really different between both oil levels.

Hypoid gear mesh friction

A first approach of hypoid losses is exposed here by comparing ISO/TR 14179-2 [START_REF]Gears -Thermal capacity Part 2 : Thermal load-carrying capacity[END_REF] and Buckingham [START_REF] Buckingham | Analytical Mechanics of Gears[END_REF] formulae. The friction coefficient is determined according to the different methods: for Buckingham, a chart allows to identify it according to sliding velocity, while it is calculated with a formula specified in Annex 1.A for the ISO technical report. For ISO formula, factor 𝑋 𝐿 = 0.8 as the oil used is a polyalfaolefin (other characteristics are reported in Table 2.3). Results are plotted on Figure 2.31 and compared with the gear friction loss deduced from measurements. It appears clearly that Buckingham formula greatly overestimates the losses. The ISO technical report formula is in good agreement with the deduced gear friction losses. This is coherent with the observation made in the previous chapter: for an identical friction coefficient, Buckingham formula is more pessimistic than the ISO. Moreover, the friction coefficient estimated according to Buckingham is higher (0.035 for spiral bevel part and 0.042 for worm part) than the ISO (from 0.020 to 0.026 depending on load).

Global measurements

Deduced gear mesh friction Figure 2.32 allows for a better comparison between ISO/TR 14179-2 and measurements. Gear mesh friction appears to be overestimated under high load and underestimated under low load. This can be due to a misevaluation of the hypoid gear mesh contact or of the friction coefficient. A more thorough study is conducted in the next chapter to address these issues.

Influence of oil level and temperature

As far as tests with an applied load are concerned, the global power loss of the axle is presented and compared with measurements on Figure 2.33 for two oil levels. The friction coefficient used for the calculation of the gear mesh losses is calculated according to ISO/TR 14179-2.

There is a good agreement between experimental data and calculations. The major power loss source varies with operating conditions: gear mesh losses are predominant at high torque; churning losses have more importance at low torque. An increase of churning losses participation is observed between normal and high oil level (respectively Figure 2.33 (a) and (b)): oil churning contribution varies with the rise of gears immersed area. Seal losses are negligible as forecasted, since the total amount of power losses is in kW.

It can be noticed again that the gear friction power loss model has some limits. The friction of hypoid gears is very specific and so a dedicated friction model must be used and validated with experiments in order to calculate accurately gear mesh losses. Both kinematic and friction aspects are addressed in the next chapter. Nevertheless, the power loss estimation presented here is satisfactory. Moreover, the calculation is noteworthy simpler and faster than with a complete elastohydrodynamic lubrication model.

Figure 2.33: Comparison of load measurements with analytical calculations for (a) normal oil level and (b) high oil level

It can be underlined that Kakavas [START_REF] Kakavas | Hypoid gear vehicle axle efficiency[END_REF] presents a different power loss breakdown: rolling-element bearing losses (both drag and friction) are the major contribution. Nevertheless, he uses a urban drive cycle, the New European Drive Cycle (NEDC), which means that the input torque is moderate and the axle remains relatively cool, as suggested by Kolekar [START_REF] Kolekar | The efficiency of a hypoid axle -a thermally coupled lubrication model[END_REF]. Indeed, during the cycle, the oil sump temperature varies between 20 and 40°C, which is really lower than the 80°C measured on long haul truck axles. In order to quantify oil temperature influence on power losses split, some calculations were conducted. Figure 2.34 shows a higher loss contribution of bearings when lubricant temperature is lower, which is in agreement with Kakavas observations.

Figure 2.34: Calculation results for oil temperature at 80°C (A, B, C) and at 40°C (A', B', C')

Moreover, it can be noticed that gear friction coefficient would also be modified by temperature, as it changes for cylindrical gears [START_REF] Diab | Prediction of Power Losses Due to Tooth Friction in Gears[END_REF].

Conclusion

Truck axle efficiency was investigated through tests and calculations. Global power losses were measured for both loaded and unloaded conditions in order to evaluate load-dependent and loadindependent losses. Each source of dissipation was then estimated separately according to the different formulae presented in the first chapter. For shaft seal friction, the formula of Simrit is kept. For Rolling Element Bearings losses, it appears that SKF formula overestimates the losses under load; more coherent results are obtained with Harris formula, which is used subsequently. For hypoid gear mesh friction, the estimation according to the ISO/TR 14179-2 formula gives a good trend, but a more thorough study is required to improve it. For oil churning, both Mauz and Jeon solutions gives unrealistic results; investigations are then conducted on churning losses of spiral bevel and hypoid gears.

Experiments were performed on a specific test bench with various spiral bevel gears and cones. Different oils and operating conditions were also tested. Results highlight the influence of some operating conditions like speed, immersion ratio and oil viscosity, but also of a main gear geometrical REB losses (Harris) Seal (Simrit) Churning (new formula) parameter which is the outer diameter. By contrast, it appears that gear teeth have little influence on churning losses. Discrepancies are observed when comparing these test results with Mauz and Jeon formulae. An improvement of the existing formula given by Jeon is then proposed to take a better account of the identified influential parameters. This new model considers two different regimes depending on Reynolds number and also the greater contribution due to meshing gear pair.

Thanks to this new formula, power loss estimation for the truck axle case is in better agreement with test results.

Introduction

The second chapter has highlighted that gear mesh friction is the major power loss source of the axle for truck conditions. Nevertheless, kinematic and friction aspects in hypoid gears are particular and not well understood. Thus, it seems important to lead a more thorough investigation.

The axle gear pair is generally a hypoid one, because of contact ratio and durability aspects. But this design increases the sliding velocity at the contact and consequently the friction. According to the literature ( [START_REF] Winter | Measurements and Optimization of the Efficiency of Hypoid Axle Drives of Vehicles[END_REF], [START_REF] Takaki | Optimization of Hypoid Gear Design for High Efficiency Drives[END_REF]), this source of dissipation depends notably on the offset, which modifies also the spiral angle. The spiral bevel design is also considered, as it is a hypoid gear without offset.

In a first part, the specificities of spiral bevel and hypoid gears are exposed. Geometrical characteristics are induced by the gear cutting process and the basic gear parameters chosen (ratio, offset). Gear mesh contact and its characteristics are described. Gear surface kinematics and its influence on oil film formation are investigated.

In a second part, the power loss due to the gear mesh is estimated for a given value of friction coefficient and according to different approaches: a global one and a local one. Indeed, the ISO/TR 14179-2 proposes an estimation at a global level, considering the gear mesh as a whole. There are generally several teeth in contact at the same time, each with different loads and sliding speeds. A calculation at contact level should be more realistic. A comparison between both approaches is used to discuss validity of the global one.

Finally, friction coefficient was identified as a major parameter of gear mesh friction loss. Different friction coefficient formulae are then reviewed and potentially influential parameters are highlighted and analysed.

Specificities of spiral bevel and hypoid gears

Macro-geometrical aspects

To design a hypoid gear, basic parameters are defined: pitch diameter, spiral angle, offset, tooth proportions… Pinion offset is the most distinctive parameter of hypoid gears. A higher offset allows a bigger pinion and so increases load capacity. Nevertheless this also increases power losses due to higher sliding velocities. Höhn et al. [START_REF] Höhn | New methods for the calculation of the load capacity of bevel and hypoid gears[END_REF] also observed a faster and bigger growth of micro pitting area on tooth flanks when increasing offset.

Contrary to spiral bevel gears and due to offset, hypoid gears do not have a pitch diameter proportional to the number of teeth. Gear teeth are asymmetrical: pressure angle is different on concave and convex sides of the teeth. Moreover, spiral angle is different on pinion and crown gear. On spiral bevel gears, spiral angle is the same for both gear pair members.

Once main parameters are defined, a fabrication process must be chosen. The two main gear cutting methods for spiral bevel and hypoid gears are face-milling and face-hobbing:  Face-milling (FM) process cuts teeth one by one, as the cutter head rotates around a fixed centre. Created flanks are thus circular arcs.  Face-hobbing (FH) process cuts all the teeth in a same movement, as the cutter head and the generating gear rotate together. Created flanks are thus extended epicycloids.

These cutting processes are illustrated on Figure 3.1 and detailed by Kolivand [START_REF] Kolivand | Development of Tooth Contact and Mechanical Efficiency Models for Face-Milled and Face-Hobbed Hypoid and Spiral Bevel Gears[END_REF]. This also have an impact on tooth height: a face-milled gear tooth has a higher heel than the toe, while a face-hobbed one has a heel and a toe of same height as represented on Figure 3.2.

Figure 3.2: (a) Face-milled and (b) face-hobbed gear tooth

On another level, because of Formate® and Generate processes, tooth profile and cutting grooves are not the same, as represented on Figure 3.3 and Figure 3.4. Indeed, gear blank is fixed (FM) or rotates with cutter head (FH) for Formate® process, while Generate process uses an additional rotation between gear blank and cradle axis. This then requires a more sophisticated machine. The pinion is always Generate to ensure a good tooth profile. The crown wheel is generally Formate® for production reasons, but sometimes also Generate. As for spur and helical gears, tooth corrections can also be applied to hypoid gears: crowning on profile and/or on length, root and tip relief… Various ease-off are thus obtained.

The whole manufacturing process of axle hypoid gears is presented on

Generate process

Tooth profile forging and cutting. Some thermochemical treatments are then performed: gear is case-carburised, quenched and tempered. Depending on cutting process, different finishing operations are done: grinding or lapping. Grinding shapes slightly tooth surfaces and helps to remove heat-treatment distortion, tooth by tooth. Lapping polishes tooth surfaces thanks to an abrasive paste.

Some optional operations can be performed after this process, like shot-peening to improve bending resistance. A manganese phosphating coating is sometimes applied, but only on one gear pair member.

Phosphating is generally used for rusting problems and must prevent scuffing in early life of gear. According to Chen et al. [START_REF] Chen | Improvement of Contact Fatigue Strength of Gears by Tooth surface Modification Processing[END_REF], this also improves pitting fatigue life when applied on only one gear of the pair due to the creation of oil reservoirs in tooth surface.

Figure 3.5: Typical manufacturing process for axle hypoid gears

It seems important to take into account all these specificities when designing a hypoid gear. Hypoid gear teeth are generally modelled thanks to Finite Elements for geometry and mechanical aspects ( [START_REF] Xu | A Model to Predict Friction Losses of Hypoid Gears[END_REF], [START_REF] Kolivand | Prediction of mechanical gear mesh efficiency of hypoid gear pairs[END_REF], [START_REF] Karagiannis | On the dynamics of lubricated hypoid gears[END_REF]- [START_REF] Astoul | New methodology to reduce the transmission error of the spiral bevel gears[END_REF]). This allows notably for performing a Tooth Contact Analysis (TCA). The analytical approach is more rarely adopted: an equivalent helical gear approximates the hypoid one ([50], [START_REF] Bahramighahnavieh | Investigation of the Hertzian Stress Distribution on the Surface of the Straight Bevel Gear[END_REF]). Moreover, Höhn et al. [START_REF] Höhn | New methods for the calculation of the load capacity of bevel and hypoid gears[END_REF] validated this approximation by comparison with a Finite Element analysis.

Figure 3.6: Finite Element model of a gear set using Barday software [22]

A meshing calculation software was developed and validated by Barday [22] at Renault VI (now part of Volvo Group). The gear surface geometry is generated according to a manufacturing process simulation. Finite Element models of one tooth of pinion and crown are then created as illustrated on This model was first developed for durability aspects. Through Transmission Error calculations, noise emission issue was also addressed [START_REF] Barday | Noise optimization of a rear axle gear set[END_REF]. It can indeed consider the ease-off topography or even a 3D measurement surface topography of an existing gear, allowing accurate estimations.

Contact aspects

A hypoid gear set is unique as pinion and crown are paired. To obtain a good contact pattern, the optimal mounting distance of gear set is checked thanks to the use of paint on teeth. Typical shape of this contact pattern is approximately a parallelogram, as illustrated on Figure 3.7. Höhn et al. [START_REF] Höhn | New methods for the calculation of the load capacity of bevel and hypoid gears[END_REF] noticed that the angles of this parallelogram depend on hypoid offset. For spiral bevel gears, this becomes a rectangle.

Figure 3.7: Representation of a typical contact pattern for a hypoid gear set

The contact pattern spreads with increasing load, as underlined on Figure 3.8.

Figure 3.8: Contact pattern spread is highlighted thanks to yellow paint for (a) low loaded case and (b) more loaded case

The hypoid gear mesh is theoretically a line contact. However, in operation, this is more an elliptical contact due to load and deformations. Many authors discretised the ellipse in several line contacts (Xu et al. [START_REF] Xu | A Model to Predict Friction Losses of Hypoid Gears[END_REF], Kolivand et al. [START_REF] Kolivand | Prediction of mechanical gear mesh efficiency of hypoid gear pairs[END_REF]). The elliptical contact is generally used to consider contact area and elastohydrodynamic lubrication (Karagiannis et al. Even if an elliptical contact is not considered through this method, an estimation of the ellipse can be done according to lengths 𝑑𝑙 and half-widths 𝑏 𝑐 . An example of contact ellipse is plotted in red dashes on Figure 3.9. Between two lines of contact, there are naturally different contact conditions. But it is also the case along a contact line itself: surface velocities, load and radius of curvature are varying and thus modifying contact width, oil film thickness and contact kinematics.

On another level, because of Generate and Formate® processes, the angle between cutting lines and contact lines can also change along contact path.

Kinematic aspects

To evaluate friction at gear mesh, sliding between surfaces in contact must be known. Classically, sliding velocity 𝑉 𝑠 , sum of surface velocities 𝑉 ∑ and entrainment velocity 𝑉 𝑒 (m/s) are defined as:

𝑉 𝑠 = 𝑉 1 -𝑉 2 (3.1) 𝑉 ∑ = 𝑉 1 + 𝑉 2 (3.2) 𝑉 𝑒 = 𝑉 ∑ 2 (3.3)
where 𝑉 1 and 𝑉 2 are respectively pinion and gear surface velocities (m/s).

Considering spiral bevel gears, pinion and crown surface velocities are identical in terms of value and direction on pitch cone only. Elsewhere, due to their conical shape and to spiral angle, their surface velocities differ in value and direction. This induces a velocity orientation angle 𝛿 between surface velocities and an entrainment orientation angle 𝛿 𝑒 between entrainment velocity and ellipse axis 𝑥, as represented on Figure 3.11. Sliding occurs only along ellipse minor axis, as for spur and helical gears. On hypoid gears, spiral angles of contacting surfaces are different between pinion and gear due to offset. This generates sliding in longitudinal direction to contact line as well. Figure 3.11 illustrates these various cases. Höhn et al. [START_REF] Höhn | New methods for the calculation of the load capacity of bevel and hypoid gears[END_REF] analysed that sliding transverse component along hypoid tooth profile is similar to helical gears. The difference in sliding conditions between hypoid and helical gears comes from the longitudinal component, which induces higher sliding velocities and more friction.

Figure 3.11: Examples of contact velocities for (a) spiral bevel gear and (b)-(c) hypoid gear

Buckingham [START_REF] Buckingham | Analytical Mechanics of Gears[END_REF] defines the pitch surfaces of hypoid gears as hyperboloids of revolution. Their contacting circles touch at pitch point but do not rotate in the same plane or in planes at right angles to each other, due to offset. This induces sliding even at pitch point.

Surface velocities vary along contact line, as represented on Figure 3.12: value and direction change. Kinematics inside contact ellipse of hypoid gears is then quite particular.

V 2 V 1 V e δe V s δ 𝑥 𝑦 V 2 V 1 V e δe V s δ 𝑥 𝑦 V 2 V 1 V e δe V s δ 𝑥 𝑦 (a) (b) (c)
Chapter 3: Hypoid gear mesh friction

Power loss estimation

Gear mesh appears to be the major power loss source in the truck axle case. In Chapter 2, this dissipation was finally evaluated according to ISO/TR 14179-2. These calculations are based on an equivalent helical gear, which is determined according to hypoid gear characteristics, as presented in Annex 1.A. This is an approximation and cannot take into account all the specificities of hypoid gears presented in the previous part.

Therefore, the validity of this friction power loss law is discussed. A local approach based on Barday software results is presented to estimate gear mesh losses. A comparison is then conducted between both approaches, with a fixed friction coefficient.

Local approach

From a purely mechanical point of view, gear mesh friction loss is due to tangent friction effort and sliding velocity between surfaces. Tangent friction effort is generated by the normal effort applied at contact and is in the direction opposite to sliding velocity.

As potential contact lines are discretised, each segment is represented by its centre 𝑀 𝑖𝑗 , with 𝑖 the contact line number and 𝑗 the segment number.

⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗

The power loss along a complete contact line 𝑖 (W) is:

𝑃 𝑖 = ∑ 𝑉 2 1 ⃗⃗⃗⃗ (𝑀 𝑖𝑗 ) • 𝑑𝑇 2 1 𝑖𝑗 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑗 (3.6)
Finally, the instantaneous power loss of the hypoid gear pair is calculated according to the simultaneous teeth pairs in contact.

Results obtained thanks to this method are exposed hereafter. Two different gear sets are considered: a hypoid one (gear set A) and a spiral bevel one (gear set B). They have the same ratio and their characteristics are listed in Table 3.1. For these examples, a fixed friction coefficient of 0.05 is used on each point: no local friction coefficient is calculated. First, contact conditions are studied on only one teeth pair contact among the simultaneous ones for a complete mesh cycle. For each successive contact line, total normal force and mean sliding velocity of the contact are calculated. Figure 3.16 presents these values for gear set A and Figure 3.17 for gear set B for various input torques.

As torque increases, normal force variations are smoother: the contact is more spread along mesh cycle and there is more overlap between teeth contact. Mean sliding velocity of the contact changes slightly as contact length increases with load. Indeed, contact shape is modified by load as illustrated on Figure 3.15.

The hypoid gear set A has a permanent sliding all along the gear mesh cycle (>1m/s), while it is close to pure rolling for spiral bevel gear set B around pitch point (<0.2m/s). This is due to hypoid offset.

Comparison between global and local approaches

The ISO/TR 14179-2 [START_REF]Gears -Thermal capacity Part 2 : Thermal load-carrying capacity[END_REF] is used in Chapter 2 for the calculation of gear mesh power loss at a global level. To discuss the validity of this approximation, a comparison is made here with the local approach presented in the previous section.

For this purpose, the two gear sets described in Table 3.1 are again considered. Gear mesh friction loss is estimated for each gear set at various input torques and speeds. As in the previous section, a fixed friction coefficient of 0.05 is used to compare both approaches. For local approach, the instantaneous power loss is averaged on a complete meshing and is considered as the gear mesh global power loss (Local approach #1). Results are plotted on

21: Comparison of power loss estimation according to local and global approaches for gear set B at an input speed of (a) 1000 rpm and (b) 2000 rpm

Both estimations give similar trends and results for each gear set: sliding velocity is directly proportionnal to input speed, so gear mesh power losses are almost doubled between the two speed conditions. Global approach always overestimates gear mesh power losses and it seems to be more pronounced for gear set B. Table 3.2 presents the difference between both estimations in terms of power loss. For gear set A, the difference is quite stable at roughly 18%. For gear set B, the difference is more important (over 30%) but decreases with higher torque. In order to compare both methods more precisely, mean values are calculated for the local approach (Local approach #2):

 Mean normal force: for each contact line, the total normal force is calculated and summed for the simultaneous teeth in contact; a mean over the complete meshing is then computed.  Mean sliding velocity: for each contact line, a mean sliding velocity is calculated and averaged for the simultaneous teeth in contact; a mean over the complete meshing is then computed.

A mean gear mesh power loss is finally calculated, as for global approach:

𝑃 ℎ𝑦𝑝𝑜𝑖𝑑 = 𝐹 𝑛 𝜈 𝑔𝑚 𝜇 𝑚 (3.7)
where 𝐹 𝑛 is the tooth normal load (N), 𝜈 𝑔𝑚 is the mean sliding speed (m/s), and 𝜇 𝑚 is the mean friction coefficient (here, still 0.05).

Difference between estimations of local approach #1 and #2 is less than 3%. The mean values calculated for local approach #2 are thus accurate and representative of local approach: they can be Concerning mean normal force, global approach underestimates it by maximum 2% and this gap decreases with increasing torque. The mismatch between local and global methods is thus mainly due to sliding velocity estimation. A comparison is made between these two sliding velocity estimations in Table 3.3 and shows an overestimation up to almost 50% on sliding speed at low input torque. Indeed, local approach considers sliding velocity only in contact zone, which depends on contact pattern and so on torque. This difference is particularly marked for gear set B, as its contact pattern is rather small in comparison to gear set A as presented on Figure 3.15. If sliding velocity is considered on complete contact lines, local mean sliding velocity is increased and difference drops under 5% for gear set B. Thus, with higher torque and wider contact pattern, local and global approaches should give identical results. However, this is an ideal case as contact pattern is rarely spread over the whole tooth. Global approach allows to obtain an approximation of gear mesh power loss, but will always give a worst case scenario, mainly due to sliding velocity overestimation.

In Chapter 2, it appears that ISO formula overestimates gear mesh friction under high load and underestimates it under low load. The same test cases are considered with local approach. Global friction coefficients are calculated according to ISO technical report: 0.020 for condition A, 0.023 for condition B and 0.026 for condition C. A better agreement is obtained with local approach. However for both methods, power loss is underestimated at low load. This can be due to friction coefficient misevaluation, as it is identical in each approach. Moreover, for local approach, it is possible to calculate a local friction coefficient. This is investigated in the next section.

Friction coefficient

Friction coefficient is a first order parameter for gear mesh power loss. It is mainly influenced by load, entrainment and sliding velocities, oil properties, and surface roughness. As seen in Chapter 2 with Buckingham formula, its calculation is very important for the accuracy of gear mesh friction estimation: overestimation of power loss is directly linked to a high friction coefficient. Several friction coefficient formulae exist and depend on one to multiple parameters among the ones listed previously. They are reviewed in the next part. Results then vary from a formula to another. According to Buckingham [START_REF] Buckingham | Analytical Mechanics of Gears[END_REF], friction coefficient varies from 0.03 to 0.07 for a spiral bevel gear and from 0.03 to 0.2 for a hypoid gear, while according to Coleman [START_REF] Coleman | Effect of mounting displacements on bevel and hypoid gear tooth strength[END_REF], it ranges from 0.01 to 0.12 for hypoid gears in general.

Parameters influencing friction coefficient are characterising the elastohydrodynamic (EHD) contact between gear teeth. This contact is defined by the presence of a lubricant film and by the nonconforming surface geometry which induces a small contact area and high contact pressures. -Boundary lubrication: the two surfaces are in contact due to asperities and friction coefficient is influenced by additive properties; -Full film lubrication: lubricant film separates completely the two surfaces and friction coefficient depends on lubricant properties only; -Mixed lubrication: friction happens through asperity contacts and lubricant film shearing at the same time.

This last regime is the most common in gear mesh applications due to gear surface roughness and high sliding encountered notably at meshing beginning and end.

Different friction coefficient formulae are reviewed just hereafter to understand how these parameters can be taken into account. Some parameters are then analysed regarding hypoid gears to lead to potential efficiency improvements.

