Power loss in planetary gear transmissions lubricated with ... · A perda de potência numa caixa...

Power loss in planetary gear transmissions

lubricated with axle oils

David Pinho Silva Dias da Costa

Porto, July of 2015

Faculdade Engenharia da Universidade do PortoDepartamento de Engenharia Mecânica

Power loss in planetary gear transmissions

lubricated with axle oils

David Pinho Silva Dias da Costa

Master thesis presented to

Faculdade de Engenharia da Universidade do Porto

Thesis supervised byDoctor Ramiro Carneiro Martins, Senior Researcher at INEGI

Professor Jorge Humberto Oliveira Seabra, Full Professor at FEUPEngineer Pedro M. T. Marques, PhD student

Porto, July of 2015

Acknowledgements

I would like to acknowledge and express my gratitude to a few people who havehelped and without whom this work couldn't have been possible.

Firstly, I would like to express my sincere thanks to my supervisors Dr. JorgeHumberto Oilveira Seabra and Dr. Ramiro Carneiro Martins for giving me the op-portunity of develop an experimental work at CETRIB (Unidade de Tribologia, Vi-brações e Manutenção Industrial).

I would like to thank to Eng. Pedro Marques for the help given at all timesthat needed, support, readiness, guidance and for all transmitted knowledge.

I am also grateful to all the people that work at CETRIB, not only for all thehelp, but also the good working atmosphere: Prof. Armando Campos, Dr. CarlosFernandes, Eng. David Gonçalves, Prof. Luís Magalhães, Prof. Jorge Castro, Dr.José Brandão, Eng. Beatriz Graça, Eng. André Gama, Anvar Maxkamov and MarouaHammami.

I would like to thank to my colleagues that are in same situation as me: PedroAires, André Falcão, Cláudio Pinto, José Sarilho and António Coimbra.

I would also like to acknowledge Faculdade de Engenharia da Universidade doPorto to opportunity given and for all resources spent with me during the course.

This thesis is a culmination of �ve years of work and I would like to thank allthe people that contributed for my success.

Finally, I would like to give my gratitude for my family and girlfriend for beingalways there.

v

Keywords

KeywordsAxle oilsPower lossE�ciencyPlanetary gearboxCoe�cient of friction

Palavras chaveÓleos para transmissão/diferencialPerdas de potênciaE�ciênciaCaixa de engrenagens planetáriaCoe�ciente de atrito

vii

Abstract

In order to reduce the operating and environmental costs it has become a ne-cessity to design machines with not only reliability concerns, but also with energetice�ciency in mind.

The di�erential is a mechanism that is not necessary in many systems, but itis needed when the input power must be splited between two outputs that rotate atdi�erent speeds. This characteristic makes it widely used in automotive vehicles. Inwhat regards power loss, di�erentials are not the most e�cient mechanisms.

Fuel economy is a very important subject with increasing global interest, soimproving automotive components e�ciency is a major contribution.

The e�ciency of vehicle power depends on each component, so the di�erentialis a mechanism with several meshing contacts including bevel or hypoid gears, forthis reason it is one of the most important power loss sources. The improvement ofperformance, even if it is quite small per vehicle can have signi�cant impact if it issummed over the millions of cars existing.

The power loss in a gearbox is dependent of the lubricant, speed and load and itcan be divided in load and no-load losses which are both dependent of the lubricantthat is used.

The easiest way to improve the e�ciency of a di�erential that is already installedwould be to act at the lubricant level. The lubricant oil plays a very important role ina mechanism, because it has the mission of reducing friction in the power transmittingcontacts, it helps to dissipate the friction generated heat, minimize the wear of thesurfaces and remove the wear particles from the contacts and to some extent preventcorrosion, thus extending the life of the equipments.

In this work three di�erent axle gear oils, two PAO's and a mineral, were selectedand their in�uence in the e�ciency of a planetary gearbox was tested. The oils werecharacterized in terms of their basic properties, mainly viscosity characterization. Thee�ciency tests were performed in a back-to-back gearbox test rig with recirculatingpower. Initially 16 tests were done with one of the lubricants aiming to understand theoperating boundaries (limiting temperature). From the 16 combinations of operatingconditions 5 were selected and performed for the three oils that were selected. Afterthe load tests, no-load tests were performed at the same operating temperatures andspeeds of the load tests.

Aiming to understand the power loss behaviour for the operating conditions, apower loss model of the planetary transmission was implemented and calibrated.

ix

Sumário

De forma a reduzir os custos de funcionamento e ambientais, tornou-se umanecessidade o projeto de máquinas não só apenas com preocupações de �abildade,mas também com a e�ciência energética em mente.

O diferencial é um mecanismo que não é necessário em todos os sistemas , mas épreciso quando a potência de entrada tem que ser dividida por duas saídas que rodama velocidades diferentes. Esta característica faz com que seja amplamente usado emveículos automóveis. No que toca a perdas de potência, os diferenciais não são dosmecanismos mais e�cientes.

A economia de combustível é um assunto muito importante com o interesseglobal em crescendo, posto isto, melhorando a e�ciência dos componentes de umveículo torna-se uma grande contribuição.

A e�ciência do veículo depende de cada componente, como o diferencial temvários engrenamentos, incluindo engrenagens cónicas ou hipóides, torna-se um dosmais importantes geradores de perdas de potência. O melhoramento do desempenho,mesmo que seja pequeno por veículo, pode ter um impacto signi�cativo se foremsomados todos os milhões de carros existentes.

A perda de potência numa caixa de engrenagens depende do lubri�cante, davelocidade e da carga, sendo que as perdas podem ser divididas por carga e semcarga, dependendo ambas do lubri�cante utilizado.

A maneira mais fácil para melhorar a e�ciência de um diferencial já instaladoseria atuar ao nível do lubri�cante. O óleo lubri�cante tem um papel muito im-portante num mecanismo, pois tem a missão de reduzir o atrito no contacto datransmissão de potência, ajudar a dissipar o calor gerado, minimizar o desgaste dassuperfícies e remover as partículas de desgaste do contacto, prevenindo também acorrosão, tornando a vida dos equipamentos mais longa.

Neste trabalho foram selecionados três óleos de diferencial, dois PAO's e ummineral, sendo testados a sua in�uência na e�ciência de uma caixa de engrenagensplanetária. Os óleos foram caracterizados em termos das suas propriedades básicas,principalmente viscosidade. Os testes de e�ciência foram feitos numa caixa de en-grenagens com con�guração back-to-back, com recirculação potência. Inicialmenteforam feitos 16 testes com um dos lubricantes utilizados, com o objectivo de perceberos limites das condições (temperaturas limite). Com as 16 combinações de condiçõesde teste, foram selecionados 5 condições e realizado os testes para os três óleos sele-cionados. Depois dos testes de carga, foram feitos os testes sem carga com as mesmastempersturas e velocidades dos ensaios de carga.

Com o objetivo de perceber o comportamento das perdas de potência com ascondições de teste, um modelo de perdas de potência da caixa planetária foi imple-mentado e calibrado.

xi

Contents

Acknowledgements v

Keywords viiTable of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiTable of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

1. Di�erentials 11.1. An introduction to di�erentials . . . . . . . . . . . . . . . . . . . . . 1

1.1.1. Types of di�erentials . . . . . . . . . . . . . . . . . . . . . . . 21.1.2. Working principle of the open di�erential . . . . . . . . . . . . 21.1.3. Locking Di�erentials . . . . . . . . . . . . . . . . . . . . . . . 41.1.4. Limited Slip Di�erentials . . . . . . . . . . . . . . . . . . . . . 4

1.2. Types of di�erential oils . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1. Axle lubricant oils speci�cations . . . . . . . . . . . . . . . . . 4

2. Lubricant types 72.1. Liquid lubricants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1. Vegetable and animal oils . . . . . . . . . . . . . . . . . . . . 72.1.2. Mineral based oils . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.3. Synthetic based oils . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2. Lubricant greases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3. Solid lubricants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4. Gaseous lubricants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5. Additives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5.1. Viscosity index improving additives . . . . . . . . . . . . . . . 102.5.2. Anti wear and extreme pressure additives . . . . . . . . . . . . 102.5.3. Anti oxidants . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5.4. Detergent additives . . . . . . . . . . . . . . . . . . . . . . . . 122.5.5. Corrosion inhibitors . . . . . . . . . . . . . . . . . . . . . . . . 122.5.6. Rust inhibitors . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5.7. Foam inhibitors . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5.8. Additives designed to lower the freeze point . . . . . . . . . . 12

3. Physical properties of the lubricant oils 133.1. Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.1. Dynamic viscosity . . . . . . . . . . . . . . . . . . . . . . . . . 133.2. Viscosity variation with temperature . . . . . . . . . . . . . . . . . . 14

3.2.1. Viscosity index . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3. Viscosity variation with pressure . . . . . . . . . . . . . . . . . . . . . 16

xiii

Contents

3.4. Viscosity variation with shear strain rate . . . . . . . . . . . . . . . . 17

4. Oils characterization 194.1. Selected gear oils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2. Engler viscometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3. Rheometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4. Densitometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.5. Vibrational viscometer . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5. Transmission test rig and test gearbox 295.1. Transmission test rig . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2. Planetary gearbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.3. Planetary gearbox loads and kinematics . . . . . . . . . . . . . . . . 35

5.3.1. Load analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.3.2. Kinematic analysis . . . . . . . . . . . . . . . . . . . . . . . . 39

6. Thermal balance of the gearbox 436.1. Heat dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.2. Power Loss model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6.2.1. Gears power loss . . . . . . . . . . . . . . . . . . . . . . . . . 446.2.2. Power loss in planetary gears . . . . . . . . . . . . . . . . . . 516.2.3. Churning loss . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.2.4. Rolling bearings power loss (Load dependent) . . . . . . . . . 526.2.5. Bearing power loss (no-Load dependent) . . . . . . . . . . . . 576.2.6. Needle roller bearings losses . . . . . . . . . . . . . . . . . . . 586.2.7. Seal losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7. Power loss tests of planetary gearbox 617.1. Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . 617.2. Operating conditions for the power loss tests . . . . . . . . . . . . . . 63

7.2.1. Preliminary test grid (16 tests grid) . . . . . . . . . . . . . . . 637.2.2. Final test grid . . . . . . . . . . . . . . . . . . . . . . . . . . . 637.2.3. No load test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.3. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 657.3.1. Sixteen test grid . . . . . . . . . . . . . . . . . . . . . . . . . 657.3.2. Five test grid . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7.4. Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.4.1. No-load results . . . . . . . . . . . . . . . . . . . . . . . . . . 777.4.2. Load results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8. Conclusion and future works 918.1. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

8.1.1. Conclusions based on experimental evidence . . . . . . . . . . 918.1.2. Conclusions based on numerical results . . . . . . . . . . . . . 92

8.2. Future works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

A. Tested oils 101A.1. 75w-90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

xiv

Contents

A.2. 75w-140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104A.3. 80w-90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

B. Reports from the experimental tests 107B.1. Load Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

B.1.1. Sixteen tests grid . . . . . . . . . . . . . . . . . . . . . . . . . 107B.1.2. Five tests grid . . . . . . . . . . . . . . . . . . . . . . . . . . . 123B.1.3. No-Load tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

C. FTIR analysis 153

D. Kissoft data 155D.1. KISSOFT analyses of the planetary gearbox . . . . . . . . . . . . . . 165

E. Detailed results of the implemented numerical approach, (Program prin-touts) 167E.1. Results before optimization [16 test grid] . . . . . . . . . . . . . . . . 167E.2. Results after optimization [5 test grid] . . . . . . . . . . . . . . . . . 171

E.2.1. 75W90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171E.2.2. 75W140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175E.2.3. 80W90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

xv

List of Figures

1.1. Example of an open di�erential. . . . . . . . . . . . . . . . . . . . . . 21.2. Example of a vehicle with one di�erential (vehicle with front-wheel

drive and rear-wheel drive, respectively). . . . . . . . . . . . . . . . . 31.3. Example of a vehicle with two di�erentials (4WD). . . . . . . . . . . 31.4. Example of a vehicle with three di�erentials (AWD). . . . . . . . . . 31.5. Viscosity classi�cation SAEJ306. . . . . . . . . . . . . . . . . . . . . 5

3.1. Laminar �ow between two planes. . . . . . . . . . . . . . . . . . . . . 133.2. Viscosity variation with shear rate: a) Grease b) Newton Fluid c) non-

Newton Fluid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1. Engler viscometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2. Kinematic viscosity variation with the temperature. . . . . . . . . . . 224.3. Rheomat 115 rheometer. . . . . . . . . . . . . . . . . . . . . . . . . . 234.4. Dynamic viscosity variation with the shear rate. . . . . . . . . . . . . 244.5. DMA 35N Densitometer. . . . . . . . . . . . . . . . . . . . . . . . . . 254.6. Density variation with the temperature obtained with measured values

(table 4.4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.7. SV-10 Vibrational viscometer. . . . . . . . . . . . . . . . . . . . . . . 264.8. Dynamic viscosity variation with the temperature. . . . . . . . . . . . 27

5.1. Test bench rig. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.2. Hydraulic cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.3. Test Rig. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.4. Central control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.5. Temperature sensors positioning in the tested gearbox. . . . . . . . . 325.6. Schematic of the planetary gearbox. . . . . . . . . . . . . . . . . . . . 335.7. Planetary gearbox. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.8. Detail of planet gearbox. . . . . . . . . . . . . . . . . . . . . . . . . . 345.9. Representation of the planetary gear (side view). . . . . . . . . . . . . 355.10. Representation of the planetary gear. . . . . . . . . . . . . . . . . . . 365.11. FBD of the carrier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.12. FBD of the planet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.13. FBD of the sun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6.1. Components of the power loss. . . . . . . . . . . . . . . . . . . . . . . 446.2. Lubricant �lm thickness in linear contacts. . . . . . . . . . . . . . . . 466.3. Lubrication regimes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.4. Stribeck curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.5. Tooth surface distress probability as function of speci�c �lm thickness

and tangential velocity at pitch point. . . . . . . . . . . . . . . . . . . 50

xvii

List of Figures

6.6. Reverse �ow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

7.1. Coupling disassembled of the test rig. . . . . . . . . . . . . . . . . . . 627.2. Heater. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627.3. Power loss vs Operating Conditions (16 test grid, 75W90). . . . . . . 667.4. Power loss vs Input Power (16 test grid, 75W90). . . . . . . . . . . . 667.5. E�ciency vs Operating Conditions (16 test grid, 75W90). . . . . . . . 677.6. Oil Temperature vs Operating Conditions (16 test grid, 75W90). . . . 687.7. ∆T vs Operating Conditions (16 test grid, 75W90). . . . . . . . . . . 687.8. Speci�c �lm Thickness vs Operating Conditions (16 test grid, 75W90). 697.9. Comparison of the power loss between 5 and 16 test grid (75W90). . . 707.10. Comparison of the oil temperature between 5 and 16 test grid (75W90). 707.11. Comparison of the temperature variation between 5 and 16 test grid

(75W90). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717.12. Power Loss vs Operating Conditions. . . . . . . . . . . . . . . . . . . 717.13. E�ciency vs Operating Conditions. . . . . . . . . . . . . . . . . . . . 727.14. Oil Temperature vs Operating Conditions. . . . . . . . . . . . . . . . 727.15. ∆T vs Operating Conditions. . . . . . . . . . . . . . . . . . . . . . . 737.16. Dynamic viscosity vs Operating Conditions. . . . . . . . . . . . . . . 747.17. Speci�c �lm thickness (Planet/Ring). . . . . . . . . . . . . . . . . . . 747.18. Speci�c �lm thickness (Sun/Planet). . . . . . . . . . . . . . . . . . . 757.19. Surface distress probability. . . . . . . . . . . . . . . . . . . . . . . . 757.20. Experimental results (no-load). . . . . . . . . . . . . . . . . . . . . . 767.21. Participation of each component in the no-load losses (75W90). . . . 787.22. Participation of each component in the no-load losses (75W140). . . . 787.23. Participation of each component in the no-load losses (80W90). . . . 797.24. Comparison between experimental and numerical power loss for 75W90

(XL=1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817.25. Comparison between experimental and numerical power loss for 75W140



(XL=1.15). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827.28. Comparison between experimental and numerical power loss for 75W140