Friction coefficient formulae

Previously in this chapter, friction coefficient formula proposed in ISO technical report 14179-2 [START_REF]Gears -Thermal capacity Part 2 : Thermal load-carrying capacity[END_REF] was used. As already mentioned, this is a mean friction coefficient:

𝜇 𝑚 = 0.048 ( 𝐹 𝑏 𝑡 ⁄ 𝑣 Σ 𝑅 𝑒𝑞 ) 0.2 𝜂 𝑜𝑖𝑙 -0.05 𝑅𝑎 0.25 𝑋 𝐿 (3.8)
where 𝐹 is the tooth normal force (N), 𝑏 𝑡 is the tooth width (mm), 𝑣 Σ is the resultant sum velocity (m/s), 𝑅 𝑒𝑞 is the equivalent radius of curvature (mm), 𝜂 𝑜𝑖𝑙 is the lubricant dynamic viscosity at operating temperature (mPa.s), 𝑅𝑎 is the arithmetic average roughness (μm) and 𝑋 𝐿 is a lubricant factor. Some parameters are determined differently depending on gear type, as equivalent gears are used for calculation of complex gear pairs. Details are given in Annex 3.B.

Despites good results shown at the end of previous section, this formula does not allow to consider local conditions along contact lines and gear mesh. It is then interesting to review local approaches for friction coefficient calculation. Concerning lubricated contacts, most of friction coefficient formulae are empirical. They are based on data from experiments conducted on EHD simulators like twin-disc machines that reproduce gear teeth contact conditions. A formula frequently used is the one of Benedict and Kelley [START_REF] Benedict | Instantaneous Coefficients of Gear Tooth Friction[END_REF]:

𝜇 = 0.
(3.9)
where 𝑊 𝑙𝑏 is the load by unit of length (lb/in), 𝜂 𝑜𝑖𝑙 is the lubricant dynamic viscosity (mPa.s), 𝑉 𝑠 𝑖𝑛 is the sliding velocity (in/s) and 𝑉 ∑ 𝑖𝑛 is the sum of surface velocities (in/s). This formula displays several important parameters influencing friction coefficient, but surface roughness is not considered. Moreover, it appears that this formula is valid only for high slide-to-roll ratio (20% to 50%) as the experiments were conducted for these conditions. Non negligible discrepancies are then obtained out of these boundaries [START_REF] Diab | Prediction of Power Losses Due to Tooth Friction in Gears[END_REF].

Diab et al. [START_REF] Diab | Prediction of Power Losses Due to Tooth Friction in Gears[END_REF] developed a formula based on Greenwood-Tripp theory (1971) completed by Mikic (1974) and on many traction curves:

𝜇 = 1 𝑝 𝑚𝑜𝑦 [ 𝐴 𝑐 𝐴 0 • 𝜏 𝑑𝑟𝑦 + (1 - 𝐴 𝑐 𝐴 0 ) • 𝜏 𝑓𝑙𝑢𝑖𝑑 ] (3.10)
where 𝑝 𝑚𝑜𝑦 is the mean contact pressure (Pa), 𝐴 𝑐 and 𝐴 0 are respectively the actual average contact area and the apparent contact area (m²), 𝜏 𝑑𝑟𝑦 and 𝜏 𝑓𝑙𝑢𝑖𝑑 are respectively the shear stress due to dry friction and to fluid friction (Pa). Here, roughness is considered through contact areas. The contribution of asperities contact and of fluid shearing are considered separately. Details are given in Annex 3.B.

Xu and Kahraman [START_REF] Xu | Prediction of friction-related power losses of hypoid gear pairs[END_REF] established an accurate elastohydrodynamic lubrication (EHL) analysis. A multiple linear regression of a large number of these analyses was done to obtain a formula that can be used directly:

𝜇 = 𝑒 𝑓(𝑆𝑅,𝑃 ℎ ,𝜂 𝑜𝑖𝑙 ,𝑅𝑎) 𝑃 ℎ 𝑏 2 |𝑆𝑅| 𝑏 3 𝑉 𝑒 𝑏 6 𝜂 𝑜𝑖𝑙 𝑏 7 𝑅 𝑏 8 (3.11)
where 𝑆𝑅 is the Slide-to-Roll Ratio, 𝑃 ℎ is the maximum Hertzian pressure (GPa), 𝜂 𝑜𝑖𝑙 is the dynamic viscosity at oil inlet (mPa.s), 𝑅𝑎 is the surface roughness (μm), 𝑉 𝑒 is the entraining velocity (m/s) and 𝑅 is the effective radius of curvature (m). Details are given in Annex 3.B. This formula takes into account many important parameters influencing friction coefficient and has the advantage to be quite simple to use.

Xu [START_REF] Xu | Development of a Generalized Mechanical Efficiency Prediction Methodology for Gear Pairs[END_REF], and after him Kolivand et al. [START_REF] Kolivand | Prediction of mechanical gear mesh efficiency of hypoid gear pairs[END_REF], used only transverse sliding for friction coefficient calculation, arguing that longitudinal sliding is multiple times smaller than transverse sliding. However, as illustrated on Figure 3.12, it may not be always the case: longitudinal sliding is sometime higher than transverse one. For this reason, it seems more reliable to consider the whole sliding. Moreover, it has not been clarified if longitudinal sliding has a particular influence on friction coefficient or not.

Influence of parameters on friction coefficient

A recurrent parameter in friction coefficient formulae is the surface roughness. In the section 2 of this chapter, it was shown that hypoid gears surface roughness is due to specific gear cutting, surface treatment and finishing. A second particularity of hypoid gears are the orientation angles. As this influences entrainment and sliding velocity, it seems worth to study.

These two potentially influential parameters are then discussed here [START_REF] Fossier | Investigations on truck axle efficiency: thermal behaviour and hypoid gears parameters influence[END_REF].

Surface roughness

As underlined by Diab et al. [START_REF] Diab | Prediction of Power Losses Due to Tooth Friction in Gears[END_REF], surface roughness is important in two ways:

-the overall roughness of surface, which is characterised by the arithmetic average roughness 𝑅𝑎 and by the profile maximum height 𝑅𝑧 and depends mainly on surface finishing; -the grooves orientation in surface roughness, which are due to surface machining during cutting and finishing process. The effect of the overall roughness is illustrated on Figure 3.24: friction coefficient is doubled if two rough surfaces are used instead of two smooth surfaces.

Concerning overall roughness, surface finishing of hypoid gears can be made with different methods, as explained in section 2.1. Different roughness are then obtained, as presented in Table 3.4. The reference is a gear surface after running-in. According to Stadtfeld [START_REF] Stadtfeld | Less Energy Consumption with High-Efficiency Bevel Gears and their Usage in the U.S[END_REF], machining grooves are first reduced during running-in phase and roughness is decreased too (𝑅𝑎 < 0.5 µm). It is then advantageous to proceed to a fine grinding after running-in in order to obtain quickly a slightly better surface finish. For each local formula, friction coefficient is calculated based on data used for the local approach presented in section 3.1 and is then averaged over meshing. Operating conditions are: gear set A, input torque of 1120 N.m, input speed of 1000 rpm and lubricant temperature of 80°C. Results are presented in Figure 3.25.

Figure 3.25: Mean friction coefficient according to several formulae for different surface roughness

As it does not take into account surface roughness, Benedict & Kelley formula gives always the same result. Moreover, it seems overestimated regarding other values. This suggests that tests were done on test samples with a high surface roughness. Diab [START_REF] Diab | Prediction of Power Losses Due to Tooth Friction in Gears[END_REF] also underlined that grooves orientation of their test samples were directed differently from reality, which could lead to wrong results.

Concerning other formulae, friction coefficient is divided at least by two between condition (a) and (d), where surface roughness is much decreased. This is in line with Diab observations. However, some discrepancies are observed between methods: on one hand, Xu and Diab formulae predict a lower friction coefficient when compared to ISO; on the other hand, Diab and ISO formulae have a similar behaviour, with a steep decrease of friction coefficient at low roughness.

According to ISO formula, gear mesh power losses are proportional to the friction coefficient. Then, it does not vary linearly with surface roughness. Friction formula from ISO is used as a reference to give an example here. An improvement from surface roughness condition (a) to condition (b) (Ra varies from 0.8 µm to 0.3 µm) decreases gear friction losses by roughly 20%. An improvement from surface roughness condition (b) to condition (d) (Ra varies from 0.3 µm to 0.05 µm) decreases it by more than 30%. However, the improvement estimation is slightly different from a formula to another.

Arithmetic average roughness gives an approximation of surface roughness influence on friction coefficient. However, grooves orientation possibly has some impact as well. For spur gears, grooves are perpendicular to the rolling direction. In the case of spiral bevel and hypoid gears, contact lines are a inclined regarding tooth tip and foot, while grooves orientation depends on cutting method, as presented in sections 2.1 and 2.2. An angle between cutting and contact lines exists and can also change along contact path.

Footprints were taken on crown gear and pinion in order to study their surface roughness on a 3D optical profilometer. Figure 3.27 gives a global view of roughness topography: parallel striations are visible on both specimens but appears more clearly on crown gear. On pinion tooth surface, it is interesting to note the presence of many remaining cavities, due to phosphating.

In order to characterise these grooves, profiles of surface roughness along tooth height are presented on Figure 3.26. A periodicity of roughly 500 µm is observed on both specimens and grooves depth is mainly between 1 and 3 µm. The resultant 𝑅𝑎 is around 0.3 µm for crown and pinion, but 𝑅𝑧 is 2.4 µm for crown and 1.6 µm for pinion.

Thanks to surface texture analysis, it appears that pinion and crown gear tooth surfaces are different.

As surface roughness influences friction coefficient and so power losses, it is important to take it into account. Friction coefficient formulae are generally limited to one roughness parameter which is not always representative of surface texture. Some tests must be done to characterise the friction of these types of surface, like the ones conducted by Diab [START_REF] Diab | Prediction of Power Losses Due to Tooth Friction in Gears[END_REF]. This phenomena was already reported in the literature. Many studies were conducted to investigate the effect of 𝛿 𝑒 on elliptical contacts. However, an elastohydrodynamic lubrication (EHL) approach is generally used: contact lubrication and estimation of lubricant film thickness are the main point of investigation. Chittenden et al. [START_REF] Chittenden | A Theoretical Analysis of The Isothermal Elastohydrodynamic Lubrication of Concentrated Contacts II General Case, with Lubricant Entrainment along either Principal Axis of the Hertzian Contact Ellipse or at some Intermediate Angle[END_REF] (1985) established empirical equations to determine minimum and central film thickness in elliptical contacts with entrainment angle. They observed a decrease in film thickness minimum when 𝛿 𝑒 increases, while central film thickness can stay constant or decrease. Omasta et al. [START_REF] Omasta | Effect of surface velocity directions on elastohydrodynamic film shape[END_REF] (2012) investigated experimentally the impact of the velocity orientation angle 𝛿 and of the angle between entrainment and sliding velocities in circular EHL contacts. They observed variations in terms of film thickness distribution depending on angles: wedge, asymmetry, dimple formation... It appears also that sliding orientation has an influence. Stahl et al. [START_REF] Stahl | Theoretical and Experimental investigations on EHL point contacts with different entrainment velocity directions[END_REF] (2013) developed a steady-state isothermal EHL model and validated it with experimental results for circular contacts on a twin-disc machine with an orientable disc (0°< 𝛿 < 60°). A comparison is then held for elliptical contacts with a roller-on-disc tribometer on which roller can be oriented (0°< 𝛿 < 45°). A good agreement is observed between simulated and measured results and major conclusion is: sum velocity (so entrainment velocity as well) magnitude and orientation is the main parameter for EHL film formation. Finally, Pu et al. [START_REF] Pu | A Theoretical Analysis of the Mixed Elastohydrodynamic Lubrication in Elliptical Contacts With an Arbitrary Entrainment Angle[END_REF] developed a mixed EHL model to investigate the impact of 𝛿 𝑒 on pressure and film thickness in elliptical contacts. This allows to extend the observation of Chittenden et al. to moderate and heavy loading cases: minimum film thickness also decreases with increasing 𝛿 𝑒 , but central film thickness decreases and then increases again. A great difference is also noticed between both film thicknesses.

All these studies focus only on lubricant film thickness for elastohydrodynamic perspectives. Film thickness reduction could be due to heat generated by a higher shear rate and then a higher friction. However, no study address the friction coefficient issue for this type of contact. Carli et al. [START_REF] Carli | Thermal point contact EHL analysis of rolling/sliding contacts with experimental comparison showing anomalous film shapes[END_REF] did not work on contacts with orientation angles but on contact with anomalous film shapes. They proposed a friction coefficient evaluation in the EHL analysis and compared it with experimental results thanks to friction force measurements. Newtonian model overestimates the friction coefficient but a good agreement is obtained with Eyring model using an Eyring shear pressure 𝜏 0 between 3 and 5 MPa. As unusual film shapes are obtain when entrainment speed is inclined, this model could be worth of interest.

Figure 3.28: Flange design for spherical roller thrust bearing and schematic representation of the spin kinematics [START_REF] Doki-Thonon | Thermal effects in elastohydrodynamic spinning circular contacts[END_REF] This phenomena can also be compared to the one observed in rolling element bearings supporting axial load, like spherical roller thrust bearings. As represented on Figure 3.28, a particular contact occurs between rib roller-end and flange: a spinning circular contact. Doki-Thonon [START_REF] Doki-Thonon | Thermal effects in elastohydrodynamic spinning circular contacts[END_REF] addressed this problematic of spinning, which is different from rolling-sliding kinematics and is due to rotational components of velocity fields. As in studies presented just before, asymmetry and reduction of film thickness were observed. Doki-Thonon also investigated experimentally the friction coefficient and split it into two components: along axis 𝑥, friction coefficient is similar but lower than for classical rolling-sliding contact; along axis 𝑦, it is almost null (<0.01).

This may be close to the hypoid kinematics observed on the global ellipse in section 2.3: a rotation of the ellipse entrainment velocity exists. However, it is hard to validate the similitude.

Thanks to the local approach of section 3.1, the two orientation angles 𝛿 and 𝛿 𝑒 are estimated for gear set A (hypoid) and gear set B (spiral bevel). Their values are presented in Table 3.5. The angle between surface velocities decrease with increasing speed, while entrainment angle is more stable. Experiments are done at two different positions: the normal one (δ=0°) and a rotated one (δ=10°). For both positions, various conditions are tested on roller surface velocity (0.5m/s, 1m/s) and on load (15N, 20N). Measurements are presented on Figure 3.30. For each condition, traction coefficient is higher for 𝛿 = 10°. For an identical Slide to Roll ratio, this phenomena is more pronounced when roller surface velocity is low (U1=0.5m/s): friction coefficient is almost doubled. On the contrary, load has the same influence whatever the angle. As friction coefficient is directly linked to gear mesh power losses, the modification of the orientation angle can have a great influence on power losses.

Conclusion

The specificities of spiral bevel and hypoid gears were presented: geometrical parameters stem from design choices, cutting process, thermomechanical treatment and surface finishing. A model developed by Barday [22] allows to take most of these parameters into account. Considering gear mesh, gears are paired in order to have a well spread and centred pattern. This pattern results from the successive elliptical contacts occurring along contact path. For modelling purposes, each successive contact is discretised in several line contacts: contact conditions (load, velocities, surface curvature…) are calculated thanks to Barday software. The kinematic of these gears is also particular. Due to conical shape and spiral angle, surface velocities are not oriented along axis of contact ellipse, neither are they aligned together: a velocity entrainment angle 𝛿 defines the angle between surface velocities; an entrainment angle 𝛿 𝑒 defines the angle between entrainment velocity and 𝑥-axis. Moreover, for hypoid gears, due to offset, sliding is not only along 𝑥-axis, a longitudinal component also exists. Finally, considering contact ellipse from a global perspective, it appears that a rotation occurs due to a variation of entrainment velocity along 𝑦-axis.

Power loss estimation of gear mesh is then investigated. A local approach based on data from Barday software is used to analyse contact conditions (load, sliding velocity) and power loss along contact path. Two gear sets are compared: a hypoid and a spiral bevel one. A decrease of sliding velocity is observed on spiral bevel gear around pitch point, while a permanent sliding is observed on the hypoid gear due to offset. As hypoid gears are special, a comparison is made between ISO global approach and local approach to discuss validity of ISO approximation. Global approach appears to always overestimate the power loss, due to sliding velocity estimation: indeed, sliding velocity varies a lot along contact line and local approach considers the sliding only on the part of the line really in contact. Thus, for higher loads, a better agreement is expected. In our case, ISO technical report estimation obtains more coherent results for the hypoid gear than the spiral bevel one. Finally, local approach is more accurate to evaluate gear mesh power loss. However, friction coefficient estimation is still a point of improvement and can be calculated locally.

For comparison purposes, the same friction coefficient was used for global and local approaches. However, local approach offers the possibility to use local friction coefficients, which can be more accurate. As it has a great influence on gear mesh friction loss, the friction coefficient estimation is reviewed through different formulae. Similar parameters are used in most of these equations: load, entrainment and sliding velocities, oil viscosity, and surface roughness. In the case of hypoid gears, two specific parameters were selected: surface roughness and velocities that are influenced by orientation angles. Surface roughness depends mainly on cutting process and on surface finishing. It is characterised by an overall roughness (defined by 𝑅𝑎 and 𝑅𝑧) and by machining grooves orientation. The influence of overall roughness is studied thanks to the friction coefficient formulae that were presented. A decrease of friction coefficient, and so of gear mesh power losses, is observed for smoother surfaces. Concerning grooves orientation, surface roughness of a hypoid gear pair was analysed to obtain more detailed on actual contact conditions. Friction tests must be performed to characterise friction coefficient for these particular surfaces. Last, orientation angles influence was considered. Literature only focused on elastohydrodynamic aspects with film thickness analysis. The closest phenomena is the one observed at the contact between rib roller-end and flange in some rolling element bearings. The magnitude of these angles were evaluated for a hypoid gear and some preliminary tests were performed to evaluate the influence of 𝛿 on friction coefficient. For 𝛿=10°, friction coefficient can be doubled for low surface velocity. This induces a great increase of gear mesh power losses too.

Introduction

Previous chapters were dedicated to the estimation of power losses, especially of hypoid gear mesh ones. The different dissipation sources generate heat at distinct points of the system, inducing local hot spots. This can influence the efficiency due to oil properties that change with temperature and can even lead to cooling and durability issues. Bearing and gear failures can happen, like scoring, pitting, or even scuffing.

In order to correctly predict these phenomena, it is important to know the real operating temperature of each component, which is generally different from oil sump temperature. However, most of the thermal investigations on axles were done through isothermal models: the whole axle is considered at oil sump temperature. Even for efficiency applications, this has some limitations: correct estimations of global power loss and oil temperature do not validate power loss breakdown between sources. Thus, a thermo-mechanical model is developed here to obtain both efficiency and temperatures. It is then necessary to understand more precisely the thermal exchanges inside the axle and with its surrounding environment.

At first, two approaches of the thermal model are presented: an isothermal one and a thermal network one. A comparison is then conducted according to power loss measurements and calculations exposed in Chapter 2 and 3.

The literature showed that the axle is not isothermal [START_REF] Xu | Thermal Mapping of an Automotive Rear Drive Axle[END_REF] and thermal network method is an appropriate tool to calculate the different bulk temperatures. An ISO technical report computes gear bulk temperatures in order to prevent durability issues. Both methods are discussed.

A test campaign is made to investigate the thermal behaviour of the axle. After a description of experiments and their results, a comparison with thermal network is done. This finally allows for exploitation of the thermo-mechanical model: the influence of parameters is investigated.

Thermal model of the axle

A thermal modelling of the axle presented in Chapter 2 is developed. An isothermal approach is first used and is compared with the thermal network approach for a stabilised condition. It is important to notice that the formulations described in Chapter 2 and Chapter 3 are used to calculate the power losses for both thermal approaches: seal friction with Simrit, rolling element bearing losses with Harris, churning losses according to the new formula of Chapter 2 and gear mesh friction according to local approach of Chapter 3 with ISO/TR 14179-2 for friction coefficient calculation.

Additional data for axle thermal behaviour

There is no cooling system, apart from the air flow due to ventilation, and no foundation. There is only suspension brackets and shaft connections which are far from axle drive head. Thus, heat conduction through input and output shafts is neglected. Therefore, it is assumed that only the housing is dissipating the heat generated by the axle. The axle housing is made of cast iron and weighs roughly 300 kg. It is modelled as a rectangular parallelepiped. The real housing shape is more complex, as represented on Figure 4.1.

To have a good representativeness, the external surface is evaluated according to the 3D-model. Dimensions of the parallelepiped represented on Figure 4.2 are then selected to have the closest approximation depending on surfaces. 4.1. Despite the parallelepiped approximation cannot reflect the bigger surface of drive head housing when compared to oil sump housing, total surface presents a good agreement.

Isothermal approach

The ISO technical report [START_REF]Gears -Thermal capacity Part 2 : Thermal load-carrying capacity[END_REF] proposes a simple heat balance with the environment for a steady state case. Details of this method are given in Annex 4.A. The whole axle is considered as a single unit at oil sump temperature. Knowing the global power loss of the axle and the environment conditions, oil temperature is calculated according to ISO/TR 14179-2. Temperature difference between oil and ambient air is defined as:

(𝑇 𝑜𝑖𝑙 -𝑇 𝑎𝑖𝑟 ) = 1 𝑈 𝐴 𝑐𝑎 𝑄 (4.1)
where 𝑈 is the heat transmission coefficient (W/(m²K)), 𝐴 𝑐𝑎 is the external housing surface (m²), and 𝑄 is the heat generated by the axle (W). 𝑈 is an overall coefficient which includes convection with the

𝐿 ℎ𝑜𝑢𝑠𝑖𝑛𝑔 𝑤 ℎ𝑜𝑢𝑠𝑖𝑛𝑔 ℎ ℎ𝑜𝑢𝑠𝑖𝑛𝑔

Drive head housing

Oil sump housing Lateral housing oil, and convection and radiation with the surrounding environment. As suggested in the technical report, the heat conduction through the housing can be neglected for metallic housings.

1 𝑈 = 1 𝛼 𝑜𝑖𝑙 𝐴 𝑐𝑎 𝐴 𝑜𝑖𝑙 + 1 𝛼 𝑐𝑜𝑛𝑣 + 𝛼 𝑟𝑎𝑑 (4.2)
where 𝛼 𝑜𝑖𝑙 , 𝛼 𝑐𝑜𝑛 and 𝛼 𝑟𝑎𝑑 are respectively the heat transfer coefficient due to oil, convection with air and radiation (W/(m²K)), and 𝐴 𝑜𝑖𝑙 is the internal housing area (m²).

𝛼 𝑜𝑖𝑙 = 200 W/(m²K) (4.3)
𝛼 𝑟𝑎𝑑 = 0.23 × 10 -6 𝜀 ( 𝑇 𝑤𝑎𝑙𝑙 + 𝑇 𝑎𝑖𝑟

where 𝜀 is the housing emissivity and 𝑇 𝑤𝑎𝑙𝑙 is the housing wall temperature (K). In the case of forced convection with air, the heat transfer coefficient of convection becomes:

𝛼 𝑐𝑜𝑛𝑣 = ( 0.0086 (𝑅𝑒 ′ ) 0.64 𝑙 𝑥 ) 𝐴 𝑎𝑖𝑟 𝐴 𝑐𝑎 𝜂 * (4.5)
where 𝑅𝑒′ is a modified Reynolds number, 𝑙 𝑥 is the flow length (m), 𝐴 𝑎𝑖𝑟 is the ventilated housing area (m²) and 𝜂 * is a temperature ratio.