(XL=1.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837.29. Power loss and its components (75W90). . . . . . . . . . . . . . . . . 857.30. Power loss and its components (75W140). . . . . . . . . . . . . . . . 867.31. Power loss and its components (80W90). . . . . . . . . . . . . . . . . 877.32. Coe�cient of friction (Planet/Ring). . . . . . . . . . . . . . . . . . . 897.33. Coe�cient of friction (Sun/Planet). . . . . . . . . . . . . . . . . . . . 89

xviii

List of Tables

1.1. Speci�cations in current use. . . . . . . . . . . . . . . . . . . . . . . . 6

3.1. Viscosity units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2. Constants of the oil types. . . . . . . . . . . . . . . . . . . . . . . . . 16

4.1. Oil properties provided by the manufacture. . . . . . . . . . . . . . . 194.2. Test oils characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . 204.3. Physical properties of the axle oils (measured). . . . . . . . . . . . . . 214.4. Physical properties of the axle oils (measured). . . . . . . . . . . . . . 254.5. Physical properties of the axle oils. . . . . . . . . . . . . . . . . . . . 264.6. Comparison of the kinematic viscosity [cSt]. . . . . . . . . . . . . . . 27

5.1. Electric motor characteristics. . . . . . . . . . . . . . . . . . . . . . 305.2. Rolling bearings and seals of the planetary gearbox. . . . . . . . . . . 345.3. Geometric characteristics of the gears used in the planetary gearbox. 345.4. Forces at 200 rpm and 2800 Nm. . . . . . . . . . . . . . . . . . . . . 395.5. Rotational speed of the gearbox components at 200 rpm and 2800 Nm. 41

6.1. Formulation of the coe�cients ai . . . . . . . . . . . . . . . . . . . . . 456.2. HV calculated with KISSsoft R©. . . . . . . . . . . . . . . . . . . . . . 466.3. Parameters of the �lm thickness. . . . . . . . . . . . . . . . . . . . . 476.4. Dependence of the �lm thickness with the parameters. . . . . . . . . 486.5. Lubrication regimes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.6. Calculation example for the planetary gearbox rolling bearings losses

at 200 rpm and 2800 Nm. . . . . . . . . . . . . . . . . . . . . . . . . 576.7. Calculation example for the tapered roller bearing losses at 200 rpm

and 77.8 oC (no-load dependence). . . . . . . . . . . . . . . . . . . . 576.8. Torque losses to no-load dependent. . . . . . . . . . . . . . . . . . . . 586.9. Calculation example for the needle roller bearing losses at 200 rpm and

2800 Nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.10. Calculation example for the seal losses at 200 rpm. . . . . . . . . . . 59

7.1. Operating conditions for tests with 75W90 [rpm/Nm]. . . . . . . . . . 637.2. Input power [kW] for tests with 75W90. . . . . . . . . . . . . . . . . . 637.3. Operating conditions for tests with all lubricants [rpm/Nm] (Load tests). 647.4. Operating conditions for tests with all lubricants [rpm/Nm] (No-Load

tests). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647.5. Operating conditions for tests with 75W90 (16 test grid). . . . . . . . 697.6. No-load power loss of the each component (75W90). . . . . . . . . . . 787.7. No-load power loss of the each component (75W140). . . . . . . . . . 797.8. No-load power loss of the each component (80W90). . . . . . . . . . . 79

xix

List of Tables

7.9. The error obtained in optimization. . . . . . . . . . . . . . . . . . . . 847.10. Power loss and its components in W (75W90). . . . . . . . . . . . . . 857.11. Power loss and its components in % (75W90). . . . . . . . . . . . . . 857.12. Power loss and its components in W (75W140). . . . . . . . . . . . . 867.13. Power loss and its components in % (75W140). . . . . . . . . . . . . 867.14. Power loss and its components in W (80W90). . . . . . . . . . . . . . 877.15. Power loss and its components in % (80W90). . . . . . . . . . . . . . 877.16. Operating conditions for tests with 75W90 (5 test grid). . . . . . . . 907.17. Operating conditions for tests with 75W140 (5 test grid). . . . . . . . 907.18. Operating conditions for tests with 80W90 (5 test grid). . . . . . . . 90

xx

List of Symbols

Symbol Units Descriptiona m Centre distanceb m Gear widthB mm Rolling bearing widthcA,B N/m Rolling bearing spring constantd mm Rolling bearing bore diameterdm m Bearing mean diameterdsh mm Shaft diameterdi m Gear reference diameterE∗ Pa Equivalent Young modulesF N ForceFa N Axial forceFbt N Tooth normal force (transverse section)Fr N Radial forceFt N Tangential forcef0 - Coe�cient dependent on bearing design and lubrication

methodF0 N Pre-load forceF1,2 - Coe�cient that takes into account the direction of load ap-

plicationGrr Nm Variable for the calculation of the rolling frictional momentGsl Nm Variable for the calculation of the sliding frictional momenth0 m Film thicknessh0T m Corrected �lm thicknessHV - Gear loss factorK W/m2 Thermal conductivityKa N Axial load on the tapered roller bearingsKball,roll - Rolling element related constantKrs - Replenishment/starvation coe�cientKZ - Bearing type related geometric constanti - Gear ratiol m Average sum of contacting lines lengthL - Thermal parameter of the lubricantlg - Parameter for the calculation a0,1,2,3,4

m m Modulemg - Parameter for the calculation a0,1,2,3,4

M Nm Total frictional moment of a bearingMA,E,ext,mot Nm Torque (index related to the application point)Mdrag Nm Frictional moment of drag losses

xxi

List of Tables

Symbol Units DescriptionMrr Nm Rolling frictional momentMseal Nm Total frictional moment of the bearing sealMsl Nm Sliding frictional momentn rpm Rotational speedN - Number of planetsng - Parameter for the calculation of a0,1,2,3,4

p Pa PressurePa W Transmitted powerPV W Total power lossPV 0 W Total no-load dependent power lossPV AUX W Auxiliary power lossPV D W Seals power lossPV L W Bearing power loss load dependentPV L0 W Bearing power loss no-load dependentPV Z0 W Churning lossPV ZP W Gear load loss.

Qcond W Heat �ow rate due to conduction.

Qconv W Heat �ow rate due to convection.

Qrad W Heat �ow rate due to radiation.

Qtotal W Total heat �ow rateRa µm Arithmetic mean roughnessrb m Base radiusRx m Equivalent radiusR1,2 - Geometric constants for rolling frictional momentTOil

oC Oil temperatureTV L Nm Total frictional moment of a needle bearingTV L0 Nm No-load component of frictional moment of a needle bearingTV L1 Nm Load component of frictional moment of a needle bearingU - Speed parameterU1,2 m/s Velocity pf each surfacev m/s Tangential speedV e - Sliding ratioV.I - Viscosity indexv∑C m/s sum velocity at pitch pointW - Load parameterY - Axial load factor for single-row bearingsZ - Number of teethα Pa−1 Piezoviscosity coe�cientαt Rad Transverse pressure angleαSKF

o Variable used to Grr

αheat W/m·K Heat transfer coe�cientβ K−1 Thermoviscous coe�cientβb Rad Base helix angle∆T (TOil −Troom)

oC Stabilized operating temperature

xxii

List of Tables

Symbol Units Descriptionεα - Transverse contact ratioε1,2 - Tip contact ratioη Pa · s Dynamic viscosityη0 Pa · s Dynamic viscosity of the oil bath temperatureΛ - Speci�c �lm thicknessµbl - Coe�cient dependent on the lubricant additive packageµmz - coe�cient of frictionµsl - Sliding friction coe�cientν cSt Kinematic viscosityΦbl - Weighting factor for the sliding friction coe�cientΦish - Inlet shear heating reduction factorΦrs - Kinematic replenishment/starvation reduction factorφT - Inlet heating in�uence factorω rad/s Rotational speedρC mm Equivalent radius of curvature at pitch point

xxiii

1. Di�erentials

1.1. An introduction to di�erentials

The di�erential (axle gears) is the �nal drive unit in road vehicles, as representedin �gure 1.1. A di�erential is an application of bevel or hypoid gearing [1]. It is amechanical gear set with di�erent functions which are of extreme importance in theroad vehicles. The functions of the di�erential are:

• Carry the engine power to the wheels;

• Act as the �nal gear reduction in the vehicle, slowing the rotational speed ofthe transmission;

• Transmit the power to the wheels while allowing them to rotate at di�erentspeeds.

When a vehicle is cornering, the inner and outer wheels travel a di�erent dis-tance (the inner wheel travels less distance). So, the average speed is equal to thedistance divided by the time it takes to go that distance, so in this case the wheelthat describes a smaller distance travels at a lower speed.

An automotive vehicle can be equipped with one, two or in special cases threedi�erentials. Normally, vehicles have one di�erential, between the set of drive wheelsand depending if vehicle is rear-wheel drive (RWD) or front-wheel drive (FWD) thedi�erentials are needed on the back or on the front, as it's in �gure 1.2.

FWD vehicles have a transaxle in which the transmission and di�erential sharethe same housing. In FWD, a helical rather than a hypoid gear pair can be used,because transaxle are typically mounted laterally across the front axle.

In a RWD vehicle a long shaft crosses the chassis and connects the transmissionto the di�erential. In this case an helical gear pair can't be used, because the shaftisn't parallel to the rear axle and the alternative is to use an hypoid gear pair.

Two di�erentials can be used on four-wheel drive (4WD).When the vehicle oper-ates in 4WD, the transfer gearbox engages the front di�erential and rear di�erentialswith same rotation, but the vehicle can operate in two wheel drive, behaving like rearwheel drive (the transfer gearbox is disengaged). Figure 1.3 shows the example offour-wheel drive vehicle.

In all-wheel-drive vehicles three di�erentials are needed one for each axle (frontand rear) and one between the front and the back wheels, as it's in �gure 1.4. Thecentral di�erential can be a conventional or loose clutch plates di�erential.

1

1. Di�erentials

1.1.1. Types of di�erentials

The types of di�erentials can be divided in three categories:

• Open/Standard di�erentials;

• Locking Di�erentials;

• Limited Slip Di�erentials.

1.1.2. Working principle of the open di�erential

When vehicles travel in straight line

When the vehicle is travelling straight, both wheels are spinning at the samespeed. The input pinion is turning the ring gear and cage and planet gears do notrotate, so as the transmission shaft turns the crown wheel (ring gear in �gure the1.1), the rotary speed is translated directly to the half-shafts and both wheels spinwith the angular velocity of the crown wheel.

When vehicles corner

When the vehicle corners, the planets rotate with respect to the crown wheel(ring gear) and turn around the sun gears (di�erential side gears in the �gure 1.1).This allows the speed of the crown gear to be delivered unevenly to the wheels.

Figure 1.1.: Example of an open di�erential [2].

2

1.1. An introduction to di�erentials

Figure 1.2.: Example of a vehicle with one di�erential (vehicle with front-wheel driveand rear-wheel drive, respectively) [3].

Figure 1.3.: Example of a vehicle with two di�erentials (4WD) [4].

Figure 1.4.: Example of a vehicle with three di�erentials (AWD) [3].

3

1. Di�erentials

1.1.3. Locking Di�erentials

A locking di�erential is a di�erential that has a mechanism that allows bothwheels to rotate at the same rate, regardless of the traction situation.

All di�erential locks are designed to lock together two or more parts of thedi�erential gear cluster by engaging adjacent sets of dog clutch teeth. All of theavailable power that is transmitted to the �nal drive will be supplied to the wheels.Even if one wheel loses grip, the opposite wheel will still receive power, enabling itto produce torque and therefore traction [5].

1.1.4. Limited Slip Di�erentials

The limited slip di�erentials send the same amount of power to both wheelswhen travelling straight, but when traction lacks in on of the wheels the limited slipdi�erentials, automatically, provide torque to the wheel with traction.

The limited slip di�erential essentially consists of an ordinary bevel gear dif-ferential arranged so that the torque from the engine friction clutches locks the halfshafts to the di�erential cage [5].

1.2. Types of di�erential oils

Di�erentials are subject to high contact pressures and high sliding speeds [6], sodi�erential oils are usually fully formulated including extreme pressure and anti-wearadditives, aiming to increase both di�erential and vehicle life.

Di�erentials can operate in a wide range of conditions:

• High speeds and high torques;

• High speeds and low torques;

• Low speeds and high torques.

1.2.1. Axle lubricant oils speci�cations

Two types of speci�cations are available for the lubricant oils:

• Viscosity speci�cations;

• Service speci�cations.

Viscosity speci�cations

Viscosity speci�cations can be established with to ends:

• Identi�cation: There are re�ning or manufacturing speci�cations that take intoaccount viscosity tolerances for certain ranges of viscosity;

• Usage: These are imposed by the consumer and are function of the usage givento lubricant oils. There are certain ranges of maximum and minimum viscosityat certain temperatures.

4

1.2. Types of di�erential oils

These classi�cations are based only on the lubricant oil viscosity.Some societies that classify axle lubricants by viscosity range are:

• SAE - Society of Automotive Engineers;

• ASTM - American Society for testing and Materials.

SAE Viscosity classi�cation

This classi�cation is exclusively based on the oil viscosity and it doesn't eval-uate the oil quality it just gives the information about the its viscosity at a certaintemperature.

The classi�cation given to the lubricant oils for automotive gearboxes and dif-ferentials is SAE J306.

Figure 1.5 shows the SAE Grade and minimum/maximum kinematic viscosityat 100oC.

Figure 1.5.: Viscosity classi�cation SAEJ306.

Service speci�cations

The most usual service classi�cation for automotive lubricant oils is:

• API - American Petroleum Institute.

5

1. Di�erentials

API Service classi�cation

Gear oils are classi�ed by the American Petroleum Institute using GL ratings.In axles, gears with di�erent designs are available for a variety of service conditions.

So, in a lubricating project is necessary have carefully on oil selecting, becauseexist some considerations to have in count, like a operational conditions, chemicaland physical characteristics.

The API has lubricant service designations for automotive manual transmis-sions, transaxles, and axles. The service speci�cations are shown in table 1.1.

Table 1.1.: Speci�cations in current use [6].API category Description

GL-4 Lubricants for axles with bevel gears operating under moderate tosever conditions of speed and load. Axles with hypoid gears undermoderate conditions of speed and load. Axles with limited-slip dif-ferentials have additional requirements that are normally de�ned bythe axle manufacturer.

GL-5 Lubricants for gears, particularly hypoid gears, in axles operat-ing under various combinations of high-speed/shock load and low-speed/high-torque conditions. Axles with limited-slip di�erentialshave additional requirements that are normally de�ned by the axlemanufacturer.

6

2. Lubricant types

Lubricant oils reduce friction, minimize surface wear, dissipate heat, removewear particles and preclude corrosion.

Lubricants should also be miscible with other chemical substances (additives).Nowadays, lubricant oils are generally made of a base and that can be synthetic

or mineral, in which additives are added. Lubricants can be divided in four greatcategories: liquid lubricants, lubricant greases, solid and gaseous lubricants. Thiswork deals with liquid lubricants, so there will be the focus. This chapter is basedon the information presented on [7].

2.1. Liquid lubricants

This type of lubricant is the most commonly used. This class can be dividedby the of the base of origin:

• Vegetable and animal;

• Mineral and synthetic.

2.1.1. Vegetable and animal oils

Certainly, the �rst oils to be used by the mankind were vegetable and animaloils, but due to the chemical inertia and requirements increase that lubricants weresubjected these were replaced by synthetic and petroleum based products. However,due to current environmental requirements more and more vegetable base lubricantsare being used.

These lubricants have some advantages compared to mineral oils, for example:

• High viscosity index;

• Low evaporation rate;

• High biodegradability.

But on the other hand, the vegetable oils oxidized quickly, and have low hightemperature resistance.

2.1.2. Mineral based oils

The petroleum based lubricants are constituted by hydrocarbons of natural ori-gin. These can be separated in three categories, which depend on the re�ning processand petroleum origin. A mineral oil can be para�nic, naphthenic and aromatic. Butnormally, the aromatic basis are undesirable.

7

2. Lubricant types

Para�n basis

The advantages of para�nic basis are the low speci�c weight, low freeze point,great oxidation resistance and low variation of the viscosity with temperature, there-fore high viscosity index, (about 100). This base is usually elastomer friendly, how-ever, the high molecular weight of certain chains can lead to oil crystallisation atambient temperature [8].

Naphthenic basis

The naphtenic basis have high speci�c weight, high freeze point, low oxidationresistance and high variation of the viscosity with temperature, therefore low viscosityindex, (about 50). This base has an aggressive behaviour towards elastomers, goodmiscibility, high volatility and good �ow characteristics at low temperatures [8].

It is possible to mix the two basis (para�n and naphtenic) to create a lubricant,that combines the following properties:

• Availability on a wide range of viscosities;

• Low volatility;

• Resistant to deterioration;

• Good protection against corrosion;

• Low price.

2.1.3. Synthetic based oils

The synthetic based oils are synthesized lubricants from hydrocarbons or theirconstituent chemical elements, the basis can be petroleum products, vegetable productsor others. Generally, synthetic lubricants have some advantage relative to mineraloils, namely:

• Higher oxidation resistance;

• Higher viscosity index;

• Lower coe�cient of friction.

The advantages are more expressive at high or low temperatures, but on theother hand, the price of synthetic based oils is higher than mineral based.

Synthetic hydrocarbons

Normally, benzenes and polyalphaole�ns are the most common types. In theirchemical structure they are almost equal to the mineral hydrocarbons. They have agood compatibility with seals, miscibility with mineral oils and good thermal stability,but needs anti-oxidation additives. They also have an excellent behaviour at lowtemperatures.

8

2.2. Lubricant greases

Polyglycols

The biggest advantage of these lubricants is the low coe�cient of friction, al-lowing it to be utilized in high sliding applications. But, the compatibility with sealsand paint is not good at operating temperatures greater than 100oC. The miscibilitywith mineral oils is also very limited.

Esters

The ester based oils are the product of a reaction between acids and alcohols, fol-lowed by water separation. The main advantages are an high thermal resistance andexcellent performance at low temperatures. Some of the esters can allow a coe�cientof friction like a polyglycol. Nowadays, an important factor is the biodegradabilityand in addition some ester oils are also quickly biodegraded.

The disadvantage of the esters is the low hydrolytic stability.

Silicones

This type of oils is chemically inert, has excellent thermal and oxidation stabilityat high temperatures, good �re resistance, non toxicity and low water miscibility.

2.2. Lubricant greases

Lubricating greases are used when continuous oil lubrication is not usable andwhen protection against possible contaminant particles is required.

This type of lubricants have its properties limited by the thickening agent, baseoil and additives. Lubricant greases result from the dispersion of a thickening agentin a lubricant oil. Most of the lubricant greases have a soap as thickener, but someorganic products can be used as thickeners too.

2.3. Solid lubricants

A solid lubricant is a solid material �lm constituted by organic or inorganicproducts.

The inorganic products used in solid lubricants are:

• Gelatinous solids;

• Soft solids mixtures;

• Surface protection by chemical reaction with the surface.

The organic solid lubricants can be divided in two categories:

• Soaps, waxes and fat;

• Polymeric �lms.

9

2. Lubricant types

2.4. Gaseous lubricants

Gaseous lubrication in some aspect analogous to liquid lubrication, since theprinciples of hydrodynamic lubrication can be applied. Both are viscous �uids, butthere are some fundamental di�erences that separate them. Gases have much lowerviscosity than the liquids and also much higher compressibility. The load carryingcapacity of a gaseous lubricant is much lower than the one of a liquid lubricant.

2.5. Additives

Additives are chemical agents which have the objective of giving speci�c prop-erties to lubricants. Some additives give new properties, that the lubricant oils didn'thave, while others improve their natural properties. Nowadays, practically every lub-ricant has some additives and the quantity varies from some hundredths up to 30 %.The objectives of additives are [9]:

• Improving the wear and friction characteristics;

• Improving the oxidation resistance;

• Control of corrosion;

• Control of contamination by reaction products, wear particles and other debris;

• Reducing excessive decrease of lubricant viscosity at high temperatures;

• Enhancing lubricant characteristics by reducing the pour point and inhibitingthe generation of foam;

• Increase viscosity index.

2.5.1. Viscosity index improving additives

This type of additives are used to improve the viscosity index on base oil, inother words, decrease viscosity at low temperatures or improving it at high temper-atures.

The viscosity index of an oil is improved with the addition of high molecularweight polymers and the most used types are butane polymers, methacrylic acidesters and fatty alcohols.

In a lubricant oil with polymeric additives, viscosity is much more modi�ed athigh temperatures than at low temperatures. This additives are used in engine oils,�uids for automatic gearboxes and hydraulic systems.

2.5.2. Anti wear and extreme pressure additives

These additives are used to reduce or eliminate friction and wear at extremelubrication conditions. This products are classi�ed in three types.

• Lubricity agents;

10

2.5. Additives

• Anti wear additives;

• Extreme pressure additives.

Lubricity agents

Generally, the lubricity agents are added to oils to decrease the coe�cient offriction at boundary lubrications conditions. This agents are used on automaticgearboxes to avoid phenomenons like "quawk" and "shatter" or in applications whenwhether "stick-slip" phenomenon is to be avoided.

Anti wear additives

Anti wear additives create a protector �lm by reaction with metallic contactsurfaces. Examples of this agents are:

• Fatty oils;

• Organic compounds;

• Phosphoric esters;

• Zinc dithiophosphates;

• Phosphorus compounds;

• Zinc compounds and others;

• Alkaline compounds.

Extreme pressure additives

These additives have the mission to avoid the adhesion between metallic surfacesat relative sliding speed on extreme pressure conditions. Extreme pressure additivescreate a protector �lm, by chemically combining with metallic surfaces. When ruptureof the �lm occurs by high contact pressure or high sliding speed, or because high localtemperatures, this protective �lm reduces the adhesion, preventing micro weldingbetween roughness peaks. These additives are constituted by sulfurous compounds,phosphorous or chlorine, or mixtures thereof and zinc dithiophosphate.

These additives are chemically active, which can create some chemical instability[10].

2.5.3. Anti oxidants

Anti oxidants have the objective to avoid, retard or modify the reaction of thehydrocarbons with oxygen in the lubricants oils. The oxidation in the oils originsacidic compounds, in it soluble, increasing the viscosity.

These are the main anti-oxidant additives:

• Sulfur compounds;

11

2. Lubricant types

• Phosphorus compounds;

• Sulfur-phosphorus compounds;

• Amines and phenol derivatives.

An antioxidant can actuate in di�erent manners, it can actuate in the oil or inthe metal parts that are in contact.

Other additives:

2.5.4. Detergent additives

This type of additive has detergent and dispersion properties. It has also themission of avoiding the formation of viscous deposits.

2.5.5. Corrosion inhibitors

Generally are also antioxidants, and have the same functions were said in (an-tioxidants) .

2.5.6. Rust inhibitors

This type of inhibitors are used to protect the ferrous components against cor-rosion, according [9]. With a chemical and physical interaction with certain metals,creates a continuous �lm that adheres to metal surfaces, preventing water contactbetween them. Some typical elements for these product are: esters of organic acids,acid esters, phosphorous and certain metal soaps.

2.5.7. Foam inhibitors

Foam inhibitors avoid the formation of the foam caused by intense agitationin high speed machines. This additive is very usually e�cient and it is used in lowquantities. The most used inhibitor additive is silicone.

2.5.8. Additives designed to lower the freeze point

Generally, these are the high molecular weight polymers that are found in par-a�ns, naphtenic oils dispense these additives because they naturally have a low freezepoint.

12

3. Physical properties of thelubricant oils

Physical properties are very important in the study of the lubricant oil beha-viour. This chapter is a small introduction to the physical properties of the lubricantoils. This chapter is based on [7] and [9].

3.1. Viscosity

From all physical and chemical properties, viscosity is the most important.Di�erent oils have di�erent viscosity that vary at di�erent rates with temperature.

3.1.1. Dynamic viscosity

Viscosity of a �uid is the resistance of a �uid against all of the internal sheardeformation. This opposing force can be calculated according to Newton's formulafor laminar �ows between a moving surface with velocity V and �xed surface, asshowed in �gure 3.1 .

Between two �at surfaces the di�erent �uid layers will be moving with di�erentvelocities v ranging from 0 to V. To y distance to the �xed surface, the �ow velocityis v and v+dv at the distance is y+dy, then the shear stress σxy, represented by τ is3.1:

τ = σxy = ηdv

dy(3.1)

where η is designed by dynamic viscosity and is a characteristic �uid coe�cient.

Figure 3.1.: Laminar �ow between two planes.

13

3. Physical properties of the lubricant oils

In a Newtonian �uids, this theory in which a proportional coe�cient existsbetween shear stress and velocity gradient it is veri�ed experimentally, but when �uidhas macromolecules or is submitted to special conditions, the proportional relationmay not be veri�ed (non-Newtonian �uid).

Table 3.1 shows the di�erent units of viscosity.

Table 3.1.: Viscosity units.Viscosity Dimension C.G.S. S.I. Correspondence

η dynamic ML−1T−1 Poise Po=g / cm · s Pascal secondPa· s=kg/m· s

1cPo=1mPa· s

ν kinematic L2T−1 Stokes=cm 2/s m2/s 1cSt=1mm2/s

3.2. Viscosity variation with temperature

The viscosity of the lubricant oils is greatly dependent of the temperature. Usu-ally, the viscosity of synthetic and mineral oils decreases when temperature increases.

Thermoviscosity

Thermoviscosity represents the viscosity behaviour with temperature.One of the expressions to calculate the viscosity variation with temperature is

the ASTM D341 equation 3.2:

LogLog(ν + a) = n−mLog(T ) (3.2)

Where:

• ν is kinematic viscosity (in cSt);

• T is Temperature (in Kelvin);

• m, n, a - constants that depend on each lubricant.

A more accurate expression is the Vogel equation 3.3:

ν = Ke[ bθ+c

] (3.3)

where:

• ν is cinematic viscosity at temperature θ;

• K, b, c are constants that depend on each lubricant;

• θ is temperature (in oC).

14

3.2. Viscosity variation with temperature

3.2.1. Viscosity index

Viscosity index is an empirical parameter which compares the kinematic viscos-ity of an oil with two reference oils which have a completely di�erent behaviour withtemperature.

According to standard ASTM D2270 [11] exists two methods to calculate theviscosity index:

• Procedure A - For oils of viscosity index up to and including 100, equation 3.4;

• Procedure B - For oils of viscosity index of 100 and greater, equations 3.5 and3.6.

Procedure A

V.I. =L− UL−H

· 100 (3.4)

Procedure B

V.I. =10N − 1

0.00715+ 100 (3.5)

Where:

N =logH − log V

log Y(3.6)

• L is the kinematic viscosity at 40 oC of an oil of 0 viscosity index having thesame kinematic viscosity at 100 oC as the oil whose viscosity index is to becalculated, in cSt;

• H is the kinematic viscosity at 40 oC of an oil of 100 viscosity index havingthe same kinematic viscosity at 100 oC as the oil whose viscosity index is to becalculated, in cSt;

• Y is the kinematic viscosity at 100 oC of the oil whose viscosity index is to becalculated in cSt;

• U is the kinematic viscosity at 40 oC of the oil whose viscosity index is to becalculated in cSt;

It should be noted that the Viscosity Index is not enough to fully characterizean oil. Generally two oils with same Viscosity Index don't exhibit the same viscosityvariation with temperature.

15

3. Physical properties of the lubricant oils

3.3. Viscosity variation with pressure

Most of the lubricating oil viscosity increases with the pressure.

Piezoviscosity

Piezoviscosity represents the viscosity behaviour with pressure.The lubricant behaviour at the extreme pressures (0.5 - 4.0 GPa), that occur in

EHD contacts is very important since piezoviscous properties are of great importanceon the �lm thickness generation. One of the expressions to calculate the viscosityvariation with pressure is Barus law 3.7:

ηS = η0e[α·p] (3.7)

where:

• ηS is dynamic viscosity at pressure p;

• η0 is dynamic viscosity at atmospheric pressure;

• α is piezoviscosity coe�cient in Pa−1.

Equation 3.7 considers that the coe�cient of piezoviscosity is pressure inde-pendent and it is de�ned at the oil temperature in contact inlet. This equation isalso proven to be quite inaccurate for pressures above 0.5 GPa and for high ambienttemperatures.

Other relations [7] between pressure and viscosity have been derived in order tosolve the problems associated with equation 3.7.

Gold [12] has proposed a lubricant and temperature dependent equation (3.8)to calculate the piezoviscosity.

α = sνt × 10−9 (3.8)

Table 3.2 shows the constants of the oil types:

Table 3.2.: Constants of the oil types.Oil type Mineral PAO

s @ 0.2 GPa 9.904 7.382t @ 0.2 GPa 0.1390 0.1390

16

3.4. Viscosity variation with shear strain rate

3.4. Viscosity variation with shear strain rate

When the shear strain rate imposed in lubricant �lm does not a�ect dynamicviscosity, it is because the �uid is Newtonian. If the dynamic viscosity a�ects theshear strain rate, �uid is considered non-Newtonian.

In some contacts, when lubricants operate at extremely high shear strain rateconditions, the viscosity can decrease.

A rheological model needs to be established in order to take into account thee�ects of pressure, temperature and shear strain rate since Newton's model 3.1, isno longer valid. Figure 3.2 shows the behaviour of the �uids with viscosity variationwith shear strain rate.

Figure 3.2.: Viscosity variation with shear rate: a) Grease b) Newton Fluid c) non-Newton Fluid.

17

4. Oils characterization

Before the beginning of the experimental tests, it was necessary to evaluatesome of the relevant physical properties of the tested oils:

• Viscosity variation with temperature;

• Density variation with temperature ;

• Viscosity variation with shear strain rate.

This process consisted in measuring the kinematic viscosity with an Englerviscometer at three di�erent temperatures (40, 70 and 100 oC). The dynamic viscosityvariation with shear rate was evaluated with a Rheomat 115 rheometer. The dynamicviscosity was evaluated with a SV-10 vibrational viscometer. The density variationwith temperature was evaluated using a DMA 35N densitometer.

4.1. Selected gear oils

The physical properties of the selected oils were listed in the manufacturer'sdata sheets (presented in Appendix A) and they are presented in table 4.1.

Table 4.2 shows the main characteristics of the three oils that were selected.The 75W90 and 75W140 lubricants are synthetic based oils, they are PAO, and the80W90 is a mineral based oil.

Table 4.1.: Oil properties provided by the manufacture.Parameter Unit Physical Properties

Designation - 75W90 75W140 80W90Base oil - PAO PAO MineralViscosity at 40 oC cSt 101 183 115Viscosity at 100 oC cSt 15 26 14.1Density at 15 oC kg/m3 866 885 886Viscosity Index - 157 178 123Pour point oC -51 -36 -33

19


Table 4.2.: Test oils characteristics.SYN FE 75W90 SYN FE 75W140 RS FE 80W90

Excellent properties atextreme pressure andanti-wear

Excellent properties atextreme pressure andanti-wear

High extreme-pressureproperties

High oxidation resistanceand high stability at highshear rates

Very high viscosity index High viscosity index

Excellent �lterability andgood anti-foam behaviour

Excellent anti-foaming,anti-corrosion and anti-foam properties

Excellent anti-foaming,anti-corrosive and anti-rust properties

Excellent stability in ser-vice

Excellent stability in op-eration

Low viscosity reductionsat high shear rateVery good therm stability

4.2. Engler viscometer

The viscosity characterization using this method is based in the time that ittakes to a certain volume of oil to pour from a recipient relative to the pouring timeof water.

The Engler viscometer has a recipient where the oil to be evaluated is poured.There is an hole at the bottom of that recipient where a wood pointer should beinserted or removed in order to stop or allow the oil to �ow from it. To keep the oil ata selected temperature, this recipient is inside another recipient with a �uid betweenthem (oil or water). This �uid is heated by a resistor thus allowing temperaturecontrol.

There are two thermometers, one in the tested oil and other in the heat carrying�uid. The measurements follow the IP 212/92 standard. Figures 4.1a and 4.1b showthe Engler viscometer.