A steady state case with running axle and constant air flow is considered. Convection with ambient air is then forced. Operating conditions reproduce efficiency tests presented in Chapter 2. Air flow velocity for each condition is given in Table 4.2. Thermal exchanges are overestimated, as already reported by Bendzulla et al. [START_REF] Bendzulla | New analysis on the heat balance of industrial gearboxes -Optimized calculation-method of a gearbox manufacturer[END_REF]. Figure 4.4 shows that the value of the overall coefficient 𝑈 is conditioned by convection and radiation exchanges with the air.

Table 4.2: Air flow velocity during efficiency tests

Oil level Normal

Figure 4.4: Heat transfer contributions for the isothermal approach

The assumption of a one part housing is not representative of the real conditions. Indeed, the air flow is not the same on every sides of the housing. This can induce less convection with the air and should be taken into account.

Thermal network approach

A more detailed model of the axle is developed to obtain a better estimation of the thermal behaviour. Figure 4.5 presents the considered components for this model, which are the main parts, such as gears, bearings, shafts and seal. Only housing and oil are missing on this illustration. The differential is still considered unused and does not generate any power loss. However, its thermal inertia is taken into account.

This system is then modelled using the thermal network method, which is explained in detail in Annex 4.B.

Method principle

Thermal network method is based on an electrical analogy with Ohm's law:

Δ𝑇 = 𝑅 𝑡ℎ × 𝑄 (4.6)
where Δ𝑇 is the temperature difference (K), 𝑅 𝑡ℎ is the thermal resistance (K/W) and 𝑄 the heat flow (W). The heat generated by the various power losses is injected in the network at the corresponding node. The evolution of each node temperature is then estimated according to the first law of thermodynamics. The internal energy of an element of the system varies due to its own power dissipation and to heat exchanges with other elements of the system:

𝑀 𝑖 𝑐 𝑖 𝑑𝑇 𝑖 𝑑𝑡 = 𝑄 𝑖 + ∑ 𝑇 𝑗 -𝑇 𝑖 𝑅 𝑡ℎ (𝑖, 𝑗) 𝑛𝑏 𝑗=1 𝑎𝑛𝑑 𝑗≠𝑖 (4.7)
where 𝑀 𝑖 is the mass of element 𝑖 (kg), 𝑐 𝑖 is its specific heat (J/kg.K), 𝑇 𝑖 and 𝑇 𝑗 are the temperature of elements 𝑖 and 𝑗 (K), 𝑡 is the time (s), 𝑄 𝑖 is the heat flow injected at node 𝑖 (W), 𝑅 𝑡ℎ (𝑖, 𝑗) is the thermal resistance between elements 𝑖 and 𝑗 (K/W) and 𝑛𝑏 is the number of thermal network nodes. This transient calculation is performed according to the algorithm described on Figure 4.6. The numerical integration is performed with Adams predictor-corrector method [START_REF] Changenet | Modélisation du comportement thermique des transmissions par engrenages[END_REF].

It is noteworthy that local oil temperatures are used instead of a global oil temperature. Also, the only input data of the thermal network are its boundary conditions: air speed and temperature.

Thermal network of a truck axle

Concerning the truck axle model, considered components are listed in Table 4. If the vehicle is running, an air flow around the axle is considered, as illustrated on Figure 4.8. Forced convection occurs on drive head and lateral housings: a normal air flow cools the drive head housing and a tangential air flow cools the lateral housing. As it is on the opposite side, sump housing does not experienced the same air flow. However, it is not free convection either, as air goes around the housing. A tangential forced air flow is then also used on this side, as represented on If the vehicle is stopped, no air flow is taken into account (wind is neglected). Free air convection takes place on every parts of the housing.

On test bench, fans can be used to generate an air flow and to cool the axle. They are located on the axle sump side, contrary to the reality on truck where air flow comes from drive head side. However, the model can be easily modified to consider one case or the other: on If the vehicle is running, gears and rolling element bearings in the axle are rotating and oil is splashed. Forced convection between oil and lubricated components happens. Power losses are also generated depending on operating conditions.

If the vehicle is stopped, the axle is also not working. Free convection with oil occurs. ). Here is a sum-up of thermal exchanges considered:

1) Conduction happens between several elements in contact, like bearings with shafts or housing, gears with their shaft, etc.; 2) Convection and radiation occur between environment and housing, which is decomposed in a set of flat plates to consider the different air flows around it; 3) Convection between oil and running gears is considered as the gears are dip lubricated;

Convection between oil and housing also occurs; 4) Striction controls the exchange between gears and their meshing point. The contact temperature increases a lot but, as the contact zone is really small in comparison to gear size, the bulk temperature is then not much influenced.

A focus is made here on the case where the vehicle is running. Some thermal resistances are detailed hereafter to give a better understanding of the thermal behaviour of the system (for more details or for other thermal resistances, see Annex 4.B).

As the housing is decomposed in a set of flat plates, a global thermal resistance is calculated for each type of housing thermal exchange (convection with air or oil and radiation): where ℎ 𝑖 is the heat transmission coefficient of plate 𝑖 (W/(m²K)) and 𝐴 𝑖 is the surface of plate 𝑖 (m²).

𝑅 𝑡ℎ ℎ𝑜𝑢𝑠𝑖𝑛𝑔 = 1 ∑ℎ 𝑖 𝐴 𝑖 (4.8)

 Radiation and forced convection between housing and ambient air:

𝑅 𝑡ℎ [START_REF]Reducing CO2 emissions from Heavy-Duty Vehicles[END_REF][START_REF] Buckingham | Analytical Mechanics of Gears[END_REF], 𝑅 𝑡ℎ [START_REF]Reducing CO2 emissions from Heavy-Duty Vehicles[END_REF][START_REF] Coleman | Effect of mounting displacements on bevel and hypoid gear tooth strength[END_REF], 𝑅 𝑡ℎ [START_REF]Reducing CO2 emissions from Heavy-Duty Vehicles[END_REF][START_REF] Mohammadpour | Transmission efficiency and noise, vibration and harshness refinement of differential hypoid gear pairs[END_REF] The radiation coefficient is determined from the Stefan-Boltzmann law [START_REF] Holman | Heat transfer[END_REF]:

ℎ 𝑟𝑎𝑑 = 𝜖 𝜎 (𝑇 𝑐𝑎 2 + 𝑇 𝑎𝑖𝑟 2 )(𝑇 𝑐𝑎 + 𝑇 𝑎𝑖𝑟 ) (4.9)
where 𝜖 is the material emissivity, 𝜎 is the Stefan-Boltzmann constant (5.67 × 10 -8 W/(m²K 4 ) ), 𝑇 𝑐𝑎 and 𝑇 𝑎𝑖𝑟 are respectively housing and ambient air temperatures (K).

The forced convection coefficient is defined according to formulae of Winter et al. [START_REF] Winter | Wärmeabführung bei Getrieben unter Quasistationären Betriebsbedingungen. Teil II : Untersuchungen zur Wärmeabführung über Stahlfundamente und Übertragung der Prüfstandsergebnisse auf die Praxis[END_REF]. For this thermal exchange, the distinction is made between surfaces normal and tangential to the air flow. For surfaces normal to the flow:

ℎ 𝑐𝑜𝑛𝑣 = 5.6 • 𝐿 -0.34 • 𝑉 0.66 (4.10)
For surfaces tangential to the flow:

ℎ 𝑐𝑜𝑛𝑣 = 7.6 • 𝐿 -0.37 • 𝑉 0.63 (4.11)
where 𝐿 is the characteristic length (m) and 𝑉 is the air flow velocity (m/s). For the particular case of forced convection with a surface normal to the flow, the characteristic length is: 𝐿 = 𝐴 𝑃 ⁄ with 𝐴 the normal surface area (m²) and 𝑃 its perimeter (m).

 Forced convection between oil and housing: 𝑅 𝑡ℎ [START_REF] Kolekar | The efficiency of a hypoid axle -a thermally coupled lubrication model[END_REF][START_REF] Buckingham | Analytical Mechanics of Gears[END_REF], 𝑅 𝑡ℎ [START_REF] Kolekar | The efficiency of a hypoid axle -a thermally coupled lubrication model[END_REF][START_REF] Coleman | Effect of mounting displacements on bevel and hypoid gear tooth strength[END_REF], 𝑅 𝑡ℎ [START_REF] Kolekar | The efficiency of a hypoid axle -a thermally coupled lubrication model[END_REF][START_REF] Mohammadpour | Transmission efficiency and noise, vibration and harshness refinement of differential hypoid gear pairs[END_REF] For convection with oil, Holman [START_REF] Holman | Heat transfer[END_REF] defines the different heat transfers with:

ℎ 𝑜𝑖𝑙 = 𝑁𝑢 𝑘 𝐿 (4.12)
where 𝑁𝑢 is the Nusselt number, 𝑘 is the lubricant thermal conductivity (W/m.K) and 𝐿 is the characteristic dimension (m). For forced convection inside the oil bath, a flow is generated by running gears. Nusselt number is then:

𝑖𝑓 𝑅𝑒 < 5 • 10 5 , 𝑁𝑢 = 0.664 √𝑅𝑒 𝑃𝑟 1 3 ⁄ (4.13) 𝑒𝑙𝑠𝑒 𝑁𝑢 = 𝑃𝑟 1 3 ⁄ (0.037 (𝑅𝑒 0.8 ) -850) (4.14)
where 𝑃𝑟 is the Prandtl number. For Reynolds calculation, the fluid velocity to take into account is due to running gears.

Oil is also projected on housing walls due to gears rotation. For this oil trickling along non-immersed surfaces, the oil displacement is due to gravity.

 Forced convection between oil and gears: 𝑅 𝑡ℎ [START_REF] Kolekar | The efficiency of a hypoid axle -a thermally coupled lubrication model[END_REF][START_REF] Skf Group | [END_REF], 𝑅 𝑡ℎ [START_REF] Kolekar | The efficiency of a hypoid axle -a thermally coupled lubrication model[END_REF][START_REF]Gears -Thermal capacity Part 1 : Rating gear drives with thermal equilibrium at 95°C sump temperature[END_REF] When gears are rotating, forced convection with oil occurs on gear flanks and teeth. Convection on gear flanks is considered as for housing. Convection with gear teeth takes place in a particular way: centrifugal projection of oil along gear teeth flanks evacuates the major amount of heat.

Oil projection is notably characterised by the angle of rotation where the gear projects oil. For dip lubricated gears, this angle 𝜃 𝑝𝑟𝑜𝑗 begins at gear teeth emergence from oil and ends at gear mesh point, as illustrated on Figure 4.9. Projection duration is also important and is defined as: t= 𝜃 𝑝𝑟𝑜𝑗 Ω ⁄ .

Figure 4.9: Definition of centrifugal projection angle

Blok [START_REF] Blok | Transmission de chaleur par projection centrifuge d'huile[END_REF] defines an evacuated heat quantity for a tooth face per width unit. This energy can be deduced from Figure 4.10, depending on a dimensionless number Ψ, defined as:

Ψ = ( 𝑅 𝑝 𝑎 𝜃 𝑝𝑟𝑜𝑗 2 𝜈 𝑙𝑢𝑏 𝐻 𝑡𝑜𝑜𝑡ℎ ) 1 4 ⁄ (4.15)
where 𝑅 𝑝 is the gear pitch radius (m), 𝑎 is the thermal diffusivity of oil (m²/s), 𝜃 𝑝𝑟𝑜𝑗 is the centrifugal projection angle (rad), 𝜈 𝑙𝑢𝑏 is the kinematic viscosity of oil (m²/s) and 𝐻 𝑡𝑜𝑜𝑡ℎ is the tooth height (m).

For the complete gear, a global heat flow is then defined as:

𝑄 = 𝑞 𝑏 𝑡𝑜𝑜𝑡ℎ × 2 𝑍 2 𝜋 Ω ⁄ (4.16)
where 𝑞 is the evacuated heat quantity for a tooth face per width unit (W), 𝑏 is the tooth width (m), 𝑍 is the gear teeth number and Ω is the gear rotational speed (rad/s).

As 𝑄 = Δ𝑇 𝑅 𝑝𝑟𝑜𝑗 ⁄ , an approximation of the curve presented on where 𝐻 𝑡𝑜𝑜𝑡ℎ is the tooth height (m), 𝜒 is the thermal effusivity of oil ( 𝜒 = √𝑘 𝜌 𝑐 N/m.s 1/2 .°C) and Δ𝑇 is the temperature difference between pinion and oil (K).  Striction: 𝑅 𝑡ℎ [START_REF] Jeon | Improving Efficiency in Drive Lines: an Experimental Study on Churning Losses in Hypoid Axle[END_REF][START_REF] Skf Group | [END_REF], 𝑅 𝑡ℎ [START_REF] Jeon | Improving Efficiency in Drive Lines: an Experimental Study on Churning Losses in Hypoid Axle[END_REF][START_REF]Gears -Thermal capacity Part 1 : Rating gear drives with thermal equilibrium at 95°C sump temperature[END_REF] Striction controls the exchange between gears and their meshing point.

During meshing, the contact temperature increases a lot. However, the heat produced does not propagate deeply in gear tooth due to small contact zone and short contact duration. This has to be compared with gear size and time until the next meshing, which are really bigger. Gear bulk temperature is then not much influenced.

A distinction is thus made between gear pair and its meshing zone. Each gear has its own bulk temperature. Meshing zone is at "flash-temperature", as defined by Blok [START_REF] Blok | Transmission de chaleur par projection centrifuge d'huile[END_REF]. The thermal exchanges between meshing zone and each gear is driven by the thermal resistance defined as:

𝑅 𝑠𝑡𝑟𝑖𝑐𝑡𝑖𝑜𝑛 = 0.767 √2 𝑏 𝑐 𝑏 𝑡𝑜𝑜𝑡ℎ 𝜒 √𝑉 𝑟 𝑖 (4.19)
where 2 𝑏 𝑐 is the width of squeezing surface (m), 𝑏 𝑡𝑜𝑜𝑡ℎ is the tooth width or mean contact length if several teeth mesh simultaneously (m), 𝜒 is the thermal effusivity of material ( 𝜒 = √𝑘 𝜌 𝑐 N/m.s 1/2 .°C) and 𝑉 𝑟 𝑖 is the mean surface velocity of gear 𝑖 (m/s).

Comparison between ISO technical reports and thermal network model

Several ISO technical reports address thermal issues like ISO/TR 14179-2 [START_REF]Gears -Thermal capacity Part 2 : Thermal load-carrying capacity[END_REF] for heat dissipation of transmissions and ISO/TR 13989-2 [START_REF]Calculation of scuffing load capacity of cylindrical, bevel and hypoid gears -Part 2: Integral temperature method[END_REF] for bulk temperature of gears. Both use only oil temperature and do not consider the system specificities (air flow, local temperatures…). Advantages of the thermal network model are highlighted hereafter.

Estimation of thermal dissipation

The isothermal method considers a one part housing with a uniform heat exchange, while the thermal network method allows for using a housing divided in three parts with various heat exchanges depending on air flow around it. Moreover concerning heat dissipation in the environment, the influencing element is convection and radiation between housing and air, as seen for the isothermal approach. For radiation, the ISO technical report uses a formulation which is close to but not in agreement with the linearization of Stefan-Boltzmann law [START_REF] Holman | Heat transfer[END_REF]. For forced convection, the formulation proposed by ISO technical report (Equation (4.5)) is comparable to the one proposed by Winter et al. [START_REF] Winter | Wärmeabführung bei Getrieben unter Quasistationären Betriebsbedingungen. Teil II : Untersuchungen zur Wärmeabführung über Stahlfundamente und Übertragung der Prüfstandsergebnisse auf die Praxis[END_REF] (Equations (4.10) and (4.11)) , but with a higher coefficient, as shown below with the simplified form of Equation (4.20). This leads to an overestimated heat flow.

𝛼 𝑐𝑜𝑛𝑣 = 0.0086 𝑅𝑒

To compare both models, the same steady state case as the one used in Part 2.2 is studied.

Figure 4.11: Comparison of heat transfer coefficient for radiation and convection with the air

More appropriate formulae and breakdown of the housing induce more accurate transfers with the air: the comparison of heat transfer coefficients on Figure 4.11 presents a decrease of roughly 40% between the two approaches. Thanks to this consideration of air flow, the heat exchanges are better taken into account. Finally, the oil sump temperature calculated with the thermal network is 80°C +/-1°C for each test condition.

Calculation of bulk temperature

The thermal network is a way to calculate bulk temperatures of the components without using a timeconsuming Finite-Element model. Standards also propose formulae in order to do so and without modelling the whole system. These two methods are compared hereafter [START_REF] Fossier | Thermal behaviour of an Axle Gear Set[END_REF].

Firstly, the gear set bulk temperature 𝜃 𝑀 (°C) is computed with the ISO/TR 13989-2 [START_REF]Calculation of scuffing load capacity of cylindrical, bevel and hypoid gears -Part 2: Integral temperature method[END_REF]:

𝜃 𝑀 = 𝜃 𝑜𝑖𝑙 + 𝐶 3 • 𝑋 𝑚𝑝 • 𝜃 𝑓𝑙𝑎𝑖𝑛𝑡 • 𝑋 𝑆 (4.21)
where 𝜃 𝑜𝑖𝑙 is the lubricating oil temperature (°C), 𝜃 𝑓𝑙𝑎𝑖𝑛𝑡 is the mean flash temperature (K), 𝐶 3 is a fixed factor, 𝑋 𝑚𝑝 is a contact factor, and 𝑋 𝑆 a lubrication factor. Some specificities of this formulation can be highlighted:

1) This method considers only the couple of gears (operating condition, gear geometry, surface roughness …) and the lubricating oil (lubrication mode, oil type…). 2) Oil temperature is required for this computation. This implies that a temperature measurement should be done, which is not the case with the thermal network. 3) Only a global temperature of the gear set 𝜃 𝑀 is calculated. Indeed, it is generally assumed in gear mechanical studies that pinion and crown are at the same temperature and conduct equal heat flows from gear mesh contact. This assumption is also made for axle gear set [START_REF] Karagiannis | On the dynamics of lubricated hypoid gears[END_REF]. However, as the crown wheel and the pinion have really different size, number of cycles and immersed surface, their temperature could be quite different, as well. This is supported by measurements made by Xu [START_REF] Xu | Thermal Mapping of an Automotive Rear Drive Axle[END_REF], observing up to 10°C difference.

The difference between bulk temperatures obtained with the standard and the ones determined with the thermal network is plotted on Figure 4.12 for different operating conditions. Similar results are obtained with both approaches on the prediction of crown wheel temperature: difference is less than 2°C. However, an important gap is observed on pinion bulk temperature between standard and thermal network estimations, up to almost 10°C. Even at mid power condition (condition B), a difference of more than 3°C can be critical for scuffing issues. The pinion is hotter than the crown wheel. This is in agreement with the observations made by Xu et al. [START_REF] Xu | Thermal Mapping of an Automotive Rear Drive Axle[END_REF] on temperature measurements under load. To explain this phenomenon, Xu et al. suggest the influence of the difference in number of cycles between pinion and crown wheel and also the lower thermal inertia of the pinion. In addition to these explanations, another influence can be highlighted: even if the heat transfer from the contact between pinion and crown is almost equivalent, the thermal resistance between pinion and oil is 3 times bigger than the one between crown and oil. This induces a better cooling of the crown and a higher temperature of the pinion. A hotter environment also exists on the input shaft around the pinion than around the crown wheel, due to closer rolling element bearings.

Conclusion on thermal network application

The thermal network method gives more coherent results both in terms of thermal exchanges and temperature evaluation. However, experimental data collected until now are not sufficient to corroborate thermal network results. Indeed, only oil temperature was monitored and it has been thoroughly underlined that temperature varies locally. A more specific test campaign is needed to obtain bulk temperature of several axle components and thus to validate the thermal network of the axle.

Investigations on axle thermal behaviour

In order to validate the thermal network model, it is necessary to measure the temperature of several components and to characterise the thermal behaviour of the axle. As those presented in Chapter 2, usual efficiency tests only monitor oil temperature, which is not sufficient to validate the power loss breakdown. New tests are then performed.

Test definition

Test rigs similar to those of Chapter 2 are used to conduct experiments with and without an applied load. This allows for efficiency measurements. For the tests with an applied load, torque range is of 5kN.m on the input and on each output with an accuracy of +/-8.5N.m. For the no-load tests, torque range on the input is of 200N.m with an accuracy of +/-0.4N.m.

For the thermal study, several components of the axle are monitored with K-type thermocouples according to the list of Table 4 The test procedure is composed of three power steps with different speeds, as detailed in Table 4.5.

Corresponding output torques and speeds are represented on Figure 4.15. Due to the use of two different test benches, the no-load step is operated separately. For each condition, a stabilisation criteria must be fulfilled before going forward: oil temperature variation must be under 0.1°C during 3 minutes. The same procedure is repeated with a higher oil level.

Experimental results

Two types of results are obtained thanks to these tests: global power loss of the axle and components temperatures. Measurement results for both efficiency and temperature aspects are summarized in Table 4.7 to Table 4.10.

Axle power loss is presented on Figure 4.16: hub losses are removed in order to have only drive head contribution. For each power step, power loss increases with speed. A higher oil level always induces higher losses. This is even more pronounced at higher speeds (conditions 2, 4 and 6) with differences from 400 W up to 1 kW. It is important to keep in mind that the operating temperature here is quite low (maximum oil sump temperature of 60°C). This is favourable for gear mesh efficiency while it is unfavourable for rolling elements bearings losses.

Temperature of components is now considered. A first observation can be made regarding axle housing. Figure 4.17 presents temperature measurements of the three sensors installed on different housing sides, for tests at normal oil level with the same speed but different loads. Drive head housing appears to be the hottest housing part in every condition. Indeed, this is a compact part of the axle with several rolling element bearings generating losses. A second observation can be made on rolling element bearings through a comparison with oil. Figure 4.18 presents temperatures of oil and bearings for tests at normal oil level too. Rolling element bearings located on input shaft are hotter than oil due to their power losses, while the differential rolling element bearing is colder mainly due to the lower rotational speed of output shaft.

Validation of axle thermal network

Thanks to the test procedure, different characteristics of the axle power losses and thermal exchanges can be determined, as reported in Table 4.11.

First, thermal exchanges with the environment must be correctly estimated. As they are conditioned by forced convection with air, it is thus necessary to validate this exchange thanks to the cooling phase, after what temperatures and power losses can be compared. A good agreement between the measured and predicted temperatures of, for example, a rolling element bearing confirms the estimation of its power loss. On the contrary, if the measured temperature is higher than the predicted one, this means that the power loss estimation is lower than in the reality.

Table 4.11: Characterisation of losses and thermal exchanges during the tests

No-load step

Load steps Cooling REB, churning and seal losses X Gear mesh friction losses X Free oil convection X Forced oil convection X X Forced air convection X

Axle cooling

As reported in Table 4.11, free oil convection occurs inside the axle and forced air convection occurs outside during the cooling. As the air flow was kept constant, temperature derivative should be constant as well. To test the model sensitivity, the heat transfer coefficient is modified by 10%. This induces an oil temperature variations of less than 1°C. The model thus gives a correct and stable approximation of axle cooling.

No-load step

During the no-load step, forced air convection also occurs on housing, which was already identified in the previous section. However, forced oil convection is also generated inside the housing due to shafts and gears rotation. As summarized in Table 4.11, this thermal exchange can be characterised here. 4.11, power losses independent from load are produced. Figure 4.24 compares these power losses according to measurement and model. Calculations underestimate the loss: there is a maximum difference of 26% with measurement point and of 11% with measurement uncertainty. This power loss underestimation explains the temperatures predicted on Figure 4.22, which are lower than measured ones.