20

4.2. Engler viscometer

Table 4.3.: Physical properties of the axle oils (measured).Parameter Unit 75W90 75W140 80W90

Physical PropertiesoEngler at 40oC - 15.27 27.25 16.76oEngler at 70oC - 5.23 8.59 4.94oEngler at 100oC - 2.63 3.73 2.33Viscosity at 40oC cSt 116.24 207.63 127.57Viscosity at 70oC cSt 36.89 65.15 35.63Viscosity at 100oC cSt 17.94 27.24 15.15Viscosity index - 171 167 122α at 40oC Pa−1 1.39× 10 −8 1.51 × 10 −8 1.94 × 10 −8

α at 70oC Pa−1 1.19 × 10 −8 1.29 × 10 −8 1.63 × 10 −8

α at 100oC Pa−1 1.09 × 10 −8 1.15 × 10 −8 1.45 × 10 −8

β at 40oC oC−1 8.45 × 10 −5 4.61 × 10 −5 7.31 × 10 −5

β at 70oC oC−1 2.76 × 10 −5 1.51 × 10 −5 2.39 × 10 −5

β at 100oC oC−1 1.36 × 10 −5 7.38 × 10 −6 1.17 × 10 −5

Vogel constantsk - 0.7632 0.0567 0.1976b - 61.3899 1497 82.2c - 509.5748 142.4155 790.6585

ASTM D341 constantsn - 2.7790 2.6932 3.2143m - 7.2511 7.0867 8.3460

(a) Engler viscometer. (b) Engler viscometer used.

Figure 4.1.: Engler viscometer.

21


40 50 60 70 80 90 100 110 1200

50

100

150

200

250

T[ºC]

Vis

cosi

ty [c

St]

Variation of the viscosity with temperature

75W9075W14080W90

Figure 4.2.: Kinematic viscosity variation with the temperature.

22

4.3. Rheometer

Table 4.3 shows the Engler degrees which allow calculate the kinematic viscosity,the viscosity calculated with Vogel equation 3.3, the piezoviscosity, the thermoviscos-ity coe�cients at 40, 70 and 100 oC, the Vogel and ASTM D341 constants.

In the �gure 4.2 it is possible observe that 75W140 has the highest viscosity atall temperatures. The 75W90 and 80W90 have a similar viscosity, although 75W90has a higher viscosity index.

4.3. Rheometer

The rehometer that was used was a Contraves Rehomat 115 as shown in the�gure 4.3.

This is a coaxial measurement system that works according to Searle's prin-ciple. The measuring shaft is rotating inside the testing �uid, and it is powered byan electrical motor. The opposing torque is measured and recorded, then using New-ton's law, from the opposing torque and angular speed the dynamic viscosity can bedetermined.

Figure 4.3.: Rheomat 115 rheometer.

23


0 200 400 600 800 1000 12000

50

100

150

200

250Dynamic Viscosity vs Shear Rate

Shear Rate [1/s]

Dyn

amic

Vis

cosi

ty [m

Pa.

s]

40ºC

70ºC

100ºC

75W9075W14080W90

Figure 4.4.: Dynamic viscosity variation with the shear rate.

Figure 4.4 shows the dynamic viscosity variation with temperature measuredwith Contraves Rehomat 115 rheometer. The measurements were done at di�erenttemperatures 40,70 and 100oC. The results showed the dynamic viscosity variationwith shear rate have a slight variation with shear rate within the measurements range,during the measurement there was a slight temperature variation, which in�uencedthe results. For this shear rate range the oils can be considered Newtonian.

4.4. Densitometer

A densitometer was used to measure the tested oils density. The densitometerused in the tests was a DMA 35N, it uses 2 ml of the oil sample and the oil temperaturemay not exceed 40 oC.

Table 4.4 displays the values measured with the DMA 35N, these results wereobtained with average of three measurements to each temperature. The �gure 4.5shows the densitometer used.

24

4.4. Densitometer

Table 4.4.: Physical properties of the axle oils (measured).Oil T [oC] Density [kg/m3]

75W9018.5 86828 861.733 858.8

75W14022.5 880.926.1 876.531.65 875.65

80W9021 882.230 87736 872.4

Figure 4.5.: DMA 35N Densitometer.

0 20 40 60 80 100810

820

830

840

850

860

870

880

890

900

T[ºC]

Den

sity

[kg/

m3 ]

Variation of the density with temperature

75W9075W14080W90

Figure 4.6.: Density variation with the temperature obtained with measured values(table 4.4).

25


Table 4.5.: Physical properties of the axle oils.Parameter Units 75W90 75W140 80W90

Physical PropertiesDensity at 40oC kg/m3 854.4 870.9 869.8Density at 70oC kg/m3 835.3 853.6 850.2Density at 100oC kg/m3 816.3 836.4 830.6

Table 4.5 shows the density of the oils at 40, 70 and 100oC.The variation of the density with temperature is presented in �gure 4.6, these

linear curves were obtained with measured values (table 4.4). All oils show a lineardecrease of density with increasing temperature, and it is possible see that the 75W90has the lower density at all temperatures and the 75W140 and 80W90 are quitesimilar.

4.5. Vibrational viscometer

The SV-10 vibrational viscometer measures the �uid viscosity by vibratingvanes. Both vanes vibrate at certain frequency and the viscosity of the �uid dampensthe vibration amplitude. In order to maintain constant amplitude, current is led tothe actuator. The current that is required by the actuator is measured, and correlatedwith the viscosity. The vibrational viscometer is shown in �gure 4.7.

Figure 4.7.: SV-10 Vibrational viscometer.

26

4.5. Vibrational viscometer

Figure 4.8.: Dynamic viscosity variation with the temperature.

The variation of the dynamic viscosity with temperature is presented in �gure4.8. Is possible to observe that 75W140 has the highest viscosity at all temperatures,which is in accordance to �gure 4.2.

With dynamic viscosity and density is possible calculate the kinematic viscosity,so to obtain more correct measurements. Table 4.6 shows the comparison betweenthe kinematic viscosity calculated with Vogel equation 3.3 and the kinematic viscositycalculated with dynamic viscosity measured with SV-10 vibrational viscometer anddensity.

Table 4.6.: Comparison of the kinematic viscosity [cSt].75W90 75W140 80W90

Temperature Vogel SV-10 Vogel SV-10 Vogel SV-10

40oC 116.24 113.08 207.63 244.69 127.57 136.0970oC 36.89 36.17 65.15 74.49 35.63 38.71100oC 17.94 15.34 27.24 31.18 15.15 14.99

27

5. Transmission test rig and testgearbox

5.1. Transmission test rig

Figure 5.1 shows a schematic of the gearbox test rig.This is a back-to-back gearbox test rig that follows the principle of recirculating

power.The output speed on the slave gearbox is the same as the input speed on the

testing gearbox, so both gearboxes work on a back-to-back con�guration. Thus, onlyreversible gearboxes can be tested.

The torque loading mechanism consists of an hydraulic cylinder that introducesan axial displacement on one of the helical gears on the gear set 2. It creates atorsional displacement and loads the test rig with a static torque. This displacementis due to axial displacement of an helical pinion that makes the wheel slightly rotate.Figure 5.2 shows this hydraulic cylinder.

The test rig can be adjusted to di�erent gearbox sizes and arrangements, throughadjustment of the platforms 12 and 14. The torque transducer (5) can have the heightand depth adjusted using mobile platform (13).

The back shafts are splitted in two pieces, to avoid critical speeds. Figure 5.3shows the test rig.

The data read by all the sensors is displayed and saved in the central control, asshown in �gure 5.4. In this con�guration the highest torque and the lowest speed isin between gearboxes. So, the working conditions that this test rig allows in currentcon�guration are:

• Input speed: [25 ; 475] rpm

• Input torque: [400 ; 5200] Nm

29

5. Transmission test rig and test gearbox

Figure 5.1.: Test bench rig.

In order to obtain the data of the experimental tests, the test bench is equippedwith several sensors. There are two torque and speed sensors and four temperaturesensors that measure:

• The input and output torques on the test gearbox;

• The input and output speeds on the test gearbox;

• The room temperature (type K thermocouple);

• The oil temperature on the test gearbox in two di�erent zones (industrial gradePT100 RTD's);

• The wall temperature on the test gearbox (industrial grade PT100 RTD's).

Figure 5.5 shows the three temperature sensors in the tested gearbox.The test rig is driven by an electric motor, and its characteristics are presented

in table 5.1.

Table 5.1.: Electric motor characteristics.Type PFMH-250M83

Nominal speed 1480 rpmNominal Power 55 kWFrequency 50 HzCos F 0,87Voltage 220/330 V

Current intensity 181/104 A

30

5.1. Transmission test rig

Figure 5.2.: Hydraulic cylinder.

Figure 5.3.: Test Rig.

31


Figure 5.4.: Central control.

(a) Oil temperature sensors. (b) Wall temperature sensor.

Figure 5.5.: Temperature sensors positioning in the tested gearbox [13].

32

5.2. Planetary gearbox

5.2. Planetary gearbox

This planetary gear set consists of a sun, three planet gears, an internal ringgear and a rigid planet carrier.

The planetary gearbox is set to work as a multiplier with ratio of 4. The inputof the gearbox is the planet carrier and the output is the sun gear, in this case thering is �xed. In other words, if the input speed is 100 rpm and input torque is 2000Nm, the outputs are 400 rpm and 500 Nm respectively.

Figure 5.6 shows a schematic representation of the planetary gearbox and �gures5.7 and 5.8 show some details of the planetary gearbox. The rolling bearings, areestimated based on the size and dimension of the gearbox, in the shaft diameter and inthe schematic representation of �gure 5.6. The deep groove ball bearing and taperedroller bearings are grease lubricated, so they have their own lubrication. Table 5.2displays the rolling bearings and seals of the planetary gearbox.

Figure 5.6.: Schematic of the planetary gearbox [13].

Figure 5.7.: Planetary gearbox [13].

33


Figure 5.8.: Detail of planet gearbox [13].

Table 5.2.: Rolling bearings and seals of the planetary gearbox.Component Quantity Reference

Deep groove ball bearing 1 6217-2ZTapered roller bearings 2 32022 X/QNeedle roller bearings 6 K 40× 48× 20Input and output seal 2 BAUM6 SLX7 140-170-

13/12 CFW A1

Table 5.3 displays the geometric characteristics of the gears used in the planetarygearbox.

Table 5.3.: Geometric characteristics of the gears used in the planetary gearbox.Symbol Units Sun Planet Ring

Number of teeth Z 36 36 -108Pro�le shit coe�cient x mm -0.0189 -0.0189 0.0566Reference diameter d mm 73.111 73.111 -219.332Base diameter db mm 68.577 68.577 -205.731Tip diameter da mm 77.035 77.035 -215.106Width b mm 42Normal module m mm 2Transverse module mt mm 2.031Pressure angle α o 20Working transverse pressure angle αwt

o 20.122Helix angle β o 10Center distance a mm 73.111Working center distance aw mm 73.035Gear ratio i - 4Arithmetic average roughness Ra µm 0.4

34

5.3. Planetary gearbox loads and kinematics


5.3.1. Load analyses

Before calculating the power loss load dependence it is necessary to determinethe forces acting in each contacting components. Properly labelled free body diagramsare presented along with the respective formulation.

Aiming for simplicity, forces of inertia, moments of inertia and gravity forceswere disregarded.

Figure 5.9.: Representation of the planetary gear (side view).

Figures 5.9 and 5.10 display the planetary gearbox where the sun is representedwith number one, the planets with number two, the ring with number three and theplanet carrier is number four.

From the input torque (applied in the planet carrier) it is possible to calculatethe forces in the planet carrier. So, �gure 5.11 is a free body diagram (FBD) of theplanet carrier. The three forces applied to this body are represented.

Equation 5.1 is the sum of the moment in point E.∑ME = 0 (5.1)

The forces F24 are calculated dividing the input torque by the number of planets(N) and the centre distance (a), equation 5.2. Equation 5.3 is the equality betweenF24 and F42.

35


F24 =Mmot

N

a(5.2)

Figure 5.10.: Representation of the planetary gear.

Figure 5.11.: FBD of the carrier.

36


|F24| = |F42| (5.3)

Figure 5.12.: FBD of the planet.

In the planets, the sum of the forces are represented by equation 5.4:∑−→F =

−→F42 +

−→F12 +

−→F32 = 0 (5.4)

Where equations 5.5 and 5.6 represent the−→F12 and

−→F32, respectively:

−→F12 =

−−→Ft12 +

−−→Fr12 (5.5)

−→F32 =

−−→Ft32 +

−−→Fr32 (5.6)

Decomposing the forces in the x and y axis and knowing that the sum of theforces in the axis is zero, the result is equations 5.7 and 5.8:∑

Fx = 0⇔ F42 = Ft12 + Ft32 (5.7)

∑Fy = 0⇔ Fr12 = Fr32 (5.8)

Where equation 5.9 allows to calculate the Ft12 and Ft32:

Ft12 = Ft32 =−F42

2(5.9)

The radial forces are calculated with equation 5.10:

37


|Fr12| = |Fr32| = |Ft12 · tanαt| (5.10)

In �gure 5.13 the forces acting in the sun gear are shown.

Figure 5.13.: FBD of the sun.

The moment in point A is calculated with equation 5.11.∑MA = AB · (

−−→F 1t21 +

−−→F 2t21 +

−−→F 3t21) +

−−→Mext =

−→0 (5.11)

Where equations 5.12 and 5.13 represent the AB and each tangential force inthe sun gear:

AB =d1

2(5.12)

|F 1t21| = |F 2

t21| = |F 3t21| (5.13)

So, equation 5.14 allows to calculate the moment in the sun gear:

−−→Mext = 3 ·

−−→Ft21 ·

d1

2(5.14)

The radial force acting on the planets is evenly distributed, so equation 5.15:

|F 1r21| = |F 2

r21| = |F 3r21| (5.15)

The reaction in A can be obtained through equation 5.16.∑−→F =

−→F 1

21 +−→F 2

21 +−→F 3

21 +−→F01 =

−→0 (5.16)

The forces acting in point A on xx direction are represented in equations 5.17and 5.18:

38


F x01 − F 1

t21 +−−−→|F 2t21| · sin(30) +

−−−→|F 3t21| · sin(30) = 0 (5.17)

F x01 = 0 (5.18)

The forces acting in point A on yy direction are represented in equations 5.19and 5.20:

F y01 −−−−→|F 2t21| · cos(30) +

−−−→|F 3t21| · cos(30) = 0 (5.19)

F y01 = 0 (5.20)

The axial force can be calculated according to equations 5.21 and 5.22:

|−−→Fa12| = |

−−→Ft12| · tan β (5.21)

|−−→Fa32| = |

−−→Ft32| · tan β (5.22)

The results of the forces with conditions 200 rpm and 2800 Nm are representedon table 5.4.

Table 5.4.: Forces at 200 rpm and 2800 Nm.Variable Result Units

F42 12361 NTangential force F i

t21 6180.7 NRadial force F i

r21 ; Fr32 2284.3 NAxial force Fa12 ; Fa32 1089.8 N

5.3.2. Kinematic analysis

The kinematic analysis is fundamental to properly formulate the power lossmodel. In this subsection the method to calculate the velocities is explained. Figure5.10 helps to understand the relation of the gears and has the numbers and the pointsrepresented.

The input speed is on the planet carrier (4), so the velocity in point C is known,because point C belongs to the planet carrier and belongs to planet, as such, theplanet velocity can be calculated.

Equation 5.23 allows to calculate the planet velocity.

−−→vC40 = −−→vC20 (5.23)

In order to obtain the rotational speed of the planet, the Mozzi equation hadto be used, so as knows that velocity in centre of planet carrier and in point D arenull, the expression used is equation 5.26.

−−→vA40 +−→ω40 ·−→AC = −−→vD20 +−→ω20 ·

−−→DC (5.24)

39


−→AC is the sum of the radius of the sun and the planet and

−−→DC is the planet

radius. 00ω40

· 0

r1 + r2

0

=

00ω20

· 0−r2

0

(5.25)

The �nal is equation 5.26, that it is the result of equations 5.24 and 5.25.

ω20 = −ω40r1 + r2

r2

(5.26)

With rotational speed of the planet it is possible to determine the rotationalspeed of the sun.

−−→vB10 = −−→vB20 (5.27)

−−→vA10 +−→ω10 ·−→AB = −−→vD20 +−→ω20 ·

−−→DB (5.28)

−→AB is the sun radius and

−−→DB is the planet diameter. 0

0ω10

· 0

r1

0

=

00ω20

· 0−2r2

0

(5.29)

ω10 = −ω202r2

r1

= ω402(r1 + r2)

r1

(5.30)

Solving equations 5.26 and 5.30 in order to the number of teeth and gear modulethe result is equations 5.35 and 5.36.