Extending model validation

The same thermal tests are performed on a second axle called SB1. Axle ratio is identical but the gear set is a spiral bevel one. General characteristics of the axle are given in Table 4.12. Housing and rolling element bearing dimensions are slightly different from axle H1. The axle thermal model is thus an efficient tool to predict both temperatures and power losses of the system. A comparison is made here between the two axles H1 and SB1. Gear mesh friction is lower on axle SB1 due to the spiral bevel gear set, while oil churning losses are higher due to a larger diameter of its crown wheel. Thus, the efficiency of these axles is similar for these test conditions.

Exploitation of the model

Axle is now represented thanks to a thermo-mechanical model: efficiency and bulk temperatures can be calculated for various operating conditions. The influence of some parameters are investigated hereafter on axle H1.

Oil properties

Various lubricants can be used in axles. They have different viscosities, as presented in Table 4.13: Mineral lubricant is the most viscous, Eco lubricant has a lower viscosity than Reference at low temperature. As Reference and Eco lubricant have the same viscosity at high temperature, no difference will be observed for a long haul condition (80km/h, 120kW) as oil temperature is around 80°C. A comparison is made here on a medium duty condition (50km/h, 60kW). Figure 4.32 displays total power loss and temperatures. A significant decrease in power loss and temperature is observed between mineral and synthetic oils: this is due to their respective viscosity, but also to the use of a coefficient depending on oil type for the friction coefficient calculation. There is a slight decrease of temperature between the two synthetic oils but no difference of efficiency. Indeed, the equilibrium between power dissipation and temperature induces a lower temperature for the Eco lubricant. Viscosity is then similar for both lubricants, and power losses too.

Gear surface roughness

In Chapter 3, it has been shown that gear tooth surface roughness has an influence on gear mesh friction. The heat produced modifies oil viscosity and can influence even more friction losses through the friction coefficient. Figure 4.33 illustrates the evolution with roughness of gear mesh power losses and of oil and gear set temperatures for a long haul condition (80km/h, 120kW). Both naturally decrease together. Table 4.14 gives the different friction coefficients obtained in these conditions with temperatures of Figure 4.33 and at 80°C. The same trends are observed as in Chapter 3, but friction coefficient is increased for medium grinding, as operating temperature here is higher than 80°C. This thus worsens even more the gear mesh conditions and induces more friction losses.

Air flow

The axle is located in the chassis of the vehicle. This protects it from air flow generated by truck displacement. For a truck running at 80km/h, the speed of air flow around the vehicle is 22m/s. However, only 5m/s of air flow circulates generally around the axle. This is a very small part of the flow.

Three air flow conditions are tested to see the impact on efficiency and temperatures. Results are presented on Figure 4.34 for a long haul condition (80km/h, 120kW). When air flow is lower (3m/s), bulk temperatures are increased by almost 15°C. This decreases oil viscosity and so power losses, but can induce component failures. When air flow is higher (8m/s), bulk temperatures drop by 10°C and power losses are increased. Apart from efficiency aspects, it is important here to notice that more power losses do not always mean higher bulk temperatures. This is right only if thermal exchanges are kept identical.

Conclusion

This final chapter presented the development of a thermal model that uses power loss estimations established in previous chapters. This model has two objectives: 1. To be able to determine bulk temperatures of critical parts of the system like gears or bearings for both efficiency and durability concerns. 2. To validate power loss estimations based on temperature measurements. It has already been

shown that oil temperature is not sufficient to confirm the power loss breakdown.

Several ISO technical reports propose formulae to address these goals. ISO/TR 14179-2 [START_REF]Gears -Thermal capacity Part 2 : Thermal load-carrying capacity[END_REF] uses an isothermal approach to consider thermal exchanges between axle and its surrounding environment. Oil operating temperature is calculated but it appears that radiation and convection with air are overestimated, as already reported in the literature [START_REF] Bendzulla | New analysis on the heat balance of industrial gearboxes -Optimized calculation-method of a gearbox manufacturer[END_REF]. ISO/TR 13989-2 [START_REF]Calculation of scuffing load capacity of cylindrical, bevel and hypoid gears -Part 2: Integral temperature method[END_REF] gives a formula to determine gear bulk temperature, based on oil operating temperature. This considers an identical bulk temperature for both meshing gears. However, pinion and crown can have different bulk temperatures due to their respective size and rotational speed. It will then be interesting to compare the aforementioned results to the thermal network method.

Most of the authors consider the axle as isothermal in their model, but it was underlined in the literature that it is not the case [START_REF] Xu | Thermal Mapping of an Automotive Rear Drive Axle[END_REF]. To take into account bulk temperatures of individual axle components, a thermo-mechanical model is developed, based on the thermal network method. The different components of the axle are considered as isothermal nodes of a network and linked by thermal resistances. Resistances manage various heat flows like conduction, convection with air or oil, radiation, and striction. Heat generated by dissipation sources is injected at the corresponding node.

Temperatures and power losses are then calculated thanks to an algorithm. Several operating conditions can be applied to the system and steady state or transient results can be obtained.

A comparison is done between thermal network and ISO technical report methods. Thermal network shows more coherent results for thermal exchanges and allows to distinguish pinion and crown gear bulk temperatures.

To validate both bulk temperature estimations and power loss breakdown, a specific test campaign has been done. An axle was monitored with many temperature sensors located on components like rolling element bearings and housing. Different operating conditions were tested in order to characterise thermal exchanges and power dissipation: loaded and unloaded running phases, but also cooling phases. Results regarding power loss are as expected: it increases with torque and speed, but also with oil level. Concerning temperature, it is confirmed that housing temperature varies between sides and that differential rolling element bearings are colder than the one located on the input shaft. Indeed, due to their lower rotational speed, they generate less power losses.

Test results are then compared to axle thermal model for the different operating conditions: cooling, no-load step and load steps. The model shows a good agreement for both temperatures and power losses. To go further, a second axle with a spiral bevel gear set was tested and modelled: the axle thermal model gives good results as well and can be used as a design tool.

This thermo-mechanical model is now used to investigate various parameters. Their influence on power losses and on bulk temperatures is analysed. Lubricant with lower viscosity provides lower bulk temperature but not necessarily better efficiency. A smoother tooth surface is an advantage for both power losses and bulk temperature. Finally, increasing air flow around axle diminishes bulk temperatures but increases power losses. The opposite is also correct. This last case also underlines that power losses and temperature do not always increase together.

Conclusion

To fulfil customer demands but also government regulations, the truck industry must decrease fuel consumption and emissions of its products. Improvement of the powertrain efficiency is then a key development. The axle is the final part of the truck driveline and can still be improved. Indeed, optimisation of axles at design stage is mostly considered regarding durability and noise aspects.

Focus is now expanding to include efficiency: this study aims to characterise the efficiency of truck axles through a thermo-mechanical model. This model is developed to estimate both power losses and thermal exchanges. A global approach is used to consider the complete system and to allow a fast increment in order to use it at design stage. However, a local approach is also investigated to evaluate meshing friction of the gear set.

The first chapter describes the truck drive axle and its gear set: a hypoid or a spiral bevel one (without offset). As most of the previous studies are on automotive axles, the differences with truck axles are presented, like dimensions (crown gear size, oil volume) and operating conditions (speed, torque). This influences power losses but efficiency stays comparable. The literature investigates axle efficiency through experimental means first and then through theoretical methods. Influential parameters are highlighted: operating conditions, such as lubricant temperature, rotational speed, load…; gear macrogeometry parameters, like hypoid offset, spiral angle, face width, diameter, ease-off… and gear microgeometry parameters, like surface roughness. The power loss sources of the axle are identified: shaft seal friction, rolling element bearing drag and friction, oil churning and gear mesh friction. For each source, power loss models exist but do not always agree with each other. Depending on application and operating conditions, great discrepancies can be observed between estimations. The thermal behaviour of the axle was investigated thanks to isothermal approaches and mainly for efficiency purposes. Models consider the axle at oil sump temperature. However, it has been shown experimentally that noticeable temperature differences exist between components [START_REF] Xu | Thermal Mapping of an Automotive Rear Drive Axle[END_REF]. Cooling and durability issues must be kept in mind: local hot spots can be created and induce failure of gears and rolling element bearings. As temperature influences lubricant viscosity, this can also modify power losses generated by each component. Moreover, knowing local temperatures allows to validate the estimation of component power losses. Indeed, different power loss breakdown can match to the same global power loss of the axle. A measurement of global power loss and of oil temperature is then not sufficient to validate the results of an efficiency model.

The second chapter attempts to estimate the efficiency of truck axle. Tests are performed to measure global power losses of an axle for loaded and unloaded conditions. This allows the evaluation of loaddependent and load-independent losses. Calculations are then conducted for each source of power loss according to the different formulae reviewed in the literature and a method is selected. For shaft seal friction, the formula of Simrit is kept. For Rolling Element Bearings losses, it appears that SKF formula overestimates the losses under load; more coherent results are obtained with Harris formula, which is then used. For hypoid gear mesh friction, the estimation according to the ISO/TR 14179-2 gives a good trend, but a more thorough study is required to improve it. For the oil churning, both Mauz and Jeon solutions gives unrealistic results. Investigations are then conducted on the churning losses of spiral bevel gears: a specific test bench is used with several spiral bevel gears and cones. Experiments are done with different oils and operating conditions. The results highlight the influence of some operating conditions like speed, immersion ratio and oil viscosity, but also of a main gear geometrical parameter which is the outer diameter. By contrast, it appears that gear teeth have little influence on churning losses. A comparison between test results and Mauz and Jeon models shows discrepancies. Based on Jeon formula who has worked on axles too, a new model is proposed to better take into account the identified influential parameters. This also considers two different regimes depending on Reynolds number and the greater contribution due to the meshing gear pair. Thanks to this new formula, the power loss estimation for the truck axle case has a better agreement with test results. For example, the influence of oil level on losses is correctly taken into account. Finally, gear mesh friction contributes a lot to losses for truck operating cases (80°C), while rolling element bearings are the first dissipation source for lower operating temperature (40°C).

As gear mesh friction appears to be a major source of dissipation for truck axles, it is thoroughly investigated in the third chapter. Indeed, spiral bevel and hypoid gears are particular. Their manufacturing process is reviewed from cutting process to surface finishing. Indeed, design choices and manufacturing determine geometrical parameters. A model developed by Barday [22] allows to take most of them into account. During gear mesh, elliptical contacts occur successively along contact path. For modelling purposes, each successive contact is discretised in several line contacts: contact conditions (load, velocities, surface curvature…) are calculated thanks to Barday software. Due to conical shape and spiral angle, the kinematic of spiral bevel and hypoid gears is different from other gears: surface velocities are not oriented along the axis of contact ellipse and they are not aligned together. Angles are defined to characterise the orientation of velocities. Moreover, entrainment velocity varies along 𝑦-axis of contact ellipse and induces a global rotation of the contact. For hypoid gears, sliding is not only transverse, as a longitudinal sliding also exists due to offset. The power loss of gear mesh is now studied with a local approach based on data from Barday software. Contact conditions and power loss along contact path are analysed. Two gear sets are compared, a hypoid and a spiral bevel one: the permanent sliding is observed on the hypoid gear, while a decrease is observed around pitch point on spiral bevel gear. A comparison is made between local approach and ISO global approach to discuss the validity of ISO technical report approximation. Global approach always overestimates the power loss, due to sliding velocity estimation: indeed, sliding velocity varies a lot along a line of contact and local approach considers the sliding only on the contacting part of the line. For higher loads, a better agreement is then expected, as contact pattern is more spread. In our case, ISO estimation obtains more coherent results for the hypoid gear than the spiral bevel one. Finally, local approach is more accurate to evaluate gear mesh power loss and gives acceptable results with a global friction coefficient calculated according to ISO technical report. However, local approach offers the possibility to use local friction coefficients: as friction coefficient influences heavily the friction loss of the gear mesh, this can be more precise. Formulae to calculate friction coefficient are based on similar parameters: load, entrainment and sliding velocities, oil viscosity, and surface roughness. In the case of hypoid gears, two specific parameters are selected: surface roughness and velocities defined by orientation angles. Surface roughness depends mainly on cutting process and on surface finishing. It is characterised by an overall roughness but also by machining grooves orientation. Friction coefficients are calculated for various overall roughness according to reviewed formulae: a decrease of friction coefficient, and so of gear mesh power losses, is observed for smoother surfaces. Orientation angles have been studied only regarding elastohydrodynamic aspects. Friction was considered for a close phenomenon: the contact between rib roller-end and flange in some rolling element bearings. Preliminary tests are performed to evaluate the influence of orientation angle: friction coefficient can be doubled for low surface velocity. This induces a great increase of gear mesh power losses too.

Finally, the fourth chapter presents the thermal model of the axle, based on the thermal network method. The different components of the axle are considered as isothermal nodes of a network and linked by thermal resistances that manage various heat flows (conduction, convection with air or oil, radiation). Power losses are estimated according to formulae established in the previous chapters and the heat generated is injected at the node corresponding to the dissipation source. Temperatures and power losses are then calculated thanks to an algorithm. Several operating conditions can be applied to the system and steady state or transient results can be obtained. A comparison is done between thermal network and ISO technical report methods. ISO/TR 14179-2 [START_REF]Gears -Thermal capacity Part 2 : Thermal load-carrying capacity[END_REF] overestimates the thermal exchanges between axle and its surrounding, as already reported in the literature [START_REF] Bendzulla | New analysis on the heat balance of industrial gearboxes -Optimized calculation-method of a gearbox manufacturer[END_REF]. Thermal network method obtains more coherent results. ISO/TR 13989-2 [START_REF]Calculation of scuffing load capacity of cylindrical, bevel and hypoid gears -Part 2: Integral temperature method[END_REF] gives a formula to determine gear bulk temperature, which considers an identical bulk temperature for both meshing gears. However, the thermal network method presents different bulk temperatures for pinion and crown. This can be explained by their respective size and rotational speed. A specific test campaign is conducted to validate the thermo-mechanical model of the axle. An axle was monitored with many temperature sensors located on components like rolling element bearings and housing. Different operating conditions were tested in order to characterise thermal exchanges and power dissipation: loaded and unloaded running phases, but also cooling phases. Results regarding power loss are as expected: it increases with torque and speed, but also with oil level. Concerning temperature, it is confirmed that housing temperature varies between sides and that differential rolling element bearings are colder than the one located on the input shaft. Indeed, due to their lower rotational speed, they generate less power losses. The thermal model shows good agreement with test results for the different operating conditions (cooling, no-load and load steps): calculated temperatures and power losses are representative. This is also validated with another design of axle with a spiral bevel gear set. The axle thermal model can then be used as a design tool. The influence of some parameters on power losses and on bulk temperatures is analysed. Lubricant with lower viscosity provides lower bulk temperature but not necessarily better efficiency. A smoother tooth surface is an advantage for both power losses and bulk temperature. Finally, increasing air flow around axle diminishes bulk temperatures but increases power losses. The opposite is also correct. This last case underlines also the fact that power losses and temperature do not always increase together.

The thermo-mechanical model of truck axles developed in this study allows to obtain both efficiency and temperature distribution. Regarding industrial application for the Volvo Group, this tool can be used at design stage to:

 Optimise efficiency of the axle. By combining it with software already used at Volvo Group, optimisation can be done regarding durability and noise as well.  Investigate on failure issues due to local hot spots on rolling element bearings or gears.

In parallel with this work, a model was also established for planetary gears [START_REF] Boni | Experimental investigations on churning power losses generated in a planetary gear set[END_REF]. Coupling it with the axle model could be considered to study hub reduction axles.

Besides model exploitation, some experimental results on complete axles must still be analysed. Indeed, two additional axles were tested during the thermal tests in order to investigate different parameters: one has a face-milled gear set and the other has a lower ratio. Two gear cutting methods (face-hobbing and face-milling) could then be compared on the axle with a hypoid gear set. Another comparison could also be done between two axle ratio (4.625 and 2.643) on the axle with a spiral bevel gear set. Test results are important to validate the consistency of the thermo-mechanical model regarding specific parameters.

The complexity of the friction in spiral bevel and hypoid gears has been underlined. Indeed, it has been shown that grooves orientation regarding rolling direction has an impact on friction coefficient [START_REF] Diab | Prediction of Power Losses Due to Tooth Friction in Gears[END_REF]. On spiral bevel and hypoid gears, this is different from spur gears: it is neither perpendicular nor parallel. Moreover, pinion and crown gear appear to have different surface roughness and this can influence friction. Tests must be done to characterise friction coefficient for these particular surfaces, on a twin disc machine for example [START_REF] Diab | Prediction of Power Losses Due to Tooth Friction in Gears[END_REF]. Various surface finishing could also be tested.

However, on twin disc machine, rolling direction is the same for both discs, while this is not the case for spiral bevel and hypoid gears. This specific configuration is reproduced on some test benches ( [START_REF] Omasta | Effect of surface velocity directions on elastohydrodynamic film shape[END_REF], [START_REF] Stahl | Theoretical and Experimental investigations on EHL point contacts with different entrainment velocity directions[END_REF]), but only film thickness is studied. Preliminary tests presented in Chapter 3 show an influence of orientation angle on friction coefficient. It would be interesting to go further, by characterising more precisely the impact on friction coefficient. Moreover, preliminary tests were done on smooth test samples only. Test samples with rough surface can be manufactured and used to conduct a procedure similar to the one used on twin disc machine. This would allow for a better characterisation of the contact of hypoid gears.

Another point of interest would be rolling element bearings. They are currently considered as isothermal elements. The model could be made finer by splitting bearings in several parts, to take into account the temperature difference between inner and outer races. This would allow for a more precise study of bearing failure or race expansion.

Annexes Annex 1.A: Detail of formulae for hypoid gear mesh friction {a} Buckingham formula

Reference [START_REF] Buckingham | Analytical Mechanics of Gears[END_REF] Hypoid gear efficiency is expressed as: {a} Tests performed by Laruelle [START_REF] Laruelle | Study of the thermal behavior of a gear unit[END_REF] Various tests were performed by Laruelle on each tested gear and with different oils.  𝑒 𝑜 : radius of the front side of the truncated cone associated to the cone of base radius 𝑅 𝑜 (m)

𝐸 ℎ𝑦𝑝𝑜𝑖𝑑 = 1 1 + 𝜙 𝑠𝑝𝑖𝑟𝑎𝑙 𝑏𝑒𝑣𝑒𝑙 + 𝜙 𝑤𝑜𝑟𝑚 (1.
Then, the back side area 𝐴1 (m²) is: {b} Peripheral area

𝐴1 = 1 2
It represents the area of the side face of the truncated cone which is immersed in the lubricant sump. Figure 21 represents the main parameters for this area.

Annex 3.A: Hertzian contacts

An equivalent geometry is generally used for Hertzian contacts: gear teeth are reduced to two parabolical bodies. Each have radii of curvature in two directions, generally 𝑥 and 𝑦 axis. The pressure distribution in such contact depends only on gap height between surfaces. Thus the body 2 can be considered as plane in a particular direction if an equivalent radius of curvature is defined for the body 1 in the same direction:

1 𝑅 𝑑 = 1 𝑅 𝑑1 + 1 𝑅 𝑑2
(3.12) with:

 𝑅 𝑑1 , 𝑅 𝑑2 : curvature radius in direction 𝑑 of body 1 or 2

An equivalent elastic modulus 𝐸 ′ (Pa) is also defined: The heat flow dissipated through the housing is: The ISO technical report recommends as a rule to determine the heat dissipation via the housing by the larger value air-side thermal resistance at the housing surface. The first two terms in the above equation can then be neglected. For high air velocities and thus good external heat transfer, it will possibly be necessary to take into account the oil-side heat transfer as well. The heat conduction through the housing should only be taken into account in special cases (double-walled housings, housings with sound insulation, non-metallic housings...).

1 𝐸′ = 1 2 ( 1 -𝜈 1 2 𝐸 1 + 1 -𝜈 2
𝑄 = 𝑈
The air-side heat transmission 𝛼 𝑐𝑎 is composed of:

𝛼 𝑐𝑎 =

Annex 4.B: Thermal network method

Reference [START_REF] Changenet | Modélisation du comportement thermique des transmissions par engrenages[END_REF], [START_REF] Changenet | Power Loss Predictions in Geared Transmissions Using Thermal Networks-Applications to a Six-Speed Manual Gearbox[END_REF] Thermal network method is detailed here for a mechanical transmission application.

{a} Method principle

The thermal network method uses a conceptual view-point for Fourier's law of heat transfer [START_REF] Holman | Heat transfer[END_REF]. A transposition can be done between an electric circuit and a thermal network, according to Table 0.4. where 𝑅 𝑠𝑒𝑟𝑖𝑒𝑠 and 𝑅 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 are the equivalent resistances respectively for series resistances and for parallel resistances.

The network is composed of nodes linked by thermal resistances. Each node is isothermal and represents an element of the modelled system. The heat generated by dissipation sources is injected in the network at the corresponding node.

According to the first law of thermodynamics, the total energy of an isolated system is constant. The internal energy of an element of the system then varies due to its own power dissipation and to heat exchanges with other elements of the system: It is noteworthy that the heat sink of the system (generally ambient air) is a particular element. If its temperature is imposed and constant, it is necessary to define its thermal inertia as infinite: 𝑀 𝑖 𝑐 𝑖 = 10 30 J/K for example. Moreover, as all the heat evacuates through this element, its 𝑄 𝑖 will be negative.

𝑀
This system of differential equations can be solved numerically thanks to Runge-Kutta or Adams methods [START_REF] Changenet | Modélisation du comportement thermique des transmissions par engrenages[END_REF].

In {b} Definition of thermal resistances

Various thermal exchanges can happen inside a mechanical transmission. Thus, different kinds of thermal resistances are defined to manage the heat flows between the nodes:

-Convection and radiation with ambient air -Convection with oil -Conduction -Striction

1) Convection and radiation with ambient air

Thermal exchange between housing and ambient air occurs through convection and radiation. The corresponding thermal resistances are then parallel. Depending on the existence of an air flow around the housing, the convection is forced or free.

The housing is decomposed in a set of flat plates to take into account the different air flows around it.

For each plate, the characteristic dimension is then plate length or height, depending on its orientation.

To consider this assembly, a global thermal resistance is calculated for each type of thermal exchange (convection and radiation):

𝑅 𝑡ℎ ℎ𝑜𝑢𝑠𝑖𝑛𝑔 = 1 ∑ℎ 𝑖 𝐴 𝑖 The radiation coefficient is determined from the Stefan-Boltzmann law [START_REF] Holman | Heat transfer[END_REF]: Free convection happens when ambient air around the housing is static. The free convection coefficient is defined after the formulae proposed by Winter et al. [START_REF] Winter | Wärmeabführung bei Getrieben unter Quasistationären Betriebsbedingungen. Teil I : Systematische Untersuchungen zur Wärmeabführung bei Natürlicher und Erzwungener Zuftströmung mit Hilfe eines Modellprüfstandes[END_REF]. The thermal exchange with air depends on the orientation of the surface, as heat dissipation is not the same depending on surface inclination.

ℎ 𝑟𝑎𝑑 =
Thus, for vertical surfaces, it is: Forced convection is due to an air flow occurring around the housing. The forced convection coefficient is also defined according to formulae of Winter et al. [START_REF] Winter | Wärmeabführung bei Getrieben unter Quasistationären Betriebsbedingungen. Teil II : Untersuchungen zur Wärmeabführung über Stahlfundamente und Übertragung der Prüfstandsergebnisse auf die Praxis[END_REF]. For this thermal exchange, the distinction is made between surfaces normal and tangential to the air flow.

ℎ 𝑐𝑜𝑛𝑣 =
For surfaces normal to the flow: For the particular case of forced convection with a surface normal to the flow, the characteristic length is: 𝐿 = 𝐴 𝑃 ⁄ with 𝐴 the normal surface area (m²) and 𝑃 its perimeter (m).

2) Convection with oil

A dip lubricated transmission is considered here. Convection then occurs between oil and different parts of the system: housing, gears, rolling element bearings… If gears are running, which is generally the case, convection with oil is forced. Otherwise convection is free.

Air is considered as inviscid, but oil is a viscous fluid. The same approximation cannot be done and fluid mechanics characteristics should be taken into account. Thus for convection with the oil, Holman [START_REF] Holman | Heat transfer[END_REF] defines the different heat transfers with:

ℎ 𝑜𝑖𝑙 = 𝑁𝑢 𝑘 𝐿 The housing is still considered as an assembly of flat plate. A global thermal resistance is again calculated according to Equation (4.38). For each plate, the characteristic dimension is also plate length or height, depending on its orientation.

Free oil convection inside the oil bath is due to density variations that induce oil motion. This also depends on surface orientation. For horizontal surfaces, Nusselt number is defined as:

Annexes

Oil is also projected on housing wall due to gears rotation. For this oil trickling along non-immersed surfaces, the oil displacement is due to gravity. A mean velocity of the fluid film is then defined: 𝑉 = 2.95 √𝐿 (4.56)

 Convection with rolling element bearings:

As rolling element bearings are partially immersed in oil bath, convection with oil must be considered. Rolling element bearings are treated in the same manner as the housing for the heat transmission coefficient calculation: free oil convection when system is at rest, forced oil convection and oil trickling when system is rotating. The characteristic dimension considered is bearing radius. Thermal resistance of convection with oil is then:

𝑅 𝑜𝑖𝑙 = 1 ℎ 𝐴 (4.57)
with 𝐴 the surface of transfer (m²), which is the bearing immersed surface for convection with oil bath or the non-immersed surface for oil trickling.

 Convection with gears:

When the system is stopped, free convection occurs between oil and gear immersed surface. This thermal exchange follows the same rules as for the housing in Equations (4.51) and (4.52). The characteristic dimension to be used is the gear immersion depth. Oil projection is notably characterised by the angle of rotation where the gear projects oil. For dip lubricated gears, this angle 𝜃 𝑝𝑟𝑜𝑗 begins at gear teeth emergence from oil and ends at gear mesh point, as illustrated on Figure 22. Projection duration is also important and is defined as: t= 𝜃 𝑝𝑟𝑜𝑗 Ω ⁄ .

𝜃 𝑝𝑟𝑜𝑗 Ω Gear mesh

Blok [START_REF] Blok | Transmission de chaleur par projection centrifuge d'huile[END_REF] defines an evacuated heat quantity for a tooth face per width unit. This energy can be deduced from Figure 23, depending on a dimensionless number Ψ, defined as:  𝜒: thermal effusivity of oil ( 𝜒 = √𝑘 𝜌 𝑐 N/m.s 1/2 .°C)  Δ𝑇: temperature difference between pinion and oil (K)

3) Conduction

Conduction happens inside elements when a temperature gradient exists. This mainly depends on element geometry. Contact conduction occurs between solids in contacts at different temperatures: for example, between drive head housing and the remaining housing, as they are bolted together. This is a complex type of conduction due to notably to surface roughness. For a metal-metal contact with air in between, thermal resistance is expressed as:

𝑅 𝑐𝑜𝑛𝑡𝑎𝑐𝑡 = 𝑥 𝑘 ̅ 𝐴

4) Striction

Striction controls the exchange between gears and their meshing point.

1 .

 1 INTRODUCTION ............................................................................................................................................. 2. DRIVE AXLE ARCHITECTURE.............................................................................................................................. 3. AXLE EFFICIENCY ...........................................................................................................................................

5 .

 5 CONCLUSION ................................................................................................................................................ 2. AXLE POWER LOSSES ............................................................................................................................. 1. INTRODUCTION ............................................................................................................................................. 2. TRUCK AXLE EFFICIENCY ...............................................................................................................................

5 .

 5 CONCLUSION .............................................................................................................................................. CONCLUSION ................................................................................................................................................ ANNEXES ...................................................................................................................................................... ANNEX 1.A: DETAIL OF FORMULAE FOR HYPOID GEAR MESH FRICTION.......................................................................

  ANNEX 3.A: HERTZIAN CONTACTS........................................................................................................................... {a} Line contact ........................................................................................................................................... {b} Elliptical contact .................................................................................................................................... ANNEX 3.B: DETAIL OF FRICTION COEFFICIENT FORMULAE ........................................................................................... {a} ISO/TR 14179-2 ..................................................................................................................................... {b} Diab formula ......................................................................................................................................... {c} Xu formula ............................................................................................................................................. ANNEX 4.A: HEAT DISSIPATION THROUGH THE HOUSING ............................................................................................. ANNEX 4.B: THERMAL NETWORK METHOD ............................................................................................................... {a} Method principle ................................................................................................................................... {b} Definition of thermal resistances .......................................................................................................... 1) Convection and radiation with ambient air .................................................................................................... 2) Convection with oil......................................................................................................................................... 3) Conduction ..................................................................................................................................................... 4) Striction ..........................................................................................................................................................REFERENCES ................................................................................................................................................. SCIENTIFIC CONTRIBUTIONS .........................................................................................................................

Figure 0 . 1 :

 01 Figure 0.1: Les principaux éléments d'un pont de camion

  a été développée pour le barbotage des engrenages hypoïdes dans les ponts automobiles. Cependant, son étude s'est basée sur deux engrenages de géométrie proche. Il est donc important d'analyser plus en détail l'influence des paramètres géométriques par de nouveaux essais. Le banc utilisé, représenté Figure 0.2, permet de tester un engrenage à la fois, de varier la vitesse et le sens de rotation et de contrôler la température du bain d'huile. Quatre géométries d'engrenage et deux de cône sont testées, ainsi que plusieurs huiles. Des essais sont établis afin de couvrir les plages de fonctionnement d'un pont de camion, en se basant sur des paramètres équivalents : la vitesse périphérique des engrenages, la viscosité et le niveau d'huile. La forme du carter du banc peut également être modifiée à l'aide de déflecteurs pour approcher celle d'un pont.

Figure 0 . 2 :

 02 Figure 0.2: Banc d'essais pour mesurer les pertes par barbotage Les essais menés permettent d'abord d'observer l'influence des conditions opératoires : vitesse de rotation Ω, ratio d'immersion ℎ 𝑙𝑢𝑏 𝑅 𝑜 ⁄ et propriétés de l'huile. Les pertes par barbotage augmentent de manière importante avec la vitesse. Ces pertes augmentent également avec le niveau d'huile : le ratio d'immersion est retenu comme facteur influent. Les pertes diminuent lorsque la viscosité 𝜈 𝑙𝑢𝑏 baisse, mais elles restent constantes à faible viscosité (sous 30 cSt) : deux régimes d'écoulement semblent exister. L'influence de la densité 𝜌 n'est pas observée ici, faute d'une différence significative de cette caractéristique entre les huiles utilisées. L'effet sur les pertes de la géométrie des engrenages est ensuite étudié : rayon extérieur 𝑅 𝑜 , largeur, largeur de denture, demi-angle et nombre de dents. Une comparaison entre un engrenage et un cône d'enveloppe extérieure identique permet de montrer que les dents ont peu d'influence sur les pertes par barbotage des spiro-coniques. Il apparaît ensuite que le diamètre extérieur est un paramètre plus influent que toutes les autres caractéristiques

Figure 0 . 3 :

 03 Figure 0.3: Comparaison des mesures et des calculs de pertes pour les conditions sans chargeCalcul des pertes dépendantes de la chargeLes pertes aux roulements ont déjà été évaluées et il reste donc à estimer les pertes par frottement à l'engrènement. Les deux formules, Buckingham et ISO, sont appliquées au couple conique du pont. Les résultats obtenus avec Buckingham surestiment beaucoup ces pertes, en partie à cause du calcul du

Figure 0 . 4 :

 04 Figure 0.4: Comparaison des mesures sous charge et des calculs de pertes pour un niveau d'huile (a) normal et (b) élevé

Figure 0 . 5 :

 05 Figure 0.5: Profil de denture et stries d'usinage pour un taillage (a) sans et (b) avec génération

6 )

 6 En dehors du cône primitif, les vitesses de surface des engrenages spiro-coniques ont des valeurs et des directions différentes. Ces directions sont définies par l'angle d'orientation des vitesses 𝛿 et l'angle d'orientation de l'entraînement 𝛿 𝑒 , comme illustré sur la Figure 0.6. Le glissement a lieu uniquement

Figure 0 . 6 :

 06 Figure 0.6: Exemples de vitesses dans le contact d'un engrenage (a) spiro-conique et (b)-(c) hypoïde La vitesse d'entraînement au sein de l'ellipse est analysée selon les composantes transverse et longitudinale à la ligne de contact. Il apparaît que la composante longitudinale est constante, mais que la composante transverse varie le long de l'ellipse, provoquant une légère rotation.

V

Figure 0 . 7 :Figure 0 . 8 :

 0708 Figure 0.7: Perte par frottement le long d'un engrènement complet (1000 tr/min et (a) 280 N.m ou (b) 1120 N.m)

Figure 0 . 9 :

 09 Figure 0.9: Coefficient de frottement moyen d'après plusieurs formules et pour différentes rugosités de surface (1000 tr/min, 1120 N.m, huile à 80°C)L'influence de la rugosité de surface sur le coefficient de frottement est d'abord étudiée. La rugosité générale dépend surtout de la finition. Elle est caractérisée par la rugosité moyenne arithmétique du profil 𝑅𝑎 et par la hauteur maximale du profil 𝑅𝑧. Diab et al.[START_REF] Diab | Prediction of Power Losses Due to Tooth Friction in Gears[END_REF] montrent que le coefficient de frottement double si deux surfaces rugueuses sont utilisées plutôt que des lisses. Une moyenne du coefficient de frottement est calculée d'après les approches locales pour différentes finitions : rectification grossière (a), rectification fine (c) et superfinition (d). Une surface rodée à l'usage (b) sert de référence. Les résultats sont comparés au coefficient de frottement global calculé avec l'ISO sur la Figure0.9. En plus de ne pas changer avec la rugosité, la formule de Benedict et Kelley prédit plus de pertes que toutes les autres estimations. Celles-ci donnent des tendances similaires : le coefficient de frottement est divisé par deux entre les conditions (a) et (d). Les pertes étant directement proportionnelles au coefficient de frottement, celles-ci sont également réduites. Cependant, la rugosité moyenne arithmétique donne seulement une approximation de la rugosité de surface et les stries d'usinage peuvent avoir un impact sur le coefficient de frottement. Contrairement au cas des engrenages cylindriques, les stries d'usinage des spiro-coniques et hypoïdes sont obliques par rapport

Figure 0 . 10 : 1 )

 0101 Figure 0.10: Réseau thermique du pont de camion Le réseau thermique du pont est représenté sur la Figure 0.10 avec les sources de perte et les différentes résistances thermiques. Ces dernières établissent les échanges thermiques entre les noeuds : 1) La conduction a lieu entre les éléments en contact, comme un roulement avec son arbre, etc. ; 2) La convection et le rayonnent ont lieu entre l'environnement et le carter, qui est décomposé en plusieurs plaques pour considérer les différents flux d'air circulant autour ; 3) La convection entre l'huile et les engrenages en rotation est prise en compte car ces derniers sont lubrifiés par bain. La convection entre l'huile et le carter a aussi lieu ; 4) La striction contrôle l'échange entre les engrenages et leur point d'engrènement. La température du contact augmente beaucoup mais, comme la zone de contact est très petite comparée à la taille de l'engrenage, la température de masse est peu influencée.

  . Enfin, augmenter le flux d'air autour du pont diminue les températures mais augmente les pertes. Ce dernier cas souligne le fait que les pertes et les températures n'augmentent de pair que si les échanges thermiques sont maintenus identiques.

Figure 0 . 12 :

 012 Figure 0.12: Température d'huile comparée à l'ambiant d'après les mesures et le modèle pour un niveau d'huile (a) normal et (b) élevé

Figure 1 . 1 :

 11 Figure 1.1: Main elements of a truck drive axle

Figure 1 . 2 :

 12 Figure 1.2: An all-wheel drive truck (6x6)

Figure 1 . 4 :

 14 Figure 1.4: Example of a drive head architecture

Figure 1 . 6 :

 16 Figure 1.6: Breakdown of estimated axle loss over the NEDC [7] Takaki et al. [37] evaluated power losses in car axles, mainly on churning, bearing and meshing friction. According to their measurements presented on Figure 1.7, more than 50% of friction losses are due to hypoid gear meshing. This observation is in contradiction to the study of Kakavas et al., as torque and speed inputs are at the same level; however, the operating temperature is not specified by Takaki et al., while it is quite low (25°C to 40°C) for Kakavas et al.. Takaki et al. then focused on gear mesh friction. They proceeded to an analysis on gear parameters such as spiral angle, offset, face width, ease off…

Figure 1 . 7 :

 17 Figure 1.7: Breakdown of loss torque on experiments of Takaki et al.[START_REF] Takaki | Optimization of Hypoid Gear Design for High Efficiency Drives[END_REF]

Figure 1 . 8 :

 18 Figure 1.8: Comparison of seal power losses for Simrit and Linke formulae (𝑑 𝑠ℎ = 80 mm)

  for a less viscous lubricant. As application boundaries are not clear for Linke formula, this can lead to incoherent results. It is then discarded for this study.

Figure 1 . 9 :

 19 Figure 1.9: Temperature comparison of seal power losses for Simrit and Linke formulae with (a) viscous oil (𝜈 40 =120mm²/s) and (b) fluid oil (𝜈 40 =40mm²/s)

Figure 1 .Figure 1 . 11 :

 1111 Figure 1.10: REB power losses according to Harris and SKF formulae

Figure 1 . 12 :

 112 Figure 1.12: Comparison of no-load and load losses for (a) SKF formula and (b) Harris formula

Figure 1 . 13 :

 113 Figure1.13: Splash oil factor according to Mauz[START_REF]Gears -Thermal capacity Part 2 : Thermal load-carrying capacity[END_REF]

Figure 1 . 17 :

 117 Figure1.17: Pinion temperature variations in the nine point loaded test with 45 mph simulated winds speed[START_REF] Xu | Thermal Mapping of an Automotive Rear Drive Axle[END_REF]

Figure 1 . 19 :

 119 Figure 1.19: Comparison of bulk measurements from thermocouples to time averaged boundary condition analytical results[START_REF] Handschuh | Thermal behaviour of spiral bevel gears[END_REF]

Figure 1 .

 1 [START_REF] Hurley | An Experimental Investigation of Thermal Behavior of an Automotive Rear Axle[END_REF] exposed analytical and experimental results on bulk temperatures and showed that top and root are the hottest part of the tooth.Wang et al. proposed a Finite Element model of a gear pair to calculate the heat generated by meshing friction, considering a constant friction coefficient (𝜇 = 0.05). For computational time saving, a transient thermal analysis of one tooth is then used with its heat accumulating with cycles. Thanks to this model, a comparative study is made for three different contact patterns as showed on Figure1.20.

Figure 1 . 20 :

 120 Figure 1.20: Tooth contact patterns for different pinion machine-setting parameters[START_REF] Wang | Investigation into the meshing friction heat generation and transient thermal characteristics of spiral bevel gears[END_REF]

  Figure 2.2 (a) was used. It allows for measurement of input and output torques and rotational speeds. Two types of dynamometers are used: torque range is of 3kN.m on the input shaft and 5kN.m on each output with an accuracy of +/-1.5N.m.

  Figure 2.2 (b) was used. The output shafts and the wheel hubs are dismantled and so just remains the drive head of the axle. Only the input torque and rotational speed are measured. The torque range of the dynamometer is of 500N.m with an accuracy of +/-0.15N.m.

Figure 2 . 2 :

 22 Figure 2.2: Test bench set-up for (a) load dependent test and (b) no load test

Figure 2 . 3 :

 23 Figure 2.3: Measurement results for load test conditions and two different oil levels

Figure 2 . 4 :

 24 Figure 2.4: Measurement results for load-independent losses test conditionsLoad-independent losses tests consist of successive speed conditions corresponding to the vehicle speed from 20km/h to 120km/h with steps of 20km/h. No torque is transmitted during these tests. Only the resisting torque is measured.The results of these tests are plotted on Figure2.3 and Figure2.4. According to the tests under load, the efficiency of this axle ranges from 94% to 98% depending on conditions. A higher oil level induces more power losses, except under high torque where losses are at the same level.

Figure 2 . 5 :

 25 Figure 2.5: Seal friction losses according to Simrit formula

  Figure 2.6 (a)), the two models give close results. Nevertheless, for the load test conditions (Figure 2.6 (b)), estimations given by SKF formula equal almost the double of Harris formula results.

Figure 2 . 6 :

 26 Figure 2.6: Comparison between total power loss measurement and REB loss calculation results according to Harris and SKF formulae for (a) no-load test conditions and (b) load test conditions

Figure 2 . 7 :

 27 Figure 2.7: Deduction of gear mesh efficiency from Harris and SKF bearing loss formulae

  mm and 220 mm). It seems important to proceed to a broader study of the spiral bevel gears geometry influence on churning losses.

Figure 2 . 9 :

 29 Figure 2.9: Oil churning test rig An electric motor drives the tested gear via a belt. The test rig allows for measurement of churning torque and gear rotational speed. The torque sensor has a range of 2 N.m with an accuracy of +/-0.002 N.m, while the speed ranges from 0 to 7000 rpm. A pump allows for modification of the oil level; heating covers and thermocouples manage the oil temperature. During tests, the oil flow can be observed through the Plexiglas face of the housing. The housing case itself measures 400 x 275 x 93 mm; this allows to avoid wall effects on oil flow. The addition of deflectors is possible as illustrated on Figure 2.10: this allows for changing housing shape.

Figure 2 . 10 :

 210 Figure 2.10: Deflector modifies housing shapeThe tested gear is supported by ball bearings, as presented on Figure2.11. Their power loss contribution is removed from the total power losses to consider only oil churning losses.

Figure 2 . 11 :

 211 Figure 2.11: Oil churning test rig scheme (cylindrical gear pair configuration)

Figure 2 . 13 :

 213 Figure 2.13: Definition of the immersion ratio

Figure 2 .

 2 Figure 2.14 also highlights the housing shape of an axle. The circular casing possibly leads to particular oil flows, as observed by Jeon [15] on Figure 2.15: with speed, the oil is moved away along the housing and so the immersion level decreases a lot.

Figure 2 . 15 :

 215 Figure 2.15: Behaviour of free surface of lubricant inside the extension housing while the ring gear was rotating forward[START_REF] Jeon | Improving Efficiency in Drive Lines: an Experimental Study on Churning Losses in Hypoid Axle[END_REF]

Figure 2 . 16 :

 216 Figure 2.16: Influence of rotational speed on churning losses (oil C, ℎ 𝑙𝑢𝑏 /𝑅 𝑜 =0.5, 40°C)

Figure 2 . 17 :

 217 Figure 2.17: Influence of immersion on churning losses (oil C, 2000 rpm, 40°C)

Figure 2 . 18 :Figure 2 . 19 :

 218219 Figure 2.18: Influence of oil properties (viscosity and density) on churning losses (1000 rpm, gear 3, ℎ 𝑙𝑢𝑏 /𝑅 𝑜 = 0.46)

Figure 2 .

 2 [START_REF] Handschuh | Thermal behaviour of spiral bevel gears[END_REF] presents churning losses generated by gear 3 as a function of speed. This figure highlights discrepancy with ISO/TR 14179-2 (Mauz) and Jeon formulae, even if results are closer to experimental results than ISO/TR 14179-1.

Figure 2 . 20 :

 220 Figure 2.20: Influence of speed on measured and calculated churning losses (oil C, gear 3, ℎ 𝑙𝑢𝑏 /𝑅 𝑜 =0.5)

Figure 2 . 21 :

 221 Figure 2.21: Influence of oil viscosity on measured and calculated churning losses (oil D, 1000 rpm, gear 3, ℎ 𝑙𝑢𝑏 /𝑅 𝑜 =0.57)

2 )

 2 It is plotted on Figure2.22 (a). Gear 1 and gear 4 obtain different results.

Figure 2 . 22 :

 222 Figure 2.22: Experimental dimensionless churning torque according to (a) Equation (2.1) and (b) Equation (2.4) (oil B, 70°C, ℎ 𝑙𝑢𝑏 /𝑅 𝑜 =0.5)However, they have similar radius, which is the only geometrical feature included in the expression of 𝐶 𝑚 according to Jeon:

Figure 2 .

 2 22 (b) for gear 1 and gear 4 in terms of dimensionless churning torque.Gear width and teeth are identified as non-influential parameters for dimensional analysis. Churning torque still depends on one gear geometrical parameter (outer radius), dynamic parameters 0, gravity), fluid parameters (viscosity, density), and oil sump geometrical parameters (oil volume, immersion level):𝐶 𝑐ℎ𝑢𝑟𝑛𝑖𝑛𝑔 = 𝑓𝑐𝑡[𝑅 𝑜 , 𝛺, 𝑔, 𝜈 𝑙𝑢𝑏 , 𝜌, 𝑉 𝑙𝑢𝑏 , ℎ 𝑙𝑢𝑏 ](2.5)

Figure 2 . 24 :

 224 Figure 2.24: Housing influence on churning (Gear 1, 2000 rpm, oil D, 40°C)

  valid for a single spiral bevel gear surrounded by a close housing and must be applied to both pinion and crown gears.

Figure 2 . 26 :

 226 Figure 2.26: Comparison between experimental results and the new formula for churning losses

Figure 2 . 27 :

 227 Figure 2.27: Comparison between no-load measurements, deduced oil churning values and oil churning calculations according to Jeon formula and new formula

Figure 2 . 29 :

 229 Figure 2.29: Comparison between no-load measurement and standard calculation (with the new churning formula)

Figure 2 . 30 :

 230 Figure 2.30: Deduced gear mesh friction for load test conditions with (..O..) normal oil level and (-Δ-) high oil level

Figure 2 . 31 :

 231 Figure 2.31: Comparison of estimated gear friction power losses according to ISO and Buckingham and measurements with (..O..) normal oil level and (-Δ-) high oil level

Figure 3 . 1 :

 31 Figure 3.1: (a) Face-milling and (b) face-hobbing cutting processes[START_REF] Kolivand | Development of Tooth Contact and Mechanical Efficiency Models for Face-Milled and Face-Hobbed Hypoid and Spiral Bevel Gears[END_REF]

Figure 3 . 3 :Figure 3 . 4 :

 3334 Figure 3.3: Formate® tooth profile and cutting grooves for (a) face-milled and (b) face-hobbed process

Figure 3 . 5 .

 35 It begins with gear

Figure 3 . 6 .

 36 A three-dimensional analysis of the gear set is conducted to determine bending and shear deformation before any gear meshing is considered.

  [START_REF] Karagiannis | On the dynamics of lubricated hypoid gears[END_REF], Mohammadpour et al.[START_REF] Mohammadpour | Elastohydrodynamic lubrication of hypoid gear pairs at high loads[END_REF]). Hertz contact theory is presented in Annex 3.A for both line and elliptical contacts.Potential line of contact are determined with the model of Barday. As illustrated on Figure3.with segments of variable length 𝑑𝑙: both ends of the lines are discretised more finely than line centre. Depending on operating conditions, some segments experience a load and are represented by rectangles on Figure3.9. A line contact is used on each of them: contact half-width 𝑏 𝑐 (mm) is calculated according to Equation (3.15) of Annex 3.A.