The ring radius is equation 5.31:

r3 = r1 + 2r2 (5.31)

The gear module is calculated with equation 5.32.

m =d

Z(5.32)

The equations 5.33 and 5.34 are to calculate the radius.

r =mZ

2(5.33)

So:

ri =mZi

2(5.34)

Where i = 1, 2 or 3

ω20 = −ω40(1 +Z1

Z2

) (5.35)

40


ω10 = ω40(2Z1 + 2Z2

Z1

) (5.36)

The rotational speed of the planetary gearbox components at 200 rpm and 2800Nm are represented in the table 5.5.

Table 5.5.: Rotational speed of the gearbox components at 200 rpm and 2800 Nm.Variable Result Units

Planet carrier w40 200 rpmPlanet w20 -400 rpmSun w10 800 rpm

41

6. Thermal balance of the gearbox

The thermal balance depends the oil temperature, so that in the system themechanical power losses and thermal evacuation of energy are balanced, as it is statedin equation 6.1.

PV = Q̇total (6.1)

In the next paragraphs the power losses (PV ) and dissipated heat (Q̇total) areexplained in detail.

6.1. Heat dissipation

In transmission mechanisms, lubricated in oil bath, the most part of the heatdissipated to exterior is to form of convention and radiation by surface of carter, buta little part of the heat is evacuated by conduction through the shafts and othercomponents. Temperature is a very important factor in order to determine the heatdissipated, resulting in expression 6.2.

.

Qtotal=.

Qrad +.

Qconv +.

Qcond (6.2)

According to Höhn [14] the total heat dissipated can be calculated by equation6.3.

.

Qtotal= αheatA(Toil − Troom) (6.3)

where:

• αheat is the heat transfer coe�cient, which takes into account heat transfer dueto radiation, convection and conduction;

• A is the external area of the gearbox;

• Toil is the oil temperature;

• Troom is the room temperature.

The αheat takes into account heat transfer due to radiation, convection andconduction, but as suggested by R.Camacho [13] this coe�cient hasn't much value,once that the air properties weren't measured. As the air properties aren't controlledthe heat transfer coe�cient would be inaccurate, because the humidity (which variesdaily) varies the αheat, so this fact might be a relevant factor in the stabilizationtemperature.

In order to show the in�uence of water vapour in the heat transfer the speci�cheat at constant pressure of dry air and water vapour are shown:

43


• cPwater vapour = 1.84 kJ/kgoC

• cPdry air = 1.01 kJ/kgoC

6.2. Power Loss model

Figure 6.1.: Components of the power loss [14].

The power loss model is based on [14] and consists in gear, seal, bearing andauxiliary losses. Gear and bearing losses are divided in two parts, load and no-load losses. No-load losses occur without torque transmission, with rotation of themechanical components, load dependent losses occur in the contact of the powertransmitting components.

Load gear power loss depends of the transmitted power, coe�cient of frictionand gear loss factor, this portion is the most important. No-load gear power loss varywith diameter, speed, immersion depth, oil viscosity and internal gearbox design.

The seals power losses varies with tangential speed and shaft diameter (inde-pendent of the transmitted power).

No-load rolling bearing power losses depend on the bearing type and size, rollingbearing arrangement, lubricant viscosity and immersion depth. Load bearing powerlosses vary with bearing type and size, operating speed and quantity of lubricant [15].

Figure 6.1 shows the power loss model and its components.

6.2.1. Gears power loss

Gear friction power loss

Gear friction power loss depends to the transmitted power, average coe�cientof friction in the gear contact and gear loss factor HV . The expression to calculatethis power loss is suggested in [14] and [16]:

PV ZP = PA · µmz ·HV (6.4)

The transmitted power is calculated using equation 6.5:

PA = Fbt · rb · ω (6.5)

44


The gear loss factor, HV can be analytically calculated obtained on the assump-tion that the coe�cient of friction is constant along the line of action, and it can becalculated according to equation 6.6, [17]:

Hv =π · (i+ 1)

z1 · i · cos(βb)· (a0 + a1 · |ε1|+ a2 · |ε2|+ a3 · |ε1| · ε1 + a4 · |ε2| · ε2) (6.6)

where:

• i is the gear ratio;

• z1 is the number of teeth of the pinion;

• βb is the base helix angle;

• ε1 ,2 are the tip contact ratios: pinion and wheel, respectively;

• a0 ,1 ,2 ,3 ,4 are the coe�cient dependent on the tip contact ratios;

• εα is the transverse contact ratio.

Based on ε1, ε2 and εα three parameters are de�ned with equations 6.7 to 6.9with lg, mg and ng ∈ Z:

ε1 ∈ [lg − 1 : lg] (6.7)

ε2 ∈ [mg − 1 : mg] (6.8)

εα ∈ [ng − 1 : ng] (6.9)

For example, considering ε1 = 1.12, ε2 = 1.03 and εα = 2.15, the parametersare lg = 2, mg = 2 and ng = 3.

a0 ,1 ,2 ,3 ,4 can be calculated according to table 6.1:

Table 6.1.: Formulation of the coe�cients ai .εα < 1 εα > 1 ; ε1 <

0 ∨ ε2 < 0εα > 1 ; ε1, ε2 > 0 ;lg +mg = ng

εα > 1 ; ε1, ε2 > 0 ; lg +mg = ng + 1

a0 0 0 2·lg ·mgng

2(lg ·mg−ng)

ng−1

a1 0 1 lg(lg−1)−mg(mg−1)−2·lg ·mgng(ng−1)

lg(lg−1)+mg(mg−1)−2(mg−lg)ngng(ng−1)

a2 0 1 −lg(lg−1)+mg(mg−1)−2·lg ·mgng(ng−1)

lg(lg−1)+mg(mg−1)−2(mg−lg)ngng(ng−1)

a31εα

0 2·mgng(ng−1)

2(mg−1)

ng(ng−1)

a41εα

0 2·lgng(ng−1)

2(lg−1)

ng(ng−1)

Equation 6.6 was derived disregarding the elasticity of the meshing tooth.So the gear loss factor was calculated with the KISSsoft R© software [18]. KISSsoft R©

software allows calculation of a many gear arrangements, including planetary gears.HV can be calculated with equation 6.4 once KISSsoft R© calculates the average powerloss with an imposed coe�cient of friction. The HV used in this work can be seen intable 6.2:

45


Table 6.2.: HV calculated with KISSsoft R©.Contact HV factor

Sun/Planet 0.1677Planet/Ring 0.0625

Coe�cient of friction

The coe�cient of friction between gear tooth has a great importance in mesh-ing analyses, because it directly interferes in power transmission e�ciency, contacttemperature and in the probability of occurrence of surface failures.

It is di�cult to de�ne the coe�cient of friction in gears, because along themeshing line all the lubrication regimes can exist: boundary �lm, mixed �lm and full�lm.

Equation 6.10 [19] allows to calculate the average coe�cient of friction betweengear tooth and is valid mostly for mixed lubrication conditions.

µmz = 0.048 · ( Fbt/l

v∑C · ρC)0.2 · η−0.05

oil ·R0.25a ·XL (6.10)

XL is the lubricant correction factor, for non-additived mineral oils it has thevalue of XL = 1.

Lubricant �lm thickness between tooth contact

According to Dowson and Higginson [20], the �lm thickness in linear elast-ohydrodynamics contacts is represented in �gure 6.2.

Figure 6.2.: Lubricant �lm thickness in linear contacts [21].

46


This theory considers an isothermal contact, smooth surfaces and abundant lub-rication, so the �lm thickness in centre of the contact h0 and minimum �lm thicknesshm can be calculated using equation 6.11 and 6.12, respectively.

h0 = 0.975 ·Rx · U0.727 ·G0.727 ·W−0.091 (6.11)

hm = 1.325 ·Rx · U0.7 ·G0.54 ·W−0.13 (6.12)

The expressions of the various parameters of the equations 6.11 and 6.12 arepresented in table 6.3.

Table 6.3.: Parameters of the �lm thickness.Parameter Designation Expression

Speed parameter U U = η0(U1+U2)2RxE∗

Material parameter G G = 2αE∗

Load parameter W W = FnRxlE∗

where:

• η0 is the dynamic viscosity;

• U1 ,2 is the velocity of each surface;

• Rx is the equivalent radius;

• E ∗ is the equivalent Young module;

• α is the piezoviscosity coe�cient;

• Fn is the normal force in the tooth;

• l is the average length of the contacting lines on a helical gear.

The average length of the contacting lines on a helical gear can be calculatedaccording to equation 6.13.

l =b · εα

cos(βb)(6.13)

Table 6.4 shows the dependence of �lm thickness of the parameters used tocalculate the groups U, V and W , so with this table it is possible to understand howthe �lm thickness varies with these parameters.

47


Table 6.4.: Dependence of the �lm thickness with the parameters.Parameter Exponent Variation range Dependence

η0 0.7 - large large very importantU = U1 + U2 0.7 - large large very important

α 0.54 - large small little importanceRx 0.43 - medium large importantE∗ 0.03 - null small not dependentFnl

0.13 - small large little importance

At the inlet zone the lubricant is submitted to high shear rates, that increasesthe lubricant temperature, thus contributing to decrease the oil viscosity and reducingthe �lm thickness.

In order to account for this e�ect the inlet shear heating factor is used and isrepresented in equation 6.14 [22].

φT = (1 + 0.1(1 + 14.8 · V 0.83e )L0.64))−1 (6.14)

where:

Ve =|U1 − U2||U1 + U2|

(6.15)

L =βη0(U1 + U2)2

Kf

(6.16)

where:

• β is the thermoviscosity coe�cient of the lubricant at initial temperature;

• Kf is the thermal conductivity of the lubricant.

Finally, the corrected �lm thickness can be calculated with equation 6.17:

h0T = φT · h0 (6.17)

Lastly, the speci�c �lm thickness can be calculated according to equation 6.18,[21], [22]:

Λ =h0T

σ(6.18)

Where, σ is the equivalent root mean square roughness and can be calculatedby expression 6.19.

σ = (σ21 + σ2

2)1/2 (6.19)

The σ1,2 (RMS) considered was 0.5 µm.The lubrication regime is a function of the speci�c �lm thickness and the three

lubrication regimes are shown in table 6.5 and in �gure 6.3 [21], [23].

48


Table 6.5.: Lubrication regimes.Boundary �lm Λ < 0, 7 The normal force is totally supported by

metal-metal contact between roughnesspeaks of the surfaces.

Mixed �lm 0, 7 < Λ < 2 The normal force is simultaneously sup-ported by metal-metal contact betweenroughness peaks of the surfaces and by thelubrication �lm.

Full �lm Λ > 2 The normal force is totally supported bythe EHD lubrication �lm that separatescompletely the surfaces, preventing metal-metal contact.

Figure 6.3.: Lubrication regimes [23].

The speci�c thickness �lm and coe�cient of friction are correlated. The Stribeckcurve presented in �gure 6.4 not only shows this correlation, but also shows that thecoe�cient of friction is minimum in mixed lubrication conditions.

Figure 6.4.: Stribeck curve [24].

49


Relation between lubrication regime and tooth damage

In EHD contacts there is a relation between the lubrication regime and thesurface distress probability [25].

Figure 6.5 shows the surface damage probability curves as a function of speci�c�lm thickness and tangential speed at the gear pitch point. At certain tangentialspeed, the probability of damage increases as speci�c �lm thickness decreases.

It can be veri�ed that 5% failure probability curve tends to a speci�c �lmthickness equal to 2, so for values greater than 2, for any tangential speed, thedamage probability is less than 5%.

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Pitch line velocity, [m/s]

Sp

ecif

ic f

ilm t

hic

knes

s, [Λ

]

Probability of wear related distress

5%

40%

80%

Figure 6.5.: Tooth surface distress probability as function of speci�c �lm thicknessand tangential velocity at pitch point [25].

The curves of �gure 6.5 can be �tted with the equations 6.20 to 6.22 [25]:

λ5% = (2.68863

vt+ 0.47767)−1 (6.20)

λ40% = (4.9179

vt+ 0.64585)−1 (6.21)

λ80% = (9.29210

vt+ 0.95507)−1 (6.22)

50


6.2.2. Power loss in planetary gears

The planetary gearbox has 3 planets, which amount to 6 meshing contacts, asshown in section 5.2.

Equation 6.4 must be applied to two di�erent engagements, one of the sun/planetand other of the planet/ring contact.

The total power loss PV ZP can be determined according equation 6.23.

PV ZP = 3 · PV ZPsun/planet + 3 · PV ZPplanet/ring (6.23)

The sun/planet and planet/ring contacts are quite di�erent, with di�erent gearloss factor and average coe�cient of friction.

6.2.3. Churning loss

These losses have been object of study throughout the years, several authorshave presented experimental and analytical studies about churning losses gears. Churn-ing losses result from �uid circulation inside the gearbox, which is strongly related tothe rotating gears immersed in lubricant oil.

According to Höhn et all [14], the main in�uencing parameters on gear churninglosses are diameter, speed, immersion depth and gear oils viscosity, additionally theinternal gearbox design can in�uence the no-load power loss of gears.

Despite the extensive bibliography about churning loss, no model was found fordetermination of churning losses in planetary gearboxes. The �uid �ow complexityand endless design possibilities of a planetary gearbox can be pointed out as thereason to the lack of churning loss models for this kind of gearboxes.

The total no-load losses can be calculated according to equation 6.24. Thetotal no-load losses depends on churning loss, no-load bearing losses (PV L0), seallosses (PV D) and auxiliary losses (PV AUX).

PV 0 = PV Z0 + PV L0 + PV D + PV AUX (6.24)

Due to the lack of proper churning loss models the total no-load loss (PV 0) wasmeasured and coupled to the numeric model. The no-load bearing losses and seallosses can be calculated as it is explained in sections 6.2.5 to 6.2.7, respectively. Sothe churning losses was calculated according expression 6.25.

PV AUX + PV Z0 = PV 0 − PV L0 − PV D (6.25)

The no-load bearing losses (PV L0) are relative to tapered roller bearings (mainlydue to the pre-load) and needle roller bearings. The auxiliary losses are due to otherscomponents which do not enter into the equation, in this case the planet carriermotion inside �uid.

Using this method it is possible to estimate the churning losses.

51


6.2.4. Rolling bearings power loss (Load dependent)

The total resistance to rotation in a rolling bearing is the result of rollingand sliding friction in the contact areas, between the rolling elements and raceways,between the rolling elements and cage, and between rolling elements and other guidingsurface.

The model used to calculate the numerical results of the groove ball bearing andtapered roller bearing losses was the SKF model [15]. The equation 6.26 presents theexpression to calculate the power loss in a rolling bearing, where M is the frictionalmoment (Nmm) and n is the rotational speed of the shaft (rpm).

PV L = M · n · π30· 10−3 (6.26)

The SKF model uses the equation 6.27 for calculating the frictional moment ofa rolling bearing.

M = Mrr +Msl +Mseal +Mdrag (6.27)

where:

• M is the total frictional moment;

• Mrr is the rolling frictional moment;

• Msl is the sliding frictional moment;

• Mseal is the frictional moment of seals;

• Mdrag is the frictional moment of drag losses, churning, splashing, etc.

Rolling frictional moment

The rolling frictional moment can be calculated using equation 6.28.

Mrr = Φish × Φrs ×Grr(ν · n)0.6 (6.28)

where:

• Φish is the inlet shear heating reduction factor;

• Φrs is the kinematic replenishment/starvation reduction factor;

• Grr is a variable depending on the bearing type, bearing mean diameter (dm[mm]),the radial force (Fr[N]) and axial load (Fa[N]);

• n is the rotational speed [rpm];

• ν is the kinematic viscosity at operating temperature of the oil [mm2/s].

52


Only a little bit of lubricant is used to form a hydrodynamic �lm, not all oflubricant can go through the contact area compared with the quantity of it availablein the bearing.

Therefore, some of the oil close to the contact inlet area is rejected and producesa reverse �ow. This reverse �ow shears the lubricant, generating heat, which lowersthe oil viscosity and reduces the �lm thickness and rolling friction. Figure 6.6 showsthis phenomena.

The inlet shear heating reduction factor is represented in equation 6.29.

Φish =1

1 + 1.84× 10−9(n · dm)1.28ν0.644(6.29)

where:

• dm is the rolling bearing mean diameter [mm] = 0.5(d+D) ;

• d is the rolling bearing bore diameter [mm];

• D is the rolling bearing outside diameter [mm].

Figure 6.6.: Reverse �ow.