Figure 3 . 9 :Figure 3 .

 393 Figure 3.9: Discretisation of a potential line of contact and representation of a contact ellipse Figure 3.10 shows an example of contact line analysis obtained thanks to Barday software. The contact centre is highlighted all along its path through gear mesh. On this illustration, three teeth pairs are potentially in contact at the same time (depending on load), at the three numbered positions.

Figure 3 . 10 :

 310 Figure 3.10: Pinion contact lines obtained by using Barday software

Figure 3 . 12 :

 312 Figure 3.12: General view of velocities for two different lines of contact in tooth tangent plane

Figure 3 Transversale

 3 Figure 3.13: Ellipse entrainment velocities

Figure 3 .

 3 14 illustrates this local contact and the related forces and velocities. A tangent friction effort 𝑑𝑇 2 1 𝑖𝑗 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ (N) is then locally defined as: 𝜇 𝑖𝑗 is the local friction coefficient, 𝑑𝑁 2 1 𝑖𝑗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ is the elementary normal effort (N) and 𝜖 𝑖𝑗 ⃗⃗⃗⃗ is the local orientation vector, which is: 𝑖𝑗 ) is the velocity vector of sliding between pinion and gear teeth.

Figure 3 . 14 :

 314 Figure 3.14: Forces and velocities at local contact

Figure 3 .

 3 Figure 3.15 highlights the difference of contact pattern between hypoid and spiral bevel gears: parallelogram for hypoid, and close to rectangle for spiral bevel. The contact zone increases similarly with load for both gears. However, it is less spread along tooth width for the considered spiral bevel gear design.

Figure 3 . 15 :Figure 3 . 16 :Figure 3 . 17 :Figure 3 . 18 :Figure 3 . 19 :

 315316317318319 Figure 3.15: Contact pattern for different input torques on gear set A and gear set B

Figure 3 .

 3 20 for gear set A and on Figure 3.21 for gear set B.

Figure 3 . 20 :Figure 3 .

 3203 Figure 3.20: Comparison of power loss estimation according to local and global approaches for gear set A at an input speed of (a) 1000 rpm and (b) 2000 rpm

  mean values of global approach.

Figure 3 .

 3 22 presents the comparison of both global and local approaches with measurements.

Figure 3 . 22 :

 322 Figure 3.22: Comparison for gear set A between measurements and gear friction power loss according to local and global approaches with (a) normal oil level and (b) high oil level

Figure 3 . 23 :

 323 Figure 3.23: Stribeck curve -Friction coefficient evolution depending on lubrication regime [58] Different regimes of lubrication are distinguished depending on the thickness of the lubricant film regarding surface roughness, as represented on Figure 3.23:

Figure 3 . 24 :

 324 Figure 3.24: Example of traction curves -Influence of roughness (TVEP, 1.2GPa, 9,425m/s, 80°C). (a) Smooth-smooth surfaces; (b) smooth-rough surfaces; and (c) rough-rough surfaces.[START_REF] Diab | Prediction of Power Losses Due to Tooth Friction in Gears[END_REF]

Figure 3 . 26 :Chapter 3 :Figure 3 . 27 :

 3263327 Figure 3.26: Tooth surface roughness profile of (a) crown gear and (b) pinion

Figure 3 . 29 :

 329 Figure 3.29: Ultra thin film measurement system with a swivelling carriage

Figure 3 . 30 :

 330 Figure 3.30: Traction coefficient measurements at δ=0° and δ=10° for (a) U1=0.5m/s and Load=15N, (b) U1=0.5m/s and Load=20N, (c) U1=1m/s and Load=15N and (d) U1=1m/s and Load=20N

Figure 4 . 1 :

 41 Figure 4.1: 3D-model of axle housing from (a) drive head side and (b) oil sump side

Figure 4 . 2 :

 42 Figure 4.2: Parallelepiped representation of axle housing Length 𝐿 ℎ𝑜𝑢𝑠𝑖𝑛𝑔 = 1450 mm, width 𝑤 ℎ𝑜𝑢𝑠𝑖𝑛𝑔 = 110 mm and height ℎ ℎ𝑜𝑢𝑠𝑖𝑛𝑔 = 350 mm give surfaces reported in Table4.1. Despite the parallelepiped approximation cannot reflect the bigger surface of drive head housing when compared to oil sump housing, total surface presents a good agreement.

Figure 4 . 3 :

 43 Figure 4.3: Oil temperature obtained with the standard formulation for air convection

Figure 4 . 5 :

 45 Figure 4.5: Modelled components of the axle

3 and the thermal network is represented on Figure 4 . 7 .

 47 The power loss sources studied in Chapter 2 are used in the network: seal friction (node n°8), rolling element bearing drag and friction (nodes n°5, 6, 9, 10, 11), oil churning (node n°2) and gear mesh friction (node n°15). The housing is divided in three parts according to Figure4.2: drive head housing, sump housing and lateral housing (nodes n°16, 17, 18).

Figure 4 . 7 :

 47 Figure 4.7: Thermal network of the axle

Figure 4 .

 4 8 (b).

Figure 4 

 4 Concerning the axle itself:

Figure 4 . 8 :

 48 Figure 4.8: Air flow around (a) real axle and (b) modelled axle

Figure 4 .( 4

 44 10 can be done to obtain directly a thermal resistance: 𝑖𝑓 Ψ < 0.68 𝑅 𝑝𝑟𝑜𝑗 = 2 𝜋 1.14 𝑏 𝑡𝑜𝑜𝑡ℎ × 2 𝑍 𝐻 𝑡𝑜𝑜𝑡ℎ √ Ω 𝜃 𝑝𝑟𝑜𝑗 χ 55 -0.6 Ψ)𝑏 𝑡𝑜𝑜𝑡ℎ × 2 𝑍 𝐻 𝑡𝑜𝑜𝑡ℎ √ Ω 𝜃 𝑝𝑟𝑜𝑗 χ (4.18)

Figure 4 . 10 :

 410 Figure 4.10: Evolution of evacuated heat quantity[START_REF] Blok | Transmission de chaleur par projection centrifuge d'huile[END_REF]

Figure 4 . 12 :

 412 Figure 4.12: Bulk temperature difference between standard and thermal network calculation: (a) pinion and (b) crown gear

. 4 .

 4 Their locations are represented on Figure 4.13 and Figure 4.14. The oil temperature sensor is located in the oil drain plug of the axle. The precision of these temperature sensors is +/-0.5 °C.

Figure 4 . 13 : 5 (drive head housing) 4 (lateral housing) 6 (air flow) 3 1FanFigure 4 . 14 :

 4135463414 Figure 4.13: Locations of external temperature sensors

Figure 4 . 15 :

 415 Figure 4.15: Output torque (on each output shaft) and output speed for the different test conditions

Figure 4 . 18 :

 418 Figure 4.18: Stabilised temperatures of oil and rolling element bearings (REBs)

Figure 4 . 19 :

 419 Figure 4.19: Oil cooling according to measurement and corresponding model on (a) load test bench and (b) no-load test bench

Figure 4 .

 4 Figure 4.19 presents oil temperature measurements and model results for the cooling phase on both test benches. A good agreement is observed on the first part of the cooling (<2500 s). Temperature decrease is then slower in the reality and takes more time to reach ambient temperature (>10000 s), but overall behaviour is well reproduced.

Figure 4 . 20 :Figure 4 .

 4204 Figure 4.20: Drive head housing cooling according to measurement and corresponding model on (a) load test bench and (b) no-load test bench

Figure 4 . 21 :

 421 Figure 4.21: Oil and REB temperatures regarding ambient according to measurement and model for no-load step conditions (1 and 2)

Figure 4 .

 4 Figure 4.21 illustrates temperature evolution of oil and rolling element bearings (REB) and compares measurement and model results. Temperature increase rate is slightly different but overall behaviour is close. Stabilised state is then considered for each no-load test condition.Figure 4.22 presents oil temperature comparison: a good agreement is obtained.Figure 4.23 exposes temperature difference between REB and oil: tendencies are similar for measurement and model.

Figure 4 .

 4 Figure 4.21 illustrates temperature evolution of oil and rolling element bearings (REB) and compares measurement and model results. Temperature increase rate is slightly different but overall behaviour is close. Stabilised state is then considered for each no-load test condition.Figure 4.22 presents oil temperature comparison: a good agreement is obtained.Figure 4.23 exposes temperature difference between REB and oil: tendencies are similar for measurement and model.

Figure 4 .

 4 Figure 4.21 illustrates temperature evolution of oil and rolling element bearings (REB) and compares measurement and model results. Temperature increase rate is slightly different but overall behaviour is close. Stabilised state is then considered for each no-load test condition.Figure 4.22 presents oil temperature comparison: a good agreement is obtained.Figure 4.23 exposes temperature difference between REB and oil: tendencies are similar for measurement and model. Still according to Table4.11, power losses independent from load are produced. Figure4.24 compares these power losses according to measurement and model. Calculations underestimate the loss: there is a maximum difference of 26% with measurement point and of 11% with measurement uncertainty. This power loss underestimation explains the temperatures predicted on Figure4.22, which are lower than measured ones.

Figure 4 . 22 :Figure 4 . 24 :

 422424 Figure 4.22: Oil temperature regarding ambient according to measurement and model for (a) normal oil level and (b) high oil level

Figure 4 . 25 :Figure 4 . 26 : 6 C

 4254266 Figure 4.25: Oil and REB temperatures regarding ambient according to measurement and model for load steps conditions(3, 4, 5 and 6)

Figure 4 .

 4 [START_REF] Chittenden | A Theoretical Analysis of The Isothermal Elastohydrodynamic Lubrication of Concentrated Contacts II General Case, with Lubricant Entrainment along either Principal Axis of the Hertzian Contact Ellipse or at some Intermediate Angle[END_REF] compares tests and calculations: results are close and present similar tendencies. This validates the axle thermal model for both temperature prediction and power loss estimation.

Figure 4 . 28 :

 428 Figure 4.28: Power loss estimation and comparison with measurement for (a) normal oil level and (b) high oil level

Figure 4 . 29 :Figure 4 . 31 :

 429431 Figure 4.29: Oil temperature of axle SB1 regarding ambient according to measurement and model for (a) normal oil level and (b) high oil level

Figure 4 . 32 :

 432 Figure 4.32: Total power losses and oil and gear set temperatures for several lubricants

Figure 4 . 34 :

 434 Figure 4.34: Total power losses and oil and gear set temperatures for different air flow speeds

Figure 1 :AnnexesFigure 2 :Figure 3 :

 123 Figure 1: Churning torque depending on rotational speed (oil A, 70°C)

Figure 4 :Figure 5 :Figure 6 :Figure 7 :

 4567 Figure 4: Churning torque depending on rotational speed (oil B, 70°C)

Figure 8 :

 8 Figure 8: Churning torque depending on rotational speed and immersion (Gear 4, oil C, 40°C)

Figure 9 :AnnexesFigure 10 :Figure 11 :Figure 12 :Figure 13 :Figure 14 :Figure 15 :

 9101112131415 Figure 9: Churning torque depending on immersion ratio (Gear 3, oil B, 40°C, 1000 rpm)

Figure 16 :Figure 17 :Figure 18 :

 161718 Figure 16: Churning torque depending on rotational speed and oil temperature (Gear 1, oil D, ℎ 𝑙𝑢𝑏 /𝑅 𝑜 =0.49, with deflector)

Figure 19 :

 19 Figure 19: Scheme from the back side of the pinion (tooth not represented) with the cone angle

Figure 20 :Annexes

 20 Figure 20: Definition of geometrical parameters  𝑅 𝑜 : gear outer radius (m)  𝜑 𝑚𝑎𝑥 : immersion angle of back side (rad)  𝜑 𝑚𝑎𝑥2 : immersion angle of front side (rad)  ℎ * : dry level (m)  ℎ 𝑙𝑢𝑏 : immersion depth of gear (m)

Figure 21 :

 21 Figure 21: Main parameters of peripheral area Finally, the peripheral area 𝐴3 (m²) can be approximated by: 𝐴 3 = (𝜑 𝑚𝑎𝑥 𝑅 𝑜 + 𝜑 𝑚𝑎𝑥2 𝑒 𝑜 ) 𝑏 (2.18) with:  𝑏: tooth width (m)

 2 Reference [ 9 ]

 29 𝜈 1 , 𝜈 2 : Poisson ratio of body 1 or 2  𝐸 1 , 𝐸 2 : elastic modulus of body 1 or 2 (Pa) Then, characteristics of Hertzian contacts can be calculated according to the following formulae. {a} Line contact Maximum Hertz pressure 𝑝 ℎ (Pa) is: 𝑤 1 : load per unit width (N/m)  𝑏 𝑐 : half-width of the contact (m) 𝑅: equivalent radius of curvature (m) {b} Elliptical contact Maximum Hertz pressure 𝑝 ℎ (Pa) is: 𝑝 ℎ = 3 𝑤 2 𝜋 𝑎 𝑏 𝑐 (3.16) Annexes with:  𝑤: load per unit width (N/m)  𝑎: half-width of the contact (m)  𝑏 𝑐 : half-width of the contact (m) 𝑘: ratio of contact semi-axes  𝜀: elliptical integral  𝑅 ̅ : equivalent radius of curvature (m) 𝑅 𝑥 : equivalent radius of curvature in direction 𝑥 (m)  𝑅 𝑦 : equivalent radius of curvature in direction 𝑦 (m) Annex 3.B: Detail of friction coefficient formulae {a} ISO/TR 14179-𝐹: tooth normal force (N)  𝑏: tooth width (mm)  𝑣 Σ : resultant sum velocity (m/s)  𝑅 𝑒𝑞 : equivalent radius of curvature (mm)  𝜂 𝑜𝑖𝑙 : lubricant dynamic viscosity at operating temperature (mPa.s)  𝑅𝑎: arithmetic average roughness (μm)  𝑋 𝐿 : oil lubricant factor 𝑋 𝐿 = 1.0 for mineral oils; 𝑋 𝐿 = 0.8 for polyalfaolefins and esters; 𝑋 𝐿 = 0.75 (6\𝑣 Σm ) 0.2 for polyglycols; 𝑋 𝐿 = 1.3 for phosphoric esters; 𝑋 𝐿 = 1.5 for traction fluids; 𝑅𝑎 = 0.5 (𝑅𝑎 1 + 𝑅𝑎 2 ) (3.22) with:  𝑅𝑎 1,2 : arithmetic average roughness (μm) For hypoid gears geometry, an equivalent helical gear is used. Specific formulae are defined: 𝐹 = 𝐹 𝑛 cos 𝛽 𝑏2 (3.23) 𝐹 𝑛 : tooth normal force, normal section (N)  𝛽 𝑏2 : helix angle at base circle (°)  𝑏 2 : crown gear tooth width (mm)  𝑣 Σ𝑚 : mean resultant sum velocity (m/s)  𝑣 Σ𝑠 : sum velocity in direction of tooth length (m/s)  𝑣 Σℎ the sum velocity in direction of tooth depth (m/s)

( 3 . 35 )

 335 𝜏 𝐿 = 𝐶 0 • exp(𝐶 𝑝 • 𝑝 𝑚𝑜𝑦 ) • exp (𝐶 𝑡 ( 𝜏 𝑁 : shear stress assuming a Newtonian fluid lubrication (Pa)  𝜏 𝐿 : limiting shear stress (Pa)  𝜂(𝑝 𝑚𝑜𝑦 , 𝑇 𝑓𝑙𝑢𝑖𝑑𝑒 ): viscosity depending on pressure and temperature (Pa.s)  𝐶 0 : limiting shear stress parameter for ambient pressure and reference temperature (Pa)  𝐶 𝑝 : limiting shear stress parameter for pressure (Pa -1 )  𝐶 𝑡 : limiting shear stress parameter for temperature (K -1 )  𝑇 𝑓𝑙𝑢𝑖𝑑𝑒 , 𝑇 0 : fluid and reference temperatures (K) {c} Xu formula Reference[START_REF] Xu | Prediction of friction-related power losses of hypoid gear pairs[END_REF] 𝜇 = 𝑒 𝑓(𝑆𝑅,𝑃



  𝑅 1,2 : radius of curvature of respectively pinion and gear (m)Annex 4.A: Heat dissipation through the housingReference: ISO/TR 14179-2[START_REF]Gears -Thermal capacity Part 2 : Thermal load-carrying capacity[END_REF]

  𝛼 𝑜𝑖𝑙 : oil-side heat transfer coefficient (W/(m²K)) Reference value: 𝛼 𝑜𝑖𝑙 = 200 W/(m²K)  𝛼 𝑐𝑎 : air-side heat transfer coefficient (W/(m²K))  𝐴 𝑜𝑖𝑙 : internal housing area (m²)  𝛿 𝑤𝑎𝑙𝑙 : mean housing wall thickness (m)  𝜆 𝑤𝑎𝑙𝑙 : thermal conductivity of housing (W/(m.K)

  : heat transmission coefficient of plate 𝑖 (W/(m²K))  𝐴 𝑖 : surface of plate 𝑖 (m²)  Radiation:



  𝐿: characteristic length (m)  𝑇 𝑐𝑎 : housing temperature (K)  𝑇 𝑎𝑖𝑟 : ambient air temperature (K)Annexes Forced convection:

ℎ 𝑐𝑜𝑛𝑣 = 5 . 6 •

 56 𝐿 -0.34 • 𝑉 0.66 (4.43) For surfaces tangential to the flow: ℎ 𝑐𝑜𝑛𝑣 = 7.6 • 𝐿 -0.37 • 𝑉 0.63 (4.44) with:  𝐿: characteristic length (m)  𝑉: air flow velocity (m/s)



  𝑁𝑢: Nusselt number  𝑘: lubricant thermal conductivity (W/m.K)  𝐿: characteristic dimension (m) Nusselt number is linked to Reynolds number 𝑅𝑒 for forced convection or Grashof number 𝐺𝑟 for free convection and to Prandtl number 𝑃𝑟. Formulae are detailed for each particular heat exchange in the following parts. 𝑔: standard gravity (=9.81 m/s²)  𝛽: volume coefficient of expansion (/K) (=0.0007 for oil; equal to the inverse of fluid temperature for an ideal gas)  Δ𝑇: temperature difference between wall and fluid (K)  𝐿: characteristic dimension (m)  𝜈: fluid kinematic viscosity (m²/s)  𝜇: fluid dynamic viscosity (Pa.s)  𝑐: specific heat (J/kg.K)  𝑘: lubricant thermal conductivity (W/m.K)  𝑉: flow velocity (m/s)  Convection with housing:

52 )For

 52 For forced convection inside the oil bath, a flow is generated by running gears. Nusselt number is then: Reynolds calculation, the fluid velocity to take into account is due to running gears:𝑉 = Ω 𝑅 𝑝 (4.55) with: Ω: gear rotational speed (rad/s)  𝑅 𝑝 : gear pitch radius (m)

Figure 22 :

 22 Figure 22: Definition of centrifugal projection angleWhen gears are rotating, forced convection with oil takes place in a particular way: centrifugal projection of oil along gear teeth flanks evacuates the major amount of heat. Convection with oil bath can be also considered, but this contributes a few.

Figure 23 :

 23 Figure 23: Evolution of evacuated heat quantity [65]For the complete gear, a global heat quantity is then defined as:𝑄 = 𝑞 𝑏 𝑡𝑜𝑜𝑡ℎ × 2 𝑍 2 𝜋 Ω ⁄

Figure 24 :Figure 25 :

 2425 Figure 24: Heat conduction through a cylindrical solid with various widths

 2 𝑘 2 

 22 𝑥: contact width depending on pressure (m) varies from 3.10 -3 to 6.10 -3 m for metal-metal contact  𝑘 ̅ : equivalent thermal conductivity (W/m.K) Calculated according to conductivities of contacting material: 𝑘 ̅ = 𝐴: apparent contact area (m²)

  Method principle ....................................................................................................................................... 2.3.2. Thermal network of a truck axle ............................................................................................................... 2.3.3. Thermal resistances................................................................................................................................... Comparison between ISO technical reports and thermal network model ..................................... 2.4.1. Estimation of thermal dissipation ............................................................................................................. 2.4.2. Calculation of bulk temperature ............................................................................................................... 2.4.3. Conclusion on thermal network application ............................................................................................. 3. INVESTIGATIONS ON AXLE THERMAL BEHAVIOUR ................................................................................................ .3.1. Axle cooling ............................................................................................................................................... 3.3.2. No-load step .............................................................................................................................................. 3.3.3. Load steps ................................................................................................................................................. 3.4. Extending model validation ........................................................................................................... 4. EXPLOITATION OF THE MODEL........................................................................................................................ 4.1. Oil properties ................................................................................................................................. 4.2. Gear surface roughness ................................................................................................................. 4.3. Air flow ..........................................................................................................................................

	2.4.

... 2.1. Considered drive axle ...................................................................................................................... 2.2. Efficiency tests ................................................................................................................................. 1. INTRODUCTION ........................................................................................................................................... 2. THERMAL MODEL OF THE AXLE ....................................................................................................................... 2.1. Additional data for axle thermal behaviour .................................................................................. 2.2. Isothermal approach ..................................................................................................................... 2.3. Thermal network approach ........................................................................................................... 2.3.1. 3.1. Test definition ................................................................................................................................ 3.2. Experimental results ...................................................................................................................... 3.3. Validation of axle thermal network ............................................................................................... 3

  ..... {a} Buckingham formula ............................................................................................................................. {b} ISO/TR 14179-2 ..................................................................................................................................... ANNEX 2.A: EXPERIMENTAL RESULTS ON OIL CHURNING TEST RIG .................................................................................. ANNEX 2.B: CALCULATION OF IMMERSED SURFACE AREA.............................................................................................

{a} Tests performed by Laruelle

[START_REF] Laruelle | Study of the thermal behavior of a gear unit[END_REF]

.......................................................................................................... {b} Complementary tests on gear 3 and oil D ............................................................................................. {c} Tests with a deflector ............................................................................................................................ {a} Front and back areas ............................................................................................................................ {b} Peripheral area......................................................................................................................................

  1)La surface mouillée 𝑆 𝑚 considère l'enveloppe extérieure de l'engrenage. Une modification est aussi apportée au calcul du couple adimensionné de barbotage 𝐶 𝑚 afin de prendre en compte les deux régimes d'écoulement d'huile identifiés. Ceux-ci se distinguent grâce au nombre de Reynolds :

	Pour Re ≤ 20 000 :	𝐶 𝑚 = 1.45 ( ℎ 𝑙𝑢𝑏 𝑅 𝑜	) 0.15	( 𝑉 𝑙𝑢𝑏 𝑅 𝑜 3 ) -0.20	𝑅𝑒 -0.25 𝐹𝑟 -0.53	(0.2)
	Pour Re > 20 000 :	𝐶 𝑚 = 0.12 ( ℎ 𝑙𝑢𝑏 𝑅 𝑜	) 0.15	( 𝑉 𝑙𝑢𝑏 𝑅 𝑜 3 ) -0.20	𝐹𝑟 -0.53	(0.3)

Cette formule peut enfin être appliquée au pont de camion. D'après Jeon, un couple d'engrenages génère plus du double de la perte de la couronne seule, à cause d'un changement du flux d'huile. Afin de prendre ce phénomène en considération, le niveau d'huile est augmenté de 30% pour un couple conique. La Figure

0

.3 présente les résultats obtenus, qui concordent bien avec les mesures.