For oil-air, oil jet, low level oil bath lubrication and grease lubrication methods,continuous over-rolling displaces excess lubricant from the raceways. The lubricantmay not have su�cient time to replenish the raceways, specially for applications whereviscosity or speeds are high, causing "kinematic starvation" e�ect. This e�ect reducesthe thickness of the hydrodynamic �lm and rolling friction. The expression thatrepresents the kinematic replenishment/starvation reduction factor can be estimatedusing equation 6.30.

Φrs =1

e

[Krs·ν·n(d+D)·

[KZ

2(D−d)

]0.5] (6.30)

where:

• Krs is the replenishment/starvation constant (for grease and oil-air lubricationis 6× 10−8 and for low level oil lubrication or jet lubrication is 3× 10−8);

• Kz is the rolling bearing type related geometric constant [15].

53


The rolling frictional variable (Grr) is dependent on the type of the rollingbearing and the load type.

Deep groove ball bearings

When Fa = 0

Grr = R1d1.96m F 0.54

r (6.31)

When Fa > 0

Grr = R1d1.96m

[Fr +

R2

sinαFFa]0.54

(6.32)

αF = 24.6(FaC0

)0.24[o] (6.33)

Tapered roller bearings

Grr = R1d2.38m (Fr +R2Y Fa)

0.31 (6.34)

The geometry constant R1,2 are listed in SKF catalogue and Y is a factor of thetapered roller bearing.

Sliding frictional moment

The sliding frictional moment can be calculated using equation 6.35.

Msl = Gslµsl (6.35)

where:

• Msl is the sliding frictional moment [Nmm];

• Gsl is the a variable depending on the rolling bearing type, the mean rollingbearing diameter [mm], the radial load [N] and the axial load [N];

• µsl is the sliding frictional coe�cient.

The sliding friction for full-�lm and mixed lubrication conditions can be estim-ated using equation 6.36.

µsl = Φbl · µbl + (1− Φbl)µEHL (6.36)

where:

• Φbl is the weighting factor for the sliding friction coe�cient represented in equa-tion 6.37;

• µbl is the coe�cient depending on the additive package of the lubricant, gener-ally is 0.1;

54


• µEHL is the sliding friction coe�cient in full-�lm conditions and the values are:

� 0.02 for cylindrical roller bearings;

� 0.002 for tapered roller bearings;

other bearings:

� 0.05 for lubrication with mineral oils;

� 0.04 for lubrication with synthetic oils;

� 0.1 for lubrication with transmission �uids;

Φbl =1

e2.6·10−8(nν)1.4dm(6.37)

The sliding frictional variable (Gsl) is di�erent to each type of the rolling bearingand the load type.

Deep groove ball bearings

When Fa = 0

Gsl = S1d−0.26m F 5/3

r (6.38)

When Fa > 0

Gsl = S1d−0.145m

[F 5r +

S2 · d1.5m

sinαFF 4a

]1/3(6.39)

Tapered roller bearings

Gsl = S1d0.82m (Fr + S2Y Fa) (6.40)

The geometry constant S1,2 are listed in a table in SKF catalogue [15].

Drag losses

When bearings lubricated by the oil bath method are partially submerged or,completely submerged, the drag losses should not be forgotten. Drag losses arein�uenced by bearing speed, oil viscosity, oil level and by the size and geometryreservoir.

In deep groove ball bearing and tapered roller bearings, this losses are zero,because these type of rolling bearings are auto-lubricated with grease, so these lossesaren't considered.

55


Pre-load

The pre-load has the purpose to provide a minimum load on the rolling bear-ing and prevent a rolling bearing damage, this can be the consequence of a slidingmovements of the rolling elements. The tapered roller bearings are assumed to bein a back-to-back con�guration and when the axial force Fa acts in the one rollingbearing, the second rolling bearing is subjected an axial displacement, so the pre-loaddoes reduce that displacement.

The pre-load imposed in the rolling bearings can be calculated by equation 6.41.

F0 = Ka · (cB

cA + cB) (6.41)

Where the Ka is the axial force in the �rst rolling bearing and cA and cB arethe spring constants of the rolling bearings. How the rolling bearings are equals inthis case, cA = cB, so the equation 6.41 can be rewritten 6.42.

F0 =1

2Ka (6.42)

The Ka is determinated based in SKF catalogue [15], for the maximum axialforce allowed on the output shaft of the planetary gearbox and the axial force causedby the maximum input torque for each test.

Calculation example

Table 6.6 shows a calculation example of a deep groove ball bearings (DGB)and the two tapered roller bearings (TRB) power losses lubricated with 75W90 oil(PAO) and with an input torque/speed of 2800 Nm and 200 rpm, the oil temperatureconsidered was 77.8 oC.

56


Table 6.6.: Calculation example for the planetary gearbox rolling bearings losses at200 rpm and 2800 Nm.

Variable ResultsDGB TRB1 TRB2

Rolling frictional moment [Nmm] Mrr 171.69 1816.3 1412.5Inlet shear heating reduction factor Φish 0.97 0.99 0.99Kinematic starvation reduction factor Φrs 0.97 0.99 0.99Variable of the rolling frictional moment Grr 0.56 12.98 10.09Sliding frictional moment [Nmm] Msl 10.73 1830.3 813.41Variable of the sliding frictional moment Gsl 846.88 26538 11794Sliding frictional moment coe�cient µsl 0.01 0.07 0.07Weighting factor for the sliding frictionalmoment

Φbl 0.11 0.68 0.68

Coef. depending on the additive packagein the lubricant

µbl 0.1 0.1 0.1

Sliding friction coe�cient µEHL 0.002 0.002 0.002Pre-load [kN] F0 0 17Drag losses [Nm] Mdrag 0 0 0Total frictional moment [Nmm] M 182.42 3646.6 2225.9Total power loss [W] PV L 15.26 123.13

6.2.5. Bearing power loss (no-Load dependent)

The tapered roller bearings have pre-load applied, that means that even withoutinput torque they will have a high torque loss.

To determine the tapered bearing power losses without dependence of the loadthe method to follow is the same of what was explained above. However, it is onlyconsidered the pre-load. So, the two tapered roller bearings have losses due pre-load.

Table 6.7 shows an example of the tapered roller bearings (TRB) no-load losseslubricated with 75W90 oil (PAO) and with an input speed of 200 rpm, the oil tem-perature considered was 77.8 oC. All values of the table 6.7 are for one bearing, onlythe power loss and the pre-load are for the total of the two bearings.

Table 6.7.: Calculation example for the tapered roller bearing losses at 200 rpm and77.8 oC (no-load dependence).

Variable ResultsTRB

Rolling frictional moment [Nmm] Mrr 1642Variable of the rolling frictional moment Grr 11.73Sliding frictional moment [Nmm] Msl 1321.9Variable of the sliding frictional moment Gsl 19166Pre-load [kN] F0 17Total frictional moment [Nmm] M 2963.8Total power loss [W] PV L0 124.29

57


6.2.6. Needle roller bearings losses

The model used to determined the power losses of the needle roller bearingswas presented by Höhn et all [14]. The SKF model is not prepared for the needleroller bearings calculation. The equation 6.43 allows the calculation of power loss inrolling bearings.

PV L = TV L ·π · n30

(6.43)

The torque loss (TV L) is divided in two parts, load and no-load.

TV L = TV L0 + TV LP1 (6.44)

The no-load dependent torque loss (TV L0) is calculated according to table 6.8.

Table 6.8.: Torque losses to no-load dependent.TV L0 = 1.6 · 10−8f0 · d3

m [Nm] νoiln < 2000 [mm2/s · min]TV L0 = 10−10f0 · (νoiln)2/3d3

m [Nm] νoiln ≥ 2000 [mm2/s · min]

The coe�cient f0 is a function of the rolling bearing type and the lubrication[14], in this case is f0 = 12.

The load dependent losses are calculated according to equation 6.45.

TV LP1 = 10−3f1 · P1dm (6.45)

The coe�cient f1 is a function of the rolling bearing type and P1 is the equivalentrolling bearing load [14], in this case f1 = 0.002 and P1 = Fr.

Table 6.9 shows a calculation example of a needle roller bearing power losseswith 75W90 oil (PAO) for an input torque/speed of 2800 Nm and 200 rpm, the oiltemperature considered was 77.8 oC. Table 6.9 presents the torque loss for each needleroller bearing and the power loss for the six needle roller bearings.

Table 6.9.: Calculation example for the needle roller bearing losses at 200 rpm and2800 Nm.

Variable Result

No-load component [Nm] TV L0 0.02Load component [Nm] TV LP1 0.03

Torque loss [Nm] TV L 0.05Power loss [W] PV L 19.23

6.2.7. Seal losses

Seal losses are the losses with less in�uence in total power loss, compared withother losses could be neglected. Nevertheless, in order to create a more re�ned model,seal losses were considered. An approximation for the determination of the seal powerlosses is given by equation 6.46 [26].

58


PV D = 7.69 · 10−6d2sh · n (6.46)

where:

• dsh is the mean diameter of the seal [mm];

• n is the rotational speed of the shaft [rpm].

Table 6.10 shows a calculation example of the seal power losses with 75W90 oil(PAO) and an input speed on the gearbox of 200 rpm, the oil temperature consideredwas 77.8 oC.

Table 6.10.: Calculation example for the seal losses at 200 rpm.Variable Result

Input seal power loss [W] P inV D 18.63

Output seal power loss [W P outV D 44.39

Seal losses [W] PV D 63.02

59

7. Power loss tests of planetarygearbox

7.1. Experimental procedure

The power loss tests were performed with the planetary gearbox lubricated indip lubrication. Each operating condition tested (input torque and input speed) wasrun until steady state operation was reached, i.e. stabilized operating temperatureof the gearbox. In previous works [13], a duration of 4 hours was enough to obtainstabilized temperature. The operating variables, speed and torque measured in theinput and output of the planetary gearbox, as well as, the temperatures of oil, roomand cage were continuously recorder with a frequency of 0.5Hz.

Only the last 30 minutes of operation were considered for analyses, once, steadystate operation conditions was expected. The ventilation of the room has the object-ive to ensure an approximately constant room temperature.

At the end of each oil tested, the gearbox test rig needs to be prepared fortesting with other oil.

The oil was drained and the gearbox was �lled with solvent and �ushed toremove particles and lubricant remains. After cleaning, the test gearbox was re�lledwith fresh oil (1 Litre).

After load tests, it was necessary modify the test rig, so to prepare the testrig for the no-load tests. The coupling between the test gearbox (4) and the torquetransducer (3) shown in section 5.1 was removed to ensure no-load conditions.

The input torque in planetary gearbox is the torque loss, so the input power ontest gearbox is the no-load power loss. Figure 7.1 shows the coupling disassembled.

Aiming to heat and isolate the test gearbox, a card box with an heater wasassembled around the planetary gearbox. Figure 7.2 shows the mechanism used toisolate and heat the gearbox.

At no-load conditions the torque transducer read a value that is not zero. Thisvalues has a linear drift with time, so the zeros are recorded before and after the testsin order to calibrate the no-load loss measurements.

61

7. Power loss tests of planetary gearbox

Figure 7.1.: Coupling disassembled of the test rig.

Figure 7.2.: Heater.

62

7.2. Operating conditions for the power loss tests

7.2. Operating conditions for the power loss tests

7.2.1. Preliminary test grid (16 tests grid)

These preliminary tests were performed with the objective of understanding theboundaries of the oil operating temperature as a function of the input torque andspeed. Operating conditions that promote an oil temperature above 100 oC are notacceptable, specially for the 80W90 mineral oil.

The 16 test grid (table 7.1) was performed with the 75W90. From the 16 tests,5 were selected based on previous studies with wind turbine gear oils [13].

Table 7.1 displays the operating conditions of the preliminary test grid with 16di�erent operating conditions. The tests were done from the lower rotation and lowertorque to the higher rotation and higher torque. Table 7.2 displays the input powerof the preliminary test grid with 16 di�erent operating conditions.

Table 7.1.: Operating conditions for tests with 75W90 [rpm/Nm].SYN Fe 75W90

100/1600 150/1600 200/1600 250/1600100/2000 150/2000 200/2000 250/2000100/2400 150/2400 200/2400 250/2400100/2800 150/2800 200/2800 250/2800

Table 7.2.: Input power [kW] for tests with 75W90.

Power [kW]Speed [rpm]

100 150 200 250

Torque[Nm]

1600 16.755 25.132 33.51 41.8872000 20.943 31.415 41.887 52.3592400 25.132 37.699 50.265 62.8312800 29.321 43.982 58.643 73.303

7.2.2. Final test grid

After preliminary tests, were performed the �ve tests with three selected oils,these tests were performed to understand the behaviour of all oils and compare thethree oils, these tests will allow compare with numerical results.

Table 7.3 displays the operating conditions of the �nal test grid with 5 di�erentoperating conditions. Initially these tests were performed with 75W90, then with75W140 and lastly with 80W90.

63


Table 7.3.: Operating conditions for tests with all lubricants [rpm/Nm] (Load tests).Oils Speed [rpm] Torque [Nm] Power [kW] Test number

SYN FE 75W90 100 2800 29.321 17; 22; 27SYN FE 75W140 150 2000 31.415 18; 23; 28RS FE 80W90 150 2400 37.699 19; 24; 29

150 2800 43.982 20; 25; 30200 2800 58.643 21; 26; 31

7.2.3. No load test

After the load tests, the no-load tests were performed according to the operatingconditions shown in table 7.3 (speed and oil temperature). For the tests with 150rpm only one test was performed, rather than three (the temperature range waslarger). The torque loss measured is the average of the values read in the range[TOil−1;TOil+1], so when the oil temperature reached TOil+1 the heater was turnedo� and the tests was over at TOil − 1.

The duration of each test wasn't �xed, because it depended of the time thatthe oil took to reach the necessary temperatures. In order to obtain better andmore accurate results, three tests for each operating condition were performed. Themeasured torque is relatively low and a small alteration on test can in�uence the �nalresults, so the three tests are averaged. If one of the tests is signi�cantly di�erentfrom the others the test is repeated and replaced by the mean of the measurements.

Table 7.4 contains the information about no-load tests and it has the operatingconditions.

Table 7.4.: Operating conditions for tests with all lubricants [rpm/Nm] (No-Loadtests).

Oil Test Test number Oil Temperature [oC] Range of tem-peratures [oC]

RS FE 80W90 100/2800 32; 35; 38 51.1 52.5/50150/2000− 2800 33, 36, 39 54.9/56.8/59.1 53.5/60

200/2800 34; 37; 40 71.1 70/72SYN FE 75W140 100/2800 41; 44; 47 53.2 52/54.2

150/2000− 2800 42; 45; 48 56.1/57.2/61 55/62200/2800 43; 46; 49 72.6 71.5/73.6

SYN FE 75W90 100/2800 50; 53; 56 53.1 52/54.1150/2000− 2800 51; 54; 57 56.5/59.5/63.8 55.5/64.8

200/2800 52; 55; 58 77.8 76.8/78.8

64

7.3. Experimental results


Once the experimental tests were concluded, the data was collected and ana-lysed. Equations 7.1 to 7.3 were used to calculate the experimental power loss.

PV = Pin − Pout (7.1)

Pin = Tin · nin ·2π

60(7.2)

Pout = Tout · nout ·2π

60(7.3)

Where:

• PV is the power loss [W];

• Pin/out is the input/output power [W];

• Tin/out is the input/output torque [Nm];

• nin/out is the input/output speed [rpm].

The e�ciency is calculated according to equation 7.4.

Efficiency[%] =PoutPin· 100 (7.4)

The experimental results were analysed in order to understand the behaviour ofthe overall e�ciency of the system at the di�erent operating conditions. The resultsare presented in the next sub chapters, �rstly, it begins with experimental analysis,initially with sixteen test grid and then the �ve test grid. The power loss, e�ciency,oil temperature, variation of the temperature, speci�c �lm thickness, viscosity andsurface distress probability were analysed. The analysis of the experiments ends withthe no-load loss results.

7.3.1. Sixteen test grid

The 16 test grid was performed aiming to understand the behaviour of di�er-ential gear oils within a broad range of operating conditions and also to establish thelimiting torque and speed (maximum oil temperature). The power loss, oil temper-ature, e�ciency and speci�c �lm thickness were analysed.

The lubricant oil that was used in these tests was the 75W90. The results thatwere obtained are presented in �gures 7.3 to 7.8.

Figure 7.3 shows the power loss as a function of input torque and speed.It is visible that there is a relation between power loss and input speed and

input torque. Power loss increases with both speed and torque. At lower speeds(100/150 rpm), the power loss is more sensitive to speed than torque. The inputtorque becomes more important at higher speeds, in other words, at higher speedsthe power loss has an higher increase with the increase of the input torque.

65


Even that the power loss increases with speed and torque, the power loss doesnot have a direct relation with input power, as represented in �gure 7.4. The powerloss always increase with input power, although the increase rate is not always con-stant, this can be justi�ed with the oil temperature reached, that in some points withlower input power attains higher oil temperature.

Figure 7.5 shows that at lower speeds the e�ciency is higher. Is possible observethat at 250 rpm the e�ciency increases with increasing torque.

1600 2000 2400 2800200

400

600

800

1000

1200

Tin

, [Nm]

Po

wer

Lo

ss, [

W]

100 rpm

1600 2000 2400 2800200

400

600

800

1000

1200

Tin

, [Nm]

Po

wer

Lo

ss, [

W]

150 rpm

1600 2000 2400 2800200

400

600

800

1000

1200

Tin

, [Nm]

Po

wer

Lo

ss, [

W]

200 rpm

1600 2000 2400 2800200

400

600

800

1000

1200

Tin

, [Nm]

Po

wer

Lo

ss, [

W]

250 rpm

Figure 7.3.: Power loss vs Operating Conditions (16 test grid, 75W90).

10 20 30 40 50 60 70 800.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Power In, [kW]

Po

wer

Lo

ss, [

kW]

Power Loss vs Input Power

100 rpm150 rpm200 rpm250 rpm

Figure 7.4.: Power loss vs Input Power (16 test grid, 75W90).

66


1600 2000 2400 280098

98.2

98.4

98.6

98.8

99

Tin

, [Nm]

Eff

icie

ncy

, [%

]

100 rpm

1600 2000 2400 280098

98.2

98.4

98.6

98.8

99

Tin

, [Nm]

Eff

icie

ncy

, [%

]

150 rpm

1600 2000 2400 280098

98.2

98.4

98.6

98.8

99

Tin

, [Nm]

Eff

icie

ncy

, [%

]

200 rpm

1600 2000 2400 280098

98.2

98.4

98.6

98.8

99

Tin

, [Nm]E

ffic

ien

cy, [

%]

250 rpm

Figure 7.5.: E�ciency vs Operating Conditions (16 test grid, 75W90).

In �gure 7.6 it can be seen that the increase in the oil temperature is moresensitive to the increase of speed than to the increase of torque.

The stabilized operating temperatures ∆T (TOil − Troom) (�gure 7.7) follow thesame general trend of the power loss. This is somewhat expected and it is in agreementwith what was presented in the thermal balance analysis (equation 6.1).

Based on section 6.2.1, the speci�c �lm thickness depends on various factors,such as the operating conditions, material and has a large dependence of the oilproperties (dynamic viscosity, thermoviscosity and piezoviscosity coe�cients) there-fore the speci�c �lm thickness depends of the temperature.

The speci�c �lm thickness was calculated for the two di�erent contacts (sun/planetand planet/ring), the results are shown in �gure 7.8. Figure 7.8 shows that Λ wasalways higher in the planet/ring contact. The speci�c �lm thickness was alwayslower than 0.7, which indicates boundary lubrication conditions. Λ is higher in theplanet/ring because of the lower contact pressure due to the larger sum of contactlines length and much higher contact radius.

Table 7.5 displays the main results of the preliminary test grid with 16 di�erentoperating conditions.

67


1600 2000 2400 280040

60

80

100

Tin

, [Nm]

TO

IL, [

ºC]

100 rpm

1600 2000 2400 280040

60

80

100

Tin

, [Nm]

TO

IL, [

ºC]

150 rpm

1600 2000 2400 280040

60

80

100

Tin

, [Nm]

TO

IL, [

ºC]

200 rpm

1600 2000 2400 280040

60

80

100

Tin

, [Nm]

TO

IL, [

ºC]

250 rpm

Figure 7.6.: Oil Temperature vs Operating Conditions (16 test grid, 75W90).

1600 2000 2400 280010

20

30

40

50

60

70

Tin

, [Nm]

∆ T

, [ºC

]

100 rpm

1600 2000 2400 280010

20

30

40

50

60

70

Tin

, [Nm]

∆ T

, [ºC

]

150 rpm

1600 2000 2400 280010

20

30

40

50

60

70

Tin

, [Nm]

∆ T

, [ºC

]

200 rpm

1600 2000 2400 280010

20

30

40

50

60

70

Tin

, [Nm]

∆ T

, [ºC

]

250 rpm

Figure 7.7.: ∆T vs Operating Conditions (16 test grid, 75W90).

68


1600 2000 2400 28000.1

0.2

0.3

0.4

0.5

Tin

, [Nm]

Sp

ecif

ic f

ilm t

hic

knes

s

100 rpm

1600 2000 2400 28000.1

0.2

0.3

0.4

0.5

Tin

, [Nm]

Sp

ecif

ic f

ilm t

hic

knes

s

150 rpm

1600 2000 2400 28000.1

0.2

0.3

0.4

0.5

Tin

, [Nm]

Sp

ecif

ic f

ilm t

hic

knes

s

200 rpm

1600 2000 2400 28000.1

0.2

0.3

0.4

0.5

Tin

, [Nm]

Sp

ecif

ic f

ilm t

hic

knes

s

250 rpm

Sun/PlanetPlanet/Ring

Figure 7.8.: Speci�c �lm Thickness vs Operating Conditions (16 test grid, 75W90).

Table 7.5.: Operating conditions for tests with 75W90 (16 test grid).nin [rpm] Tin [Nm] Pin [kW] Ploss [W] Toil [

oC] ∆ T [oC] Eff. [%] Λ (S/P; P/R)

102.47 1548.9 16.621 237.0 41.23 16.47 98.57 0.274 0.419101.98 1934.5 20.66 249.3 42.78 18.64 98.79 0.252 0.385102.15 2321.9 24.84 343.7 46.22 21.09 98.62 0.218 0.332101.59 2708.0 28.81 371.0 50.58 24.31 98.71 0.183 0.279152.59 1549.0 24.75 377.1 49.34 23.60 98.48 0.270 0.413153.93 1935.4 31.19 469.4 53.05 26.35 98.49 0.235 0.359153.94 2322.3 37.44 580.4 56.89 29.45 98.45 0.205 0.313153.95 2709.1 43.68 652.2 67.82 38.06 98.51 0.149 0.227200.24 1549.0 32.48 521.7 61.28 30.69 98.39 0.226 0.346200.12 1935.3 40.56 640.8 62.73 35,22 98.42 0.213 0.325199.97 2320.6 48.59 868.9 74.59 44.59 98.21 0.155 0.236200.00 2708.1 56.72 1071.0 86.55 56.49 98.11 0.118 0.180250.49 1549.2 40.46 794.5 68.59 39.34 98.04 0.219 0.334250.47 1935.5 50.77 902.5 75.67 44.85 98.22 0.181 0.276250.46 2321.9 60.89 943.7 85.02 53.71 98.45 0.145 0.222250.44 2709.0 71.05 1044.1 95.73 62.61 98.53 0.117 0.179

69


7.3.2. Five test grid

Load Tests

Figures 7.9 to 7.11 compare the 16 and 5 test grid results.Figure 7.9 shows that the power loss results follow a similar trend, nevertheless

there are some di�erences. The di�erences between the majority of the points arewithin what is reasonable and achievable with the gearbox test rig.

At 200 rpm and 2800 Nm the power loss is completely di�erent, but ∆T (�gure7.11) is consistent with this result, so something went wrong with either the test setup or the gearbox.

2000 2400 2800

400

500

600

700

800

900

1000

1100

Torque, [Nm]

Po

wer

Lo

ss, [

W]

Power Loss at 150 rpm

5 Test grid16 Test grid

(a) 150 rpm.

100 150 200

400

500

600

700

800

900

1000

1100

Speed, [rpm]

Po

wer

Lo

ss, [

W]

Power Loss at 2800 Nm


(b) 2800 Nm.

Figure 7.9.: Comparison of the power loss between 5 and 16 test grid (75W90).

2000 2400 280050

55

60

65

70

75

80

85

Torque, [Nm]

TO

il, [ºC

]

Oil Temperature at 150 rpm


(a) 150 rpm.

100 150 20050

55

60

65

70

75

80

85

Speed, [rpm]

TO

il, [ºC

]

Oil Temperature at 2800 Nm


(b) 2800 Nm.

Figure 7.10.: Comparison of the oil temperature between 5 and 16 test grid (75W90).

70


2000 2400 280020

25

30

35

40

45

50

55

60

Torque, [Nm]

∆ T

, [ºC

]

∆ T at 150 rpm


(a) 150 rpm.

100 150 20020

25

30

35

40

45

50

55

60

Speed, [rpm]

∆ T

, [ºC

]

∆ T at 2800 Nm


(b) 2800 Nm.

Figure 7.11.: Comparison of the temperature variation between 5 and 16 test grid(75W90).

The 5 test grid results were compared to the selected gear oils. These analysesshow the behaviour of all oils and compare the three oils, so with these comparisonsallow know the best oil at the various parameters.

Figure 7.12 shows the power loss to the 5 test grid the general trend is the samefor all lubricants, at 2800 Nm the power loss always increases with speed. At �xedspeed the power loss increases with load. The 75W90 promotes more power loss thanthe other oils, except in one test, the test with less load.

Regarding e�ciency, �gure 7.13 shows that the 80W90 is more e�cient at themost severe operating conditions. At 150 rpm, the e�ciency of the 75W90 decreaseswith increment of the input torque, on the contrary, the e�ciency of the 75W140 and80W90 increase with increment of the input torque. At 2800 Nm the oil with lowere�ciency is the 75W90.

2000 2400 2800350

400

450

500

550

600

650

700

750

800

850

Torque, [Nm]

Po

wer

Lo

ss, [

W]

Power Loss at 150 rpm

75W9075W14080W90

(a) 150 rpm.

100 150 200350

400

450

500

550

600

650

700

750

800

850

Speed, [rpm]

Po

wer

Lo

ss, [

W]

Power Loss at 2800 Nm

75W9075W14080W90

(b) 2800 Nm.

Figure 7.12.: Power Loss vs Operating Conditions.

71


2000 2400 280098.45

98.5

98.55

98.6

98.65

98.7

98.75

98.8

98.85

98.9

Torque, [Nm]

Eff

icie

ncy

, [%

]

Efficiency at 150 rpm

75W9075W14080W90

(a) 150 rpm.

100 150 20098.45

98.5

98.55

98.6

98.65

98.7

98.75

98.8

98.85

98.9

Speed, [rpm]

Eff

icie

ncy

, [%

]

Efficiency at 2800 Nm

75W9075W14080W90

(b) 2800 Nm.

Figure 7.13.: E�ciency vs Operating Conditions.

2000 2400 280050

55

60

65

70

75

80

Torque, [Nm]

TO

il, [ºC

]

Oil Temperature at 150 rpm

75W9075W14080W90

(a) 150 rpm.

100 150 20050

55

60

65

70

75

80

Speed, [rpm]

TO

il, [ºC

]Oil Temperature at 2800 Nm

75W9075W14080W90

(b) 2800 Nm.

Figure 7.14.: Oil Temperature vs Operating Conditions.

72


2000 2400 280020

25

30

35

40

45

50

Torque, [Nm]

∆ T

, [ºC

]

∆ T at 150 rpm

75W9075W14080W90

(a) 150 rpm.

100 150 20020

25

30

35

40

45

50

Speed, [rpm]

∆ T

, [ºC

]

∆ T at 2800 Nm

75W9075W14080W90

(b) 2800 Nm.

Figure 7.15.: ∆T vs Operating Conditions.

Figures 7.14 and 7.15 show the oil and the stabilization temperatures.The 75W140 and 80W90 oil temperatures have a similar behaviour and the

75W90 has in three points at a slightly higher temperature, at the most severe con-ditions. With these results, the temperature can being related with power loss, asshowed in �gure 7.12 at the most severe tests the power loss of the 75W90 is higher,beyond that, temperature contribute to modify the �lm thickness, surface distressprobability and also coe�cient of friction. Therefore, for various reasons, should beavoided the high temperatures, with the aim of protect the all components of thegearbox and avoid the extremely high power losses.

Figure 7.16 shows the dynamic viscosity of the three oils at the operating con-ditions. This property in�uences the coe�cient of friction between tooth contact.The viscosity varies with temperature and as seen in �gure 7.14 the oil temperat-ure increases with speed/torque increase, so the dynamic viscosity also has a bigdependence of the operating conditions.

Figures 7.17 and 7.18 show the speci�c �lm thickness in planet/ring and sun/planetcontact. The speci�c �lm thickness has the same behaviour than in 16 test grid, inother words, the planet/ring contact has the highest speci�c �lm thickness, at all con-ditions and for the three oils. The 75W140 always has higher speci�c �lm thicknessthan others two lubricants. Even though a mineral oil, the 80W90 has an higher spe-ci�c �lm thickness than 75W90, having the same viscosity, because the coe�cient ofpiezoviscosity for the mineral oils is higher. For the all oils, the speci�c �lm thicknessstayed below 0.7, which indicate boundary lubrication conditions in both contacts.

As shown in �gures 7.17, 7.18 and 7.12 the 75W90 has the lowest speci�c �lmthickness and has the highest power loss (in four tests), so the power loss and speci�c�lm thickness are related, this relation will be discussed when the coe�cient of frictionis analysed.

The speci�c �lm thickness does not have a direct relation with dynamic viscosity,since that the speed increases (the dynamic viscosity decreases), the speci�c �lmthickness doesn't always decrease, like in points at 100/150 rpm and 2800 Nm.

73


2000 2400 280020

30

40

50

60

70

80

90

100

110

Torque, [Nm]

Dyn

amic

vis

cosi

ty, [

kg/(

m.s

)]

Dynamic viscosity at 150 rpm

75W9075W14080W90

(a) 150 rpm.

100 150 20020

30

40

50

60

70

80

90

100

110

Speed, [rpm]

Dyn

amic

vis

cosi

ty, [

kg/(

m.s

)]

Dynamic viscosity at 2800 Nm

75W9075W14080W90

(b) 2800 Nm.

Figure 7.16.: Dynamic viscosity vs Operating Conditions.

2000 2400 28000

0.1

0.2

0.3

0.4

0.5

0.6

Torque, [Nm]

Sp

ecif

ic f

ilm t

hic

knes

s

Contact Planet/Ring (150 rpm)

75W9075W14080W90

(a) 150 rpm.

100 150 2000

0.1

0.2

0.3

0.4

0.5

0.6

Speed, [rpm]

Sp

ecif

ic f

ilm t

hic

knes

sContact Planet/Ring (2800 Nm)

75W9075W14080W90

(b) 2800 Nm.

Figure 7.17.: Speci�c �lm thickness (Planet/Ring).

74


2000 2400 28000

0.1

0.2

0.3

0.4

0.5

0.6

Torque, [Nm]

Sp

ecif

ic f

ilm t

hic

knes

s

Contact Sun/Planet (150 rpm)

75W9075W14080W90

(a) 150 rpm.

100 150 2000

0.1

0.2

0.3

0.4

0.5

0.6

Speed, [rpm]

Sp

ecif

ic f

ilm t

hic

knes

s

Contact Sun/Planet (2800 Nm)

75W9075W14080W90

(b) 2800 Nm.

Figure 7.18.: Speci�c �lm thickness (Sun/Planet).

Figures 7.19 show the surface distress probability of both of the contacts.The results indicate that the operating conditions are quite severe, because

there is a signi�cant amount of points below the Λ5%.The sun/planet contact always promotes the most severe conditions, mainly

due the high line loads and smaller equivalent contact radius.The 75W140 promotes the highest Λ and 75W90 the lowest. Even with 75W140

such operating conditions should be avoided due to associated risk of tooth surfacedistress phenomena.

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


Sp

ecif

ic f

ilm t

hic

knes

s, [Λ

]

Probability of wear related distress (Planet/Ring)

5%

40%

80%

75W9075W14080W90

(a) Planet/Ring.

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


Sp

ecif

ic f

ilm t

hic

knes

s, [Λ

]

Probability of wear related distress (Sun/Planet)

5%

40%

80%

75W9075W14080W90

(b) Sun/Planet.

Figure 7.19.: Surface distress probability.

75


No-Load Tests

Figure 7.20 shows the no-load power loss. These tests allowed to calculate thechurning loss and also to understand the in�uence of the no-load losses.