  Des profilométries optiques sont réalisées sur le pignon et la couronne : il apparaît que leurs surfaces ont des rugosités différentes. Des essais comme ceux effectués par Diab[START_REF] Diab | Prediction of Power Losses Due to Tooth Friction in Gears[END_REF] permettraient de mieux caractériser le frottement de ce type de surface. L'influence des angles d'orientation 𝛿 et 𝛿 𝑒 sur le coefficient de frottement est enfin étudiée, car ils caractérisent les vitesses de surfaces et d'entraînement des spiro-coniques et des hypoïdes. La plupart des études menées sur ces angles d'orientation dans les contacts elliptiques s'intéressent uniquement à la lubrification élastohydrodynamique et à l'épaisseur de film de lubrifiant ([28]-[START_REF] Pu | A Theoretical Analysis of the Mixed Elastohydrodynamic Lubrication in Elliptical Contacts With an Arbitrary Entrainment Angle[END_REF]). Par des essais ou des modèles empiriques, il apparaît que l'amplitude et l'orientation de la vitesse d'entraînement modifient la formation du film. La diminution de l'épaisseur de film pourrait être causée par un plus grand taux de cisaillement dû à un frottement important. Cependant, aucune étude ne s'est penchée sur cet aspect. Carli et al.[START_REF] Carli | Thermal point contact EHL analysis of rolling/sliding contacts with experimental comparison showing anomalous film shapes[END_REF] étudient le frottement dans les contacts circulaires avec un film de lubrifiant anormal. Comme ces anomalies sont dues à une vitesse d'entraînement orientée, le modèle établi pourrait être transposé aux contacts elliptiques. Ce phénomène est aussi comparable à celui

	𝑀 𝑖 𝑐 𝑖	𝑑𝑇 𝑖 𝑑𝑡	= 𝑄 𝑖 +	∑ 𝑛𝑏 𝑗=1 𝑎𝑛𝑑 𝑗≠𝑖	𝑇 𝑗 -𝑇 𝑖 𝑅 𝑡ℎ (𝑖, 𝑗)	(0.8)

observé dans certains roulements, entre le collet du rouleau et la bague du roulement : un contact circulaire pivotant. Doki-Thonon

[START_REF] Doki-Thonon | Thermal effects in elastohydrodynamic spinning circular contacts[END_REF]

analyse le frottement dans ce contact, mais la similitude avec l'ellipse pivotante des hypoïdes n'est pas facile à valider.

Des essais préliminaires sont réalisés pour mesurer le coefficient de frottement en fonction des angles d'orientation. Deux positions sont testées (δ=0° et δ=10°), ainsi que différentes conditions de charge et de vitesse de surface. Le coefficient de frottement obtenu est toujours plus élevé pour δ=10°, d'autant plus que la vitesse de surface est faible. La charge a le même effet quel que soit l'angle. Ainsi, une modification de l'angle d'orientation modifie le coefficient de frottement et donc les pertes.

Chapitre 4 -Modèle thermique du pont

Modèle thermique du pont

Le modèle thermique se base sur les formules de pertes présentées dans les chapitres précédents et calcule également les échanges thermiques. Le carter est simplifié par un parallélépipède rectangle décomposé en trois parties : carter latéral, côté tête et côté cuve. Dans un premier temps, l'approche isotherme proposée par le rapport technique ISO/TR 14179-2

[START_REF]Gears -Thermal capacity Part 2 : Thermal load-carrying capacity[END_REF]

est appliquée au pont de camion. La température du pont, qui correspond à celle de la cuve d'huile, est ainsi calculée en fonction des échanges thermiques avec l'environnement : rayonnement et convection forcée entre l'air et le carter, convection forcée avec l'huile. Il apparaît que la température calculée est plus basse que celle mesurée à cause d'une surestimation des échanges thermiques. Celle-ci peut être due à une mauvaise prise en compte par l'ISO du flux d'air autour du carter. Dans un second temps, la méthode des réseaux thermiques sert à modéliser le pont de camion. Basée sur une analogie électrique, elle reprend la loi d'Ohm pour lier l'écart de température Δ𝑇 au flux de chaleur 𝑄 par une résistance thermique 𝑅 𝑡ℎ :

Δ𝑇 = 𝑅 𝑡ℎ × 𝑄 (0.7)

La chaleur générée par les pertes est injectée dans le réseau au noeud 𝑖 correspondant. L'évolution de chaque température de noeud 𝑇 𝑖 est ensuite estimée selon la première loi de la thermodynamique :

  Grâce à ces résistances, différents fonctionnements du camion sont modélisés. Si le véhicule roule, l'air environnant est en convection forcée avec le carter: le flux arrive côté tête et est considéré différemment selon les faces du carter. L'huile est aussi en convection forcée avec les éléments internes du pont, car les engrenages et les roulements sont en rotation. Si le véhicule est à l'arrêt, la convection est libre à l'intérieur et à l'extérieur.Le réseau thermique obtient des résultats plus proches des mesures grâce aux flux distincts selon les faces du carter et à des formules plus adaptées aux échanges thermiques. Un autre intérêt du réseau thermique est de connaître les températures de masse des éléments du pont, sans nécessiter un modèle Eléments-Finis complexe. Le rapport technique ISO/TR 13989-2[START_REF]Calculation of scuffing load capacity of cylindrical, bevel and hypoid gears -Part 2: Integral temperature method[END_REF] propose lui aussi de calculer la température de masse du couple conique, avec quelques spécificités : 1) Seul le couple d'engrenages et le lubrifiant sont pris en compte ; 2) La température d'huile est requise pour le calcul, cela implique une mesure de température, qui n'est en revanche pas nécessaire pour le réseau thermique ; 3) Seule une température globale du couple conique est calculée : il est considéré que le Le modèle obtient une décroissance en température similaire aux essais pour l'huile et le carter de tête. Les autres phases peuvent ensuite être étudiées : le palier sans charge (0 kW) et ceux sous charge (60 kW, 120 kW). L'évolution des températures a la même tendance d'après le modèle et les mesures. Les températures stabilisées calculées selon le modèle sont proches des mesures, comme le montre la Figure0.11 pour les roulements et la Figure 0.12 pour l'huile. Les pertes sont bien estimées aussi par le modèle, comme présenté sur la Figure 0.13. Un second pont avec un couple spiro-conique est testé et modélisé. Les résultats obtenus sont cohérents : le modèle peut servir d'outil de design.

pignon et la couronne sont à la même température, alors que les mesures réalisées par Xu et al.

[START_REF] Xu | Thermal Mapping of an Automotive Rear Drive Axle[END_REF]

suggèrent le contraire. Des résultats similaires sont obtenus par les deux approches pour la température de masse de la couronne. En revanche, une différence importante est observée sur celle du pignon : le réseau thermique estime jusqu'à 10°C supplémentaires, ce qui peut être critique pour du grippage. Le pignon est plus chaud que la couronne, ce qui est en accord avec les mesures de Xu et al.

[START_REF] Xu | Thermal Mapping of an Automotive Rear Drive Axle[END_REF]

. Cela s'explique par un nombre de cycles et une inertie différents, mais aussi par un échange plus important entre l'huile et la couronne, ce qui la refroidit mieux. De plus, les roulements de l'arbre d'entrée, proches du pignon, sont très chauds.

Investigations sur le comportement thermique du pont

Le réseau thermique du pont doit être validé à l'aide d'une campagne d'essais spécifiques. En effet, les précédentes mesures ne sont pas suffisantes car seule la température d'huile était enregistrée. De nouveaux essais sont donc réalisés : des bancs similaires à ceux du Chapitre 2 sont utilisés et une dizaine de thermocouples sont placés en différents points du pont (huile, roulements, carter, environnement). Trois paliers de puissance (0 kW, 60 kW, 120 kW) avec chacun deux vitesses de fonctionnement sont réalisés, ainsi qu'une phase de refroidissement. Une ventilation constante est maintenue durant la totalité des essais pour caractériser la convection forcée avec l'air.

Des résultats en termes de perte globale du pont et de températures des composants sont obtenus. Les pertes de puissance évoluent comme attendu : elles augmentent avec la vitesse et le niveau d'huile. Il faut noter que la température d'huile est basse (<60°C), ce qui favorise le rendement du couple conique tout en augmentant les pertes des roulements. Concernant la température des composants, les différentes zones du carter sont à des températures différentes, ce qui soutient une décomposition du carter en plusieurs éléments. Les roulements de l'arbre d'entrée sont plus chauds que l'huile, tandis que les roulements du différentiel sont plus froids. Ceci est principalement dû à la faible vitesse de rotation des arbres de roue.

Grâce aux essais, il est possible de valider les échanges thermiques et les pertes de puissance. Il s'agit d'abord d'estimer correctement les échanges avec l'environnement, car ils conditionnent tout le reste, températures et pertes. La phase de refroidissement permet de caractériser la convection forcée avec l'air. Figure 0.11: Température des roulements comparée à l'huile d'après les mesures et le modèle

  comparison is then conducted on hypoid efficiency formulae with friction coefficient at 0.05, as proposed by Coleman. Efficiency is calculated for an input torque increasing from 200 N.m to 2000 N.m. Main characteristics of the hypoid gear pair considered are given in Table1.2. Kakavas et al.[START_REF] Kakavas | Hypoid gear vehicle axle efficiency[END_REF] and Kolekar et al.[START_REF] Kolekar | The efficiency of a hypoid axle -a thermally coupled lubrication model[END_REF] used an approach similar to the ISO technical report or to Buckingham with average values too.

	Table 1.2: Gear set characteristics
				Pinion	Crown gear
	Number of teeth			8	37
	Outer diameter (mm)		134.6	428
	Face angle (°)			15.18	74.57
	Pressure angle (°)			21.35	/
	Spiral angle (°)			45	34.42
	Offset (mm)			35	/
	100
		98
	Efficiency (%)	94 96 92		Buckingham Coleman ISO/TR 14179-2
		90
		88
		0	500	1000 1500 2000
			Input torque (N.m)
	Figure 1.16: Hypoid gear efficiency according to Buckingham, Coleman and ISO/TR 14179-2 formulae
	(fixed friction coefficient = 0.05)
	Although their behaviour is similar, Buckingham and ISO technical report formulae show great
	discrepancy. This can be explain by the fixed friction coefficient: for Buckingham, it should rely only on
	sliding velocity, while load, oil viscosity and surface roughness also have an influence for ISO technical
	report.

Results are plotted on Figure

1

.16 and are valid whatever the speed. Buckingham and ISO/TR 14179-2 formulae give a constant efficiency, while Coleman formula increases with load. This variation of efficiency with load is not in agreement with the previous analysis, as the friction coefficient is fixed. Coleman formula is then discarded.

Table 2 .

 2

	1: Axle gear set characteristics
		Pinion (A)	Crown gear (B)
	Number of teeth		8	37
	Outer diameter (mm)	134.6	428
	Width (mm)		85	62
	Face width (mm)	74.4	63
	Face angle (°)	15.18	74.57
	Pressure angle (°)	21.35	/
	Spiral angle (°)		45	34.42
	Hand of spiral		LH	RH
	Offset (mm)		35	/
	Table 2.2: Rolling Element Bearings characteristics
		Tail REB	Head REB	Pilot REB	Differential REB
		(1)	(2)	(3)	(4) and (5)
	Bearing designation	31312	31314	NJK308	32218F
	Roller type	tapered	tapered	cylindrical	tapered
	Inner diameter (mm)	60	70	40	90
	Outer diameter (mm)	130	150	90	160
	Width (mm)	33.5	38	23	42.5
	Preload (kN)	20	0	0	4 / 0

Table 2 .

 2

	3: Oil characteristics
	Kinematic viscosity @ 40°C (cSt)	120
	Kinematic viscosity @ 100°C (cSt)	15.9
	Viscosity Index	140
	Density @ 15°C	0.860
	Oil type	Synthetic

Table 2 .

 2

	Condition	A	B	C
	Vehicle speed (km/h)		80
	Power input (kW)	60	120	240

4: Conditions of load-dependent losses tests

Table 2 . 5 :

 25 Factors used for Harris formula

		Tail REB	Head REB	Pilot REB	Differential REB
	f0	3	3	5	8
	f1	0.0004	0.0004	0.00055	0.0004
	Measurements of the global power loss are also represented on Figure 2.6 to give an order of
	magnitude of the power loss source contribution.
	In order to discriminate among the solutions, a comparison is made between experimental and
	calculation results for different tests. The no-load test at 80 km/h is used as a reference: under these
	conditions, the estimated bearing losses are around 550 W for both models. As rotational speed, oil
	level and temperature are the same for the loaded cases, the load-independent losses should be
	identical. This includes:
		-550 W of REBs losses (calculated)
		-100 W of seal friction (calculated in Part 2.3.1)
		-1100 W of oil churning (deduced from Figure 2.6)

Table 2 . 6 :

 26 Tested gears characteristicsIt can be noticed that the outer diameter is used here instead of the pitch diameter. The outer diameter is also representative of the gear general shape, as represented on Figure2.12, and easy to know. By the way, outer diameters and pitch diameters are almost similar on Jeon's tested gears (respectively 213.2 and 210 mm for the first crown gear, and 222 and 220 mm for the second).Various oils are available to investigate the influence of oil properties. Their characteristics are gathered in Table2.7.

		Gear 1	Gear 2	Gear 3	Gear 4	Cone 1	Cone 2
	Number of teeth	41	37	41	37	/	/
	Outer diameter (mm)	157	130	188	154	157	160
	Width (mm)	22	27	30	30	22	27
	Face width (mm)	27	24,5	32	27,5	27	24,5
	Face angle (°)	72,4	58,1	72,4	58,1	72,4	58,1
	Pressure angle (°)	20	20	20	20	/	/
	Spiral angle (°)	35	35	35	35	/	/
	Hand of spiral	LH	LH	LH	LH	/	/

Figure 2.12: Definition of the gear main geometrical parameters

Table 2 .

 2

	8: Peripheral speeds on truck application and on test rig
	TRUCK APPLICATION Vehicle speed (km/h)	20	50	80	100
	Peripheral speed (m/s)	Crown gear Pinion gear	2.5 3.6	6.2 9.0	9.9 14.4	12.4 18.0
	TEST RIG	Input speed (rpm)	1000	1500	2000
		Gear 1	8.2	12.3	16.4
	Peripheral speed (m/s)	Gear 2 Gear 3	6.8 9.8	10.2 14.8	13.6 19.7
		Gear 4	8.1	12.1	16.1

Table 3 .

 3

	1: Characteristics of gear sets
		Gear set A	Gear set B
		Pinion	Crown gear	Pinion	Crown gear
	Number of teeth	8	37	8	37
	Outer diameter (mm)	134.6	428	135.5	478
	Width (mm)	85	62	77.7	60.7
	Face width (mm)	74.4	63	75.5	75.5
	Face angle (°)	15.11	74.34	16.42	79.33
	Pressure angle (°)	21.21	/	20	/
	Spiral angle (°)	45	34.25	38	38
	Hand of spiral	LH	RH	LH	RH
	Offset (mm)	35	/	0	/

Table 3 . 2 :

 32 Estimation difference between local and global approaches

	Input torque (N.m)	280	560	1120
	Power loss difference (%)	Gear set A Gear set B	17.7 42.8	18.3 36.4	18.7 30.5

Table 3 .

 3 3: Difference of mean sliding velocity between local and global approaches

	Input torque (N.m)	280	560	1120
	Sliding velocity	Gear set A	19.2	15.5	16.9
	difference (%)	Gear set B	48.0	38.2	33.3

Table 3 .

 3 Based on the arithmetic average roughness 𝑅𝑎, mean friction coefficient is calculated according to formulae (3.8), (3.9), (3.10), and(3.11). For ISO technical report formula, global approach is still used.

	4: Gear surface roughness after various surface finishing
	Condition	Surface finishing	Ra (µm)	Rz (µm)
	a	Grinding (medium)	> 0.8	> 3
	b	Reference	0.2 -0.3	1.5 -2.5
	c	Lapping, grinding (fine)	0.2	3 -4
	d	Superfinishing	0.05 -0.1	1 -2

Table 3 .

 3 5: Velocity and entrainment orientation angles for gear sets A and B

	Gear set A	Gear set B
	Input speed
	(rpm)

δ (°) δe (°) δ (°) δe (°)

min mean max min mean max min mean max min mean max 500

		0	6	23	21	43	76	0	11	15	16	33	68
	1000	0	4	14	22	41	70	0	6	10	19	35	65
	1500	0	3	10	22	41	68	0	5	7	21	37	64
	2000	0	2	8	22	41	67	0	4	6	21	37	64

Table 4 . 1 :

 41 Housing surfaces according to 3D-model and parallelepiped approximation (𝐴 𝑐𝑎 )

			Surface (mm²)
		3D-model	Parallelepiped	Difference
	Drive head housing	583,231	507,500	-13.0%
	Oil sump housing	442,410	507,500	14.7%
	Lateral housing	382,892	396,000	3.4%
	Total	1,408,533	1,411,000	0.2%

Table 4 .

 4 4: List of temperature sensors

	Sensor number	Monitored component	Sensor number	Monitored component
	1	Ambient air	6	Lateral housing
	2	Oil	7	Head rolling element bearing
	3	Air flow	8	Tail rolling element bearing
	4	Drive head housing	9	Pilot rolling element bearing
	5	Sump housing	10	Differential rolling element bearing

Table 4 .

 4 Even during the cooling after the tests, the fan is blowing and temperatures are recorded. Air flow is kept constant during the whole procedure in order to characterise the forced convection between air and housing. It is characterised thanks to an anemometer: its speed is of 22 m/s on load test bench and of 12 m/s on no-load test bench.The axle and the oil used for thermal tests are the same as the one used for the efficiency tests of Chapter 2. Specification of this axle and oil volumes are presented inTable 4.6.

	5: Test conditions
	Test condition	No-load step 1 2	3	Load steps 4 5	6
	Input power (kW)	~ 0		60		120
	Vehicle speed (km/h)	50	80	50	80	50	80
	A fan prevents the axle to overheat and helps a quicker temperature stabilisation by inducing forced
	air convection.

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2018LYSEI019/these.pdf © [C. Fossier], [2018], INSA Lyon, tous droits réservés 148 Thermal model of the axle

Table 4 .

 4

	6: Specification of tested axle
		Axle	H1
		Ratio	4.625
		Type	Hypoid
	Gear set	Cutting method	Face-hobbed
		Number of teeth	8 x 37
	Oil volume (L)	Normal level High level	14 25

Table 4 .

 4

		7: Efficiency measurements at normal oil level
			INPUT			OUTPUT
	Cond.	Speed Torque Power Speed Torque Power Power loss Efficiency (rpm) (N.m) (kW) (rpm) (N.m) (kW) (kW) (%)
	1	1278	16.5	2.20	/	/	/	2.20	/
	2	2045	17.7	3.79	/	/	/	3.79	/
	3	1276	445	59.5	276	1969	56.9	2.6	95.6
	4	2044	276	59.0	443	1192	55.2	3.7	93.7
	5	1276	899	120.1	276	4032	116.5	3.5	97.1
	6	2044	557	119.2	443	2481	115.0	4.2	96.5
	Table 4.8: Efficiency measurements at high oil level
			INPUT			OUTPUT
	Cond.	Speed Torque Power Speed Torque Power Power loss Efficiency (rpm) (N.m) (kW) (rpm) (N.m) (kW) (kW) (%)
	1	1278	19.0	2.54	/	/	/	2.54	/
	2	2045	19.5	4.18	/	/	/	4.18	/
	3	1276	445	59.4	276	1956	56.6	2.9	95.2
	4	2044	276	59.0	443	1172	54.3	4.7	92.1
	5	1276	897	119.8	276	4017	116.2	3.7	96.9
	6	2044	556	119.1	443	2467	114.3	4.8	96.0
	Table 4.9: Temperature measurements regarding ambient temperature (°C) at normal oil level
	Cond	Oil	Head REB	Tail REB	Pilot REB	Differential REB	Drive head housing	Sump housing	Lateral housing
	1	17.4	20.3	20.9	18.0	17.2	14.9	12.1	9.1
	2	28.1	30.4	31.2	30.0	28.6	24.2	20.1	16.7
	3	22.7	24.9	24.6	23.9	21.7	18.6	14.9	12.8
	4	30.6	32.9	32.8	32.9	30.5	25.3	20.4	16.9
	5	31.4	33.3	32.8	32.7	29.5	25.7	20.5	16.7
	6	35.7	37.6	37.3	38.0	34.6	29.8	24.7	21.5

Table 4 .

 4 10: Temperature measurements regarding ambient temperature (°C) at high oil level

	Cond.	Oil	Head REB	Tail REB	Pilot REB	Differential REB	Drive head housing	Sump housing	Lateral housing
	1	18.5	19.9	20.6	19.0	18.2	15.9	12.3	11.6
	2	28.9	30.8	31.3	30.2	28.3	25.0	20.1	18.9
	3	22.4	23.3	23.3	23.2	21.4	18.4	14.8	13.7
	4	31.0	32.8	32.7	32.4	29.9	26.0	21.0	20.0
	5	29.2	29.9	29.8	30.2	27.3	23.5	18.5	17.0
	6	36.3	37.9	37.5	37.8	34.5	30.3	24.1	22.4

Table 4 .

 4 12: Comparison of tested axle characteristics

		Axle	H1	SB1
		Ratio	4.625	4.625
		Type	Hypoid	Spiral bevel
	Gear set	Cutting method	Face-hobbed	Face-milled
		Number of teeth	8 x 37	8 x 37
	Oil volume	Normal level	14	10
	(L)	High level	25	19.5
	Experimental and calculated results are presented hereafter. Figure 4.29 and Figure 4.30 expose oil
	and REB temperature comparison: a good agreement is obtained. A point of discrepancy is solely
	observed on pilot REB, which is colder than oil and even than differential REB. It is unlikely that this
	bearing produces almost no losses, so an instrumentation issue is considered. Power loss
	measurements and estimations are compared successfully on Figure 4.31.

Table 4 .

 4 13: Oil properties of considered lubricants

	Oil	Type	Density	Viscosity at 40°C	Viscosity at 100°C
				(cSt)	(cSt)
	Mineral	Mineral	0.895	298	25.2
	Reference	Synthetic	0.86	120	15.9
	Eco	Synthetic	0.869	103	15.3

Table 4 .