A direct connection between the no-load loss and speed cannot be devised, butthe results show that the no-load loss attains the highest value at 150 rpm. Theviscosity is very important in this losses, so temperature plays a very important role.

The load tests at 100 and 150 rpm yielded similar temperatures and so theincrease in speed is what drives the no-load loss in this range. At 200 rpm theoperating temperatures dramatically increases resulting in much lower operating vis-cosities lowering the churning losses. At 200 rpm the 75W140 has the highest viscosityresulting in much higher churning losses than the 75W90/80W90.

Theoretically, the losses due the rolling bearings and seals are same for thedi�erent oils, so the di�erence between oils is the relative di�erence of the churningloss. The 75W140 has the worst behaviour without load, this behaviour is justi�edwith higher viscosity than other two oils.

At highest input torque (2800 Nm) and with speed increase the 75W90 and80W90 show similar dynamic viscosity and power loss.

2000 2400 280040

60

80

100

120

140

160

Torque, [Nm]

Pow

er L

oss,

[W]

No−Load Power Loss at 150 rpm

75W9075W14080W90

(a) 150 rpm.

100 150 20040

60

80

100

120

140

160

Speed, [rpm]

Pow

er L

oss,

[W]

No−Load Power Loss at 2800 Nm

75W9075W14080W90

(b) 2800 Nm.

Figure 7.20.: Experimental results (no-load).

76

7.4. Numerical results


In this section the numerical and experimental results will be compared. Thepower loss distribution will be presented and analysed aiming to understand thein�uence of each power loss component in the global power loss.

7.4.1. No-load results

Participation of each component in no-load power loss

Aiming to understand the experimental no-load loss the numerical no-load com-ponents were calculated and compared with the experimental results

Figures 7.21 to 7.23 show the di�erence between no-load losses and numericallosses of each component without load, these losses not consider any loads, onlypre-load accounted for, therefore the input torque is zero 1. Tapered, needle rollerbearings and seals losses increase with increasing speeds, independently of the oil.

In tables 7.6 to 7.8 it is possible conclude that the losses due the tapered rollerbearings are the double than seal losses, the tapered roller bearings have a greatdependence of the pre-load and the seal losses only have dependence of the speed.

Figures 7.21 to 7.23 also show that at all points, the numerical no-load powerlosses are higher than experimental no-load power losses. It is also noticed in somepoints that tapered roller bearings losses alone are higher than experimental losses.This clearly indicates that no-load tapered roller bearings losses are being overestim-ated.

These results make impossible calculate the churning loss, as the initial objectiveof the no-load tests.

1The input torque in x axis is a referential for performed tests.

77


2800 2000 2400 2800 28000

50

100

150

200

250

100 rpm

150 rpm

200 rpm

Input torque, [Nm]

Po

wer

Lo

ss, [

W]

75W90

PVL0TRB

PVL0NRB

PVD

Experimental no−load

Figure 7.21.: Participation of each component in the no-load losses (75W90).

Table 7.6.: No-load power loss of the each component (75W90).75W90

Components 100/2800 150/2000 150/2400 150/2800 200/2800 Units

P TRBV L0 66.49 98.36 96.95 95.24 124.29 W

PNRBV L0 3.73 6.33 5.89 5.35 6.58 WPV D 32.67 47.25 47.25 47.25 63.02 W

Experimental PV 0 61.29 100.08 104.62 123.39 47.27 WError -67.84 -51.87 -43.47 -19.81 -310.16 %

2800 2000 2400 2800 28000

50

100

150

200

250

100 rpm

150 rpm

200 rpm

Input torque, [Nm]

Po

wer

Lo

ss, [

W]

75W140

PVL0TRB

PVL0NRB

PVD



78




P TRBV L0 65.58 101.41 97.97 99.03 128.19 W

PNRBV L0 5.45 9.98 9.29 8.83 10.72 WPV D 32.27 48.46 47.24 48.46 63.83 W


2800 2000 2400 2800 28000

50

100

150

200

250

100 rpm

150 rpm

200 rpm

Input torque, [Nm]

Po

wer

Lo

ss, [

W]

80W90

PVL0TRB

PVL0NRB

PVD





P TRBV L0 66.29 101.25 100.19 98.88 129.83 W

PNRBV L0 3.98 6.94 6.58 6.17 7.49 WPV D 32.27 48.12 48.11 48.04 64.27 W


79


7.4.2. Load results

Comparison between experimental power loss and sum of the numeric loadlosses and experimental no-load loss

After the numerical no-load losses, a comparison between experimental andnumerical power loss was done.

As the churning loss became impossible to calculate, it stops making sense tocompare the sum of each component numerically calculated with the experimentalpower loss. So the comparison of the experimental power loss was done with sum ofthe numeric load and experimental no-load loss. Initially, to have the notion of trendthe lubricant factor was XL=1.

Figures 7.24 to 7.26 show the comparison between experimental and numericalpower loss with XL=1. As can be seen in �gures 7.24 and 7.25, the sum of numericload and no load experimental has a small deviation, being the experimental resultshigher than numerical results. Aiming to optimized, the XL factor was adjustedmanually until the points of the experimental power loss have overlap, this wayallows a good approximation of the XL factor, once that the higher percentage ofload component is the friction losses, so the error of the other components are low.

To the mineral oil (80W90) the XL factor in more accordance is 1. Normally,this value is considering for mineral oils, but without additives, and this oils havemany additives.

Figures 7.27 and 7.28 show the plots with a better approximation with experi-mental power loss. This method allows an optimization of the gears losses and showsthat the selection of the oil can change the power loss. It is noticed that the ap-proximation of the plots is good only in four points, the severe point (200 rpm/ 2800Nm) has a big error. This result has various reasons, already for the sixteen test gridthis point had occurred something wrong, it indicates that something related withgearbox or with oils behaviour at severe conditions, the tapered roller bearings mayhad caused an unexpected increased of temperature or also the additives of the oilsmay be related with this signi�cant di�erence.

For 75W90 the XL considered was 1.15 and for the 75W140 was 1.1. Theseresults, prove that these two oils worsen the coe�cient of friction. Based in chapter4, these oils have an excellent properties at extreme pressure and anti-wear, theseadditives create a protector �lm by reaction with metallic contact surfaces and thisfact can worsen the coe�cient of friction between contact surfaces.

As is shown in table 7.9 the error obtained in optimization for the four tests toeach oil is minimum.

80


2000 2400 2800350

400

450

500

550

600

650

700

750

800

850

Torque, [Nm]

Po

wer

Lo

ss, [

W]

150 rpm (75W90)

Experimental Power LossNumeric Load + No Load Power Loss

(a) 150 rpm.

100 150 200350

400

450

500

550

600

650

700

750

800

850

Speed, [rpm]

Po

wer

Lo

ss, [

W]

2800 Nm (75W90)


(b) 2800 Nm.

Figure 7.24.: Comparison between experimental and numerical power loss for 75W90(XL=1).

2000 2400 2800350

400

450

500

550

600

650

700

750

800

850

Torque, [Nm]

Po

wer

Lo

ss, [

W]

150 rpm (75W140)


(a) 150 rpm.

100 150 200350

400

450

500

550

600

650

700

750

800

850

Speed, [rpm]

Po

wer

Lo

ss, [

W]

2800 Nm (75W140)


(b) 2800 Nm.


81


2000 2400 2800350

400

450

500

550

600

650

700

750

800

850

Torque, [Nm]

Po

wer

Lo

ss, [

W]

150 rpm (80W90)


(a) 150 rpm.

100 150 200350

400

450

500

550

600

650

700

750

800

850

Speed, [rpm]

Po

wer

Lo

ss, [

W]

2800 Nm (80W90)


(b) 2800 Nm.


2000 2400 2800350

400

450

500

550

600

650

700

750

800

850

Torque, [Nm]

Po

wer

Lo

ss, [

W]

150 rpm and XL=1.15(75W90)


(a) 150 rpm.

100 150 200350

400

450

500

550

600

650

700

750

800

850

Speed, [rpm]

Po

wer

Lo

ss, [

W]

2800 Nm and XL=1.15 (75W90)


(b) 2800 Nm.

Figure 7.27.: Comparison between experimental and numerical power loss for 75W90(XL=1.15).

82


2000 2400 2800350

400

450

500

550

600

650

700

750

800

850

Torque, [Nm]

Po

wer

Lo

ss, [

W]

150 rpm and XL=1.1(75W140)


(a) 150 rpm.

100 150 200350

400

450

500

550

600

650

700

750

800

850

Speed, [rpm]

Po

wer

Lo

ss, [

W]

2800 Nm and XL=1.1 (75W140)


(b) 2800 Nm.

Figure 7.28.: Comparison between experimental and numerical power loss for 75W140(XL=1.1).

83


Table 7.9.: The error obtained in optimization.Oil 100/2800 150/2000 150/2400 150/2800 200/2800 Units

75W90 -2.19 -5.52 -2.25 -0.37 17.15 %75W140 -11.21 4.63 -0.50 -1.21 7.05 %80W90 3.73 3.55 0.76 -9.35 16.73 %

Participation of each component in power loss (optimized)

After the discussion on the no-load losses the total power loss was analysed.The no-load loss results indicated that there was a potential overestimation of thetapered roller bearings no-load loss. Following these conclusions the total power losswas calculated as the sum of the numerical load loss with the measured no-load losses.

Figures 7.29 to 7.31 and tables 7.10 to 7.15 show the power loss of each com-ponent in W and the respective relative weight on the total loss in %.

According to �gures 7.29 to 7.31 it is possible to observe that the gear frictionlosses have the biggest in�uence in the total power loss. The gear friction losses evolvewith the operating conditions, i. e. it increases with load and more importantly withthe input speed, the transmitted power (PA) increases. The frictional related gearpower loss accounts for 65 to 90 % of the total power loss.

The no-load losses also play a very important role in the total power loss witha relative weight in the total losses that range between 6 and 29 %. It was alreadydiscussed that the tapered roller bearings drive the no-load loss mainly due to theexisting pre-load.

It is noticed that the lowest percentage of gear friction losses are at 150/2000,once that the load is more lower and in this test the churning loss win relevance.In other hand, the percentage of the gear friction losses are higher at 200/2800,at severe conditions, logically, once that these losses vary with speed/torque, thechurning losses also lose relevance.

The tapered roller bearings total loss is relatively independent of the load thatis imposed by the helical gears in the planetary gearbox simply because this axialload is very small when compared with the pre-load. The tapered roller bearingsare, in total, the second most important power loss component in the gearbox. Thetapered rolling bearing load loss was calculated as the di�erence between the loadedand pre-loaded loss which resulted in a negative value as displayed in tables 7.10 to7.15.

Relating the section 7.4.1, is possible conclude that, in the total, the seal lossesare the third most important power loss component in the gearbox.

The components with the least signi�cant contribution on the total power lossare the deep groove ball and needle roller bearings (below 10 % of the total loss).The deep groove ball bearing increase due to speed and when the speed is �xed thepower loss increases slightly with torque (except for the 75W90). The power loss ofthe needle roller bearings also increase with speed and torque.

84


2800 2000 2400 2800 28000

100

200

300

400

500

600

700

100 rpm

150 rpm

200 rpm

Input torque, [Nm]P

ow

er L

oss

, [W

]

75W90 (XL=1.15)

PVZP

PVLDGB

PVLNRB


PVLTRB*

Figure 7.29.: Power loss and its components (75W90).

Table 7.10.: Power loss and its components in W (75W90).75W90 (XL=1.15)


PV ZP 351.96 318.07 397.98 482.02 619.78 WPDGBV L 9.14 12.46 12.71 12.59 15.26 W

P TRB∗V L -0.73 -0.59 -0.79 -0.99 -1.16 WPNRBV L 6.56 6.78 8.13 9.48 12.64 WPV 0 61.30 100.08 104.62 123.39 47.27 W

Total numeric 428.23 436.79 522.64 626.49 693.81 W

Table 7.11.: Power loss and its components in % (75W90).75W90 (XL=1.15)


PV ZP 82.19 72.81 76.15 76.94 89.33 %PDGBV L 2.13 2.85 2.43 2.01 2.19 %

P TRB∗V L -0.17 -0.13 -0.15 -0.16 -0.17 %PNRBV L 1.53 1.55 1.56 1.51 1.82 %PV 0 14.31 22.91 20.02 19.69 6.81 %

85


2800 2000 2400 2800 28000

100

200

300

400

500

600

700

800

100 rpm

150 rpm

200 rpm

Input torque, [Nm]

Po

wer

Lo

ss, [

W]

75W140 (XL=1.1)

PVZP

PVLDGB

PVLNRB


PVLTRB*


Table 7.12.: Power loss and its components in W (75W140).75W140 (XL=1.1)


PV ZP 323.19 300.57 367.46 454.15 577.86 WPDGBV L 8.96 13.05 13.34 13.77 16.68 W


Total numeric 431.94 445.29 506.35 602.53 728.73 W

Table 7.13.: Power loss and its components in % (75W140).75W140 (XL=1.1)


PV ZP 74.82 67.50 72.57 75.37 79.29 %PDGBV L 2.07 2.93 2.63 2.29 2.29 %

P TRB∗V L -0.16 -0.14 -0.17 -0.18 -0.18 %PNRBV L 1.50 1.56 1.62 1.61 1.76 %PV 0 21.77 28.15 23.36 20.91 16.84 %

86


2800 2000 2400 2800 28000

100

200

300

400

500

600

700

100 rpm

150 rpm

200 rpm

Input torque, [Nm]P

ow

er L

oss

, [W

]

80W90 (XL=1)

PVZP

PVLDGB

PVLNRB


PVLTRB*


Table 7.14.: Power loss and its components in W (80W90).80W90 (XL=1)


PV ZP 300.97 279.03 348.51 420.79 542.96 WPDGBV L 9.4 13.25 13.83 14.12 17.25 W


Total numeric 381.59 418.19 493.64 555.39 616.44 W

Table 7.15.: Power loss and its components in % (80W90).80W90 (XL=1)


PV ZP 78.87 66.72 70.60 75.76 88.08 %PDGBV L 2.46 3.17 2.80 2.54 2.79 %

P TRB∗V L -0.19 -0.15 -0.18 -0.21 -0.22 %PNRBV L 1.69 1.65 1.68 1.74 2.09 %PV 0 17.16 28.61 25.10 20.16 7.25 %

87


Figures 7.32 and 7.33 show the coe�cient of friction that was calculated con-sidering the operating conditions that were recorded during the tests as well as theadjusted lubricant parameter.

Observing �gures 7.32 and 7.33 it is possible conclude that in planet/ring con-tact the coe�cient of friction is always lower than the sun/planet contact, this dif-ference is due the higher equivalent radius and lower line load in the planet/ringcontact.

The oil with the highest coe�cient of friction is the 75W90, and in the otherhand, the oil with lowest coe�cient of friction is the 80W90. As previously statedthe gear losses are the main power loss component. The overall power loss parameterby the 75W90 is then heavily in�uenced by the coe�cient of friction. This justi�esthe fact that 75W90 has the worst power loss behaviour.

The tendency of the coe�cient of friction is same in two contacts, so when thetorque increase the coe�cient of friction increases too and when the speed increases,the coe�cient of friction decreases. This decrease can be justi�ed with equation 6.10,in these cases the speed increase overcomes the viscosity decrease with increased oiltemperature.

The speci�c �lm thickness and the coe�cient of friction are correlated andaccording to chapter 6.2.1 this correlation can be observed on stribeck curve, atconstant speed the speci�c �lm thickness decrease with torque increases, according bystribeck curve (boundary �lm) is possible see that the coe�cient of friction increases,this fact can be veri�ed in these tests. But, at constant torque the speci�c �lmthickness behaviour isn't linear with speed increases, for 100/150 rpm the speci�c�lm thickness increases and for 150/200 rpm the speci�c �lm thickness decreases,so for 150/200 rpm the relation of stribeck curve is not veri�ed. This fact can bejusti�ed with the speci�c �lm thickness being very small (boundary �lm).

At �xed speed, when torque increases the coe�cient of friction increases andgear losses (PV ZP ) increase too, as shown in tables 7.10 to 7.14. However, when thespeed increases the coe�cient of friction decreases and PV ZP increases, so this factshows that the transmitted power (PA) overcomes the coe�cient of friction.

88


2000 2400 28000.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

Torque, [Nm]

Co

effi

cien

t o

f fr

icti

on

Coefficient of friction at 150 rpm (Planet