 4 14: Gear surface roughness and friction coefficient according to ISO/TR 14179-2

	Surface finishing	Ra (µm)	Friction coefficient	Friction coefficient
			(Figure 4.33)	at 80°C
	Grinding (medium)	0.8	0.03	0.026
	Reference	0.3	0.023	0.023
	Lapping	0.2	0.021	0.020

Friction loss factor of spiral bevel gear 𝝓

  𝜙 𝑠𝑝𝑖𝑟𝑎𝑙 𝑏𝑒𝑣𝑒𝑙 : friction loss factor of spiral bevel gear  𝜙 𝑤𝑜𝑟𝑚 : friction loss factor of worm gear 𝒔𝒑𝒊𝒓𝒂𝒍 𝒃𝒆𝒗𝒆𝒍 :  𝑚: gear ratio of equivalent helical gears  𝛽 𝑎 , 𝛽 𝑟 : arc of approach and recess of equivalent helical gears  𝑓 𝑠𝑏 : average coefficient of friction for spiral bevel gear from experiments 𝑅 𝑝,𝑔 : pitch radius of hypoid pinion and gear at large ends (in.)  𝑅 𝑝,𝑔 ′ : pitch radius of equivalent bevel pinion and gear (in.) 𝑅 ℎ𝑝,ℎ𝑔 : radius of base cylinder of pinion and gear hypoid (in.)  𝑅 𝑣𝑝,𝑣𝑔 : pitch radius of equivalent helical pinion and gear (in.)  𝛾 𝑝,𝑔 : generatrix angle of pinion and gear hypoidAnd 𝑉 is the pitch-line velocity (ft/min):

	Where:
	𝑅 𝑝 = √𝑅 𝑝 ′ 2 + 𝑅 ℎ𝑝	2	(1.39)
	𝑅 𝑔 = √𝑅 𝑔 ′ 2 + 𝑅 ℎ𝑔	2	(1.40)
	𝑅 𝑣𝑝 =	cos 𝛾 𝑝 𝑅 𝑝 ′			(1.41)
	𝑅 𝑣𝑔 =	cos 𝛾 𝑔 𝑅 𝑔 ′			(1.42)
	𝑚 =	𝑅 𝑣𝑝 𝑅 𝑣𝑔				(1.43)
	with:
	with:  𝑉 = 0.5236 × 𝑅 𝑚 𝑛 with:  𝑅 𝑚 : mean pitch radius of hypoid pinion (in.)  𝜙 𝑠𝑝𝑖𝑟𝑎𝑙 𝑏𝑒𝑣𝑒𝑙 = cos 𝜙 cos 𝜙 𝑛 + cos 𝜓 [ 1 + ( 1 𝑚 ) (𝛽 𝑎 + 𝛽 𝑟 ) cos 𝜓  𝑛: rpm of driving pinion (rpm)	] ( 𝑓 𝑠𝑏 2	) (𝛽 𝑎	2 + 𝛽 𝑟	2 )	35) (1.44) (1.36)
	with:
	Where:
	tan 𝜙 =	tan 𝜙 𝑛 cos 𝜓			(1.37)
	𝑓 𝑠𝑏 is determined from a table based on an average sliding velocity 𝑉 𝑠 ′ (ft/min):
	𝑉 𝑠 ′ = 𝑉 cos 𝜙 (1 +	𝑅 𝑣𝑝 𝑅 𝑣𝑔	) (	𝛽 𝑎 + 𝛽 𝑟 4	)	(1.38)
							Annexes

 𝜙: pressure angle in plane of rotation of equivalent helical gear (°)  𝜙 𝑛 : normal pressure angle at middle of gear face (°)  𝜓: spiral angle of gear at middle tooth face (°)

Friction loss factor of worm gear 𝝓 𝒘𝒐𝒓𝒎 :

	𝜙 𝑤𝑜𝑟𝑚 𝑔𝑒𝑎𝑟 =	2 × 𝑓 𝑤 cos 𝜙 𝑛 sin 2𝛾 𝑝 + 2 × 𝑓 𝑤	(1.45)
	with:



  𝑓 𝑤 : average coefficient of friction for worm gear from experiments  𝜙 𝑛 : normal pressure angle at middle of gear face (°)  𝛾 𝑝 : angle of generatrix of pinion (°) 𝑓 𝑤 is determined from a table based on an average sliding velocity 𝑉 𝑠 (ft/min):An equivalent helical gear substitutes for the hypoid one for gear mesh calculation. Specifically for hypoid gears, the ISO Technical Report defines:

	𝑉 𝑠 = Hypoid gear mesh power loss is defined as: 0.5236 × 𝑅 𝑏𝑝 𝑛 sin 𝛾 𝑝 with: 𝑃 ℎ𝑦𝑝𝑜𝑖𝑑 = 𝐹 𝑛 𝜈 𝑔𝑚 𝜇 𝑚 with:  𝐹 𝑛 : tooth normal load (N)  𝜈 𝑔𝑚 : mean sliding speed (m/s)  𝜇 𝑚 : mean friction coefficient The following indices stand for:  1: pinion  2: crown gear Mean sliding speed 𝒗 𝒈𝒎 : 𝑣 𝑔𝑚 = 𝑣 𝑔𝑠 + (𝑣 𝑔𝛾1 -𝑣 𝑔𝑠 ) 2 + (𝑣 𝑔𝛾2 -𝑣 𝑔𝑠 ) 2 2(𝑣 𝑔𝛾1 + 𝑣 𝑔𝛾2 -2 × 𝑣 𝑔𝑠 ) with:  𝑣 𝑔𝑠 : helical speed (m/s)  𝑣 𝑔𝛾1,2 : total surface speed at tooth tip (m/s) Mean friction coefficient 𝝁 𝒎 : 𝜇 𝑚 = 0.048 ( 𝐹 𝑛 cos 𝛽 𝑏2 (0.85 × 𝑏 2 ) ⁄ 𝑣 Σm 𝑅 𝑒𝑞,𝑛 ) 0.2 𝜂 𝑜𝑖𝑙 -0.05 𝑅𝑎 0.25 𝑋 𝐿 with:  𝛽 𝑏2 : helix angle at base circle (°)  𝑏 2 : tooth width (mm)  𝑣 Σ𝑚 : mean resultant sum velocity (m/s)  𝑅 𝑒𝑞,𝑛 : equivalent radius of curvature, normal section (mm)  𝜂 𝑜𝑖𝑙 : dynamic viscosity of oil at operating temperature (mPa.s)  𝑋 𝐿 : oil lubricant factor 𝑋 𝐿 = 1.0 for mineral oils; 𝑋 𝐿 = 0.8 for polyalfaolefins and esters; 𝑋 𝐿 = 0.75 (6\𝑣 Σm ) 0.2 for polyglycols; 𝑋 𝐿 = 1.3 for phosphoric esters; 𝑋 𝐿 = 1.5 for traction fluids;  𝑅𝑎 1,2 : arithmetic average roughness 𝑣 Σ𝑚 = √𝑣 Σ𝑠 2 + 𝑣 Σℎ 2 with:  𝑣 Σ𝑠 : sum velocity in direction of tooth length (m/s)  𝑣 Σℎ : sum velocity in direction of tooth depth (m/s) 𝑣 Σ𝑠 = 𝑣 𝑡1 (sin 𝛽 𝑚1 + sin 𝛽 𝑚2 cos 𝛽 𝑚1 cos 𝛽 𝑚2 ) 𝑣 Σℎ = 2 × 𝑣 𝑡1 cos 𝛽 𝑚1 sin 𝛼 𝑛 with:  𝛽 𝑚1,2 : helix angle at medium circle (°)  𝑣 𝑡1 : peripheral speed at pitch circle (m/s)  𝛼 𝑛 : normal pressure angle (°)  𝑅 {b} ISO/TR 14179-2 Reference [9] 𝑅𝑎 = 0.5 (𝑅𝑎 1 + 𝑅𝑎 2 ) with: Annex 2.	(1.46) (1.47) (1.48) (1.49) (1.51) (1.52) (1.53) (1.50)
		Annexes

𝑏𝑝 : radius of base cylinder of pinion hypoid (in.)  𝑛: number of rpm of driving pinion (rpm)

A: Experimental results on oil churning test rig

Table 0 . 1 :

 01 Tested gears characteristics

		Gear 1	Gear 2	Gear 3	Gear 4	Cone 1	Cone 2
	Number of teeth	41	37	41	37	/	/
	Outer diameter (mm)	157	130	188	154	157	160
	Width (mm)	22	27	30	30	22	27
	Face width (mm)	27	24,5	32	27,5	27	24,5
	Face angle (°)	72.4	58.1	72.4	58.1	72.4	58.1
	Pressure angle (°)	20	20	20	20	/	/
	Spiral angle (°)	35	35	35	35	/	/
	Hand of spiral	LH	LH	LH	LH	/	/
	Table 0.2: Tested oils characteristics
			Oil A		Oil B	Oil C	Oil D
	Kinematic viscosity @ 40°C (cSt)	220		35	45.1	120
	Kinematic viscosity @ 100°C (cSt)	19		7.5	7.7	15.9
	Density @ 15°C		0.895		0.870	0.885	0.860
	Oil type		Mineral	Mineral	Mineral	Synthetic



  𝜌 𝑛 : equivalent radius of curvature, normal section (mm) 𝑣 Σℎ = 2 × 𝑣 𝑡1 cos 𝛽 𝑚1 sin 𝛼 𝑛 (3.28) with:  𝛽 𝑚1,2 : helix angle at medium circle (°)  𝑣 𝑡1 : peripheral speed at pitch circle (m/s)  𝛼 𝑛 : normal pressure angle (°) 𝜙 𝑇 : thermal correction factor [Gupta et al., 1992]  ℎ 𝑐 : central lubricant film thickness (m)  𝑚 0 : average square value of the height of roughness (corresponds to rms²) (m²) 𝑊 𝑑𝑟𝑦 : total applied load on the asperities (N)  𝑐𝑓 𝑑𝑟𝑦 : Coulomb friction coefficient, determined from experiments  𝑆 𝑐𝑟𝑖𝑡 : critical slide-to-roll ratio  Δ𝑈 = 𝑈 1 -𝑈 2 : sliding velocity (m/s)  𝑈 = 𝑈 1 + 𝑈 2 : rolling velocity (m/s)  𝐸′: equivalent Young's modulus (Pa)  𝑚 0 : average square value of the height of roughness (corresponds to rms²) (m²)  𝑚 2 : average square value of the slope of roughness profile  𝑎: half width of contact (m) 𝜏 𝑁 = 𝜂(𝑝 𝑚𝑜𝑦 , 𝑇 𝑓𝑙𝑢𝑖𝑑𝑒 ) Δ𝑈 𝜙 𝑇 • ℎ 𝑐

		𝑣 Σ𝑠 = 𝑣 𝑡1 (sin 𝛽 𝑚1 + sin 𝛽 𝑚2 𝑆 𝑐𝑟𝑖𝑡 = 0.1 𝑎 𝑚 2 √ 𝑚 0	cos 𝛽 𝑚1 cos 𝛽 𝑚2	)	(3.27) (3.33)
	with:
	{b} Diab formula Reference [26]  Fluid friction shear stress 𝝉 𝒇𝒍𝒖𝒊𝒅 :
	with:	𝜇 =	1 𝑝 𝑚𝑜𝑦 𝜏 𝑓𝑙𝑢𝑖𝑑𝑒 = 𝜏 𝐿 • sinh -1 ( [ 𝐴 𝑐 𝐴 0 • 𝜏 𝑑𝑟𝑦 + (1 -𝐴 𝑐 𝐴 0 𝜏 𝑁 ) • 𝜏 𝑓𝑙𝑢𝑖𝑑 ] 𝜏 𝐿 )	(3.29) (3.34)
	 𝑝 𝑚𝑜𝑦 = 𝑊 𝐴 0
	Area ratio	𝑨 𝒄 𝑨 𝟎 ⁄ :
			𝐴 𝑐 𝐴 0	=	1 2	[ 1 2	(1 -erf (	𝜙 𝑇 • ℎ 𝑐 √2 • 𝑚 0	))]	(3.30)
	with:
	 Dry friction shear stress 𝝉 𝒅𝒓𝒚 :
		𝜏 𝑑𝑟𝑦 =	𝑊 𝑑𝑟𝑦 𝐴 𝑐	• 𝑐𝑓 𝑑𝑟𝑦 • [1 -exp ( 𝑆 𝑐𝑟𝑖𝑡 1	•	Δ𝑈 𝑈	)]	(3.31)
						𝑊 𝑑𝑟𝑦 𝐴 𝑐	= 0.1 • 𝐸 ′ √	𝜋 𝑚 2	(3.32)
									Annexes

⁄ : mean contact pressure (Pa)  𝐴 𝑐 : actual average contact area (m²)  𝐴 0 : apparent contact area (m²)  𝜏 𝑑𝑟𝑦 : shear stress due to dry friction (Pa)  𝜏 𝑓𝑙𝑢𝑖𝑑 : shear stress due to fluid friction (Pa)  𝑊: total load on the contact (N)

  ℎ ,𝜂 𝑜𝑖𝑙 ,𝑅𝑎) 𝑃 ℎ 𝑏 2 |𝑆𝑅| 𝑏 3 𝑉 𝑒 𝑏 6 𝜂 𝑜𝑖𝑙 𝑏 7 𝑅 𝑏 8 (3.37) 𝑤𝑖𝑡ℎ 𝑓 = 𝑏 1 + 𝑏 4 |𝑆𝑅| 𝑃 ℎ log 10 ( 𝜂 𝑜𝑖𝑙 ) + 𝑏 5 𝑒 -|𝑆𝑅| 𝑃 ℎ log 10 ( 𝜂 𝑜𝑖𝑙 ) + 𝑏 9 𝑒 𝑅𝑎 (3.38)  𝑆𝑅: slide-to-Roll Ratio  𝑃 ℎ : maximum Hertzian pressure (GPa)  𝜂 𝑜𝑖𝑙 : dynamic viscosity at oil inlet under ambient pressure (Pa.s)  𝑅𝑎: surface roughness (μm)  𝑉 𝑒 : entrainment velocity (m/s)  𝑅: effective radius of curvature (m)  𝑏 𝑖 : coefficients of EHL-based formula, see Table 0.3 𝐸′ : effective modulus of elasticity (Pa)  𝐸 1,2 : modulus of elasticity of respectively pinion and gear (Pa)  𝜈 1,2 : Poisson's ratio of respectively pinion and gear (-)

	with:
	Table 0.3: Coefficients of EHL-based formula
	b1	-8.916465						b6		-0.100601
	b2	1.03303						b7		0.752755
	b3	1.036077						b8		-0.390958
	b4	-0.354068						b9		0.620305
	b5	2.812084
			𝑃 ℎ = √ 2 𝜋 𝑅 𝑊 ′ 𝐸′	× 10 -9	(3.39)
		1 𝐸′	=	1 2	(	1 -𝜈 1 𝐸 1	2	+	1 -𝜈 2 𝐸 2	2	)	(3.40)
	with:
	 𝑊′: load per length (N/m)
	 𝑉 𝑒 =	2 𝑈 1 + 𝑈 2	(3.41)
			𝑆𝑅 = 2	𝑉 𝑠 𝑉 𝑟	= 2	𝑈 1 + 𝑈 2 𝑈 1 -𝑈 2	(3.42)
	with:
	 𝑈 1,2 : surface velocity of respectively pinion and gear (m/s)
					𝑅 =	𝑅 1 𝑅 2 𝑅 1 + 𝑅 2
												Annexes

  𝐴 𝑐𝑎 (𝑇 𝑜𝑖𝑙 -𝑇 𝑎𝑖𝑟 ) (4.22) with:  𝑄: heat flow across housing surface (W)  𝑈: overall heat transmission coefficient (W/(m²K))  𝐴 𝑐𝑎 : external housing surface (m²)  𝑇 𝑜𝑖𝑙 : oil temperature (K)  𝑇 𝑎𝑖𝑟 : ambient air temperature (K)

	1 𝑈	=	1 𝛼 𝑜𝑖𝑙	𝐴 𝑐𝑎 𝐴 𝑜𝑖𝑙	+	𝛿 𝑤𝑎𝑙𝑙 𝜆 𝑤𝑎𝑙𝑙	𝐴 𝑐𝑎 𝐴 𝑜𝑖𝑙	+	1 𝛼 𝑐𝑎

  𝛼 𝑐𝑜𝑛𝑣 + 𝛼 𝑟𝑎𝑑 (4.24) with:  𝛼 𝑐𝑜𝑛𝑣 : convection heat transfer coefficient (W/(m²K))  𝛼 𝑟𝑎𝑑 : radiation heat transfer coefficient (W/(m²K)) 𝜀: housing emission ratio  𝑇 𝑤𝑎𝑙𝑙 : housing wall temperature (K)  𝛼 𝐾,𝑓𝑟𝑒𝑒 : heat transfer coefficient due to free convection (W/(m²K))  𝛼 𝐾,𝑓𝑜𝑟𝑐𝑒𝑑 : heat transfer coefficient due to forced convection (W/(m²K))  𝐴 𝑎𝑖𝑟 : ventilated housing area (m²)  𝜂 * : temperature ratio 𝑇 𝑓𝑙𝑜𝑤 : cooling air temperature (K) Here, only housings without thermal finning are considered. For free convection (𝑣 𝑎𝑖𝑟 < 1.5 m/s): Grashoff number  𝑣 𝑓𝑙𝑜𝑤 : cooling air flow velocity (m/s)  𝑣 𝑟𝑒𝑓 : impingement velocity (= 15.6 × 10 -6 m²/s)  𝑔: standard gravity (=9.81 m/s²)

	with:
	 𝜂 * =	𝑇 𝑤𝑎𝑙𝑙 -𝑇 𝑎𝑖𝑟 𝑇 𝑤𝑎𝑙𝑙 -𝑇 𝑓𝑙𝑜𝑤	(4.27)
	with:
	 𝛼 𝐾,𝑓𝑟𝑒𝑒 = 18 ℎ 𝑐𝑎	-0.1 ( 𝑇 𝑤𝑎𝑙𝑙 -𝑇 𝑎𝑖𝑟 𝑇 𝑎𝑖𝑟	) 0.3	(4.28)
	For forced convection (𝑣 𝑎𝑖𝑟 > 1.5 m/s):
	𝛼 𝐾,𝑓𝑜𝑟𝑐𝑒𝑑 =	𝑙 𝑥 0.0086 (𝑅𝑒 ′ ) 0.64	(4.29)
	with:
	 ℎ 𝑐𝑎 : overall height of housing (m)
	 𝑙 𝑥 : flow length (m)
	𝑅𝑒 ′ = √ 𝑅𝑒 2 +	2.5 𝐺𝑟	(4.30)
		𝑅𝑒 =	𝑣 𝑟𝑒𝑓 𝑣 𝑓𝑙𝑜𝑤 𝑙 𝑥	(4.31)
	𝐺𝑟 =	𝑇 𝑎𝑖𝑟 𝑣 𝑓𝑙𝑜𝑤 𝑔 ℎ 𝑐𝑎 2 (𝑇 𝑤𝑎𝑙𝑙 -𝑇 𝑎𝑖𝑟 ) 2	(4.32)
	with:
	 𝑅𝑒: Reynold's number
	 𝐺𝑟:
	𝛼 𝑟𝑎𝑑 = 0.23 × 10 -6 𝜀 ( 𝑇 𝑤𝑎𝑙𝑙 + 𝑇 𝑎𝑖𝑟 2	) 3	(4.25)
	𝛼 𝑐𝑜𝑛𝑣 = 𝛼 𝐾,𝑓𝑟𝑒𝑒 (1 -		𝐴 𝑎𝑖𝑟 𝐴 𝑐𝑎	) + 𝛼 𝐾,𝑓𝑜𝑟𝑐𝑒𝑑	𝐴 𝑎𝑖𝑟 𝐴 𝑐𝑎	𝜂 *	(4.26)
							Annexes

Table 0 .

 0 4: Analogy from an electric circuit to a thermal networkLike Ohm's law for electric circuit, the Fourier equation then becomes:Δ𝑇 = 𝑅 𝑡ℎ × 𝑄 (4.33)This analogy is helpful to solve network problems. Indeed, the relationships for series and parallel thermal resistances are also valid:

	Electric circuit						Thermal network
	Potential difference	ΔU (V)				Temperature difference ΔT	(K)
	Electrical resistance	R (Ω)				Thermal resistance	Rth (K/W)
	Current	I	(A)				Heat flow	Q	(W)
		𝑅 𝑠𝑒𝑟𝑖𝑒𝑠 = 𝑅 1 + 𝑅 2	(4.34)
		1 𝑅 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙	=	1 𝑅 1	+	𝑅 2 1	(4.35)

  𝑡: time (s)  𝑄 𝑖 : heat flow injected at node 𝑖 (W)  𝑅 𝑡ℎ (𝑖, 𝑗): thermal resistance between elements 𝑖 and 𝑗 (K/W)  𝑛𝑏: number of thermal network nodes

	𝑖 𝑐 𝑖	𝑑𝑇 𝑖 𝑑𝑡	= 𝑄 𝑖 +	∑ 𝑛𝑏 𝑗=1 𝑎𝑛𝑑 𝑗≠𝑖	𝑇 𝑗 -𝑇 𝑖 𝑅 𝑡ℎ (𝑖, 𝑗)	(4.36)
	with:
	 𝑀

𝑖 : mass of element 𝑖 (kg)  𝑐 𝑖 : specific heat (J/kg.K)  𝑇 𝑖 , 𝑇 𝑗 : temperature of element 𝑖 or 𝑗 (K) Annexes 

  case of steady state conditions, Equation (4.36) simplifies to:

	𝑄 𝑖 =	𝑗=1 𝑎𝑛𝑑 𝑗≠𝑖 ∑ 𝑛𝑏	𝑅 𝑡ℎ (𝑖, 𝑗) 𝑇 𝑖 -𝑇 𝑗	(4.37)

  𝜖 𝜎 (𝑇 𝑐𝑎 2 + 𝑇 𝑎𝑖𝑟 2 )(𝑇 𝑐𝑎 + 𝑇 𝑎𝑖𝑟 ) (4.39) with:  𝜖: material emissivity (some values are given in Table 0.5)  𝜎: Stefan-Boltzmann constant (5.67 × 10 -8 W/(m²K 4 ) )  𝑇 𝑐𝑎 : housing temperature (K)  𝑇 𝑎𝑖𝑟 : ambient air temperature (K) Table 0.5: Emissivity properties of some materials

	Material	State	Emissivity
	Steel	Polished Oxydised	0.066 0.8
	Iron	Polished Oxydised	0.14-0.38 0.61
		Polished	0.21
	Cast iron	Oxydised	0.64
		Casing	0.81
	Aluminum	Polished Oxydised	0.09 0.20-0.31
	 Free convection:

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2018LYSEI019/these.pdf © [C. Fossier], [2018], INSA Lyon, tous droits réservés

Based on publications[START_REF] Hobson | Axle Efficiency -Test Procedures and Results[END_REF],[START_REF] Xu | Effects of Bearing Preload, Oil Volume, and Operating Temperature on Axle Power Losses[END_REF],[START_REF] Kakavas | Hypoid gear vehicle axle efficiency[END_REF],[START_REF] Jeon | Improving Efficiency in Drive Lines: an Experimental Study on Churning Losses in Hypoid Axle[END_REF],[START_REF] Takaki | Optimization of Hypoid Gear Design for High Efficiency Drives[END_REF]

Based on Volvo Group data[START_REF] Group | Experimental investigations and analysis on churning losses of splash lubricated spiral bevel gears[END_REF] Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2018LYSEI019/these.pdf © [C. Fossier],[2018], INSA Lyon, tous droits réservés

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2018LYSEI019/these.pdf © [C. Fossier], [2018], INSA Lyon, tous droits réservés Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2018LYSEI019/these.pdf © [C. Fossier], [2018], INSA Lyon, tous droits réservés

Remerciements

Remerciements

It is a CIFRE convention of the National

Ces travaux de thèse sont le résultat d'une Il s'agit d

Thermal model of the axle

Annexes

During meshing, the contact temperature increases a lot. However, the heat produced does not propagate deeply in gear tooth due to small contact zone and short contact duration. This has to be compared with gear size and time until the next meshing, which are really bigger. Gear bulk temperature is then not much influenced.

A distinction is thus made between gear pair and its meshing zone. Each gear has its own bulk temperature. Meshing zone is at "flash-temperature", as defined by Blok [START_REF] Blok | Transmission de chaleur par projection centrifuge d'huile[END_REF]. The thermal exchanges between meshing zone and each gear is driven by the thermal resistance defined as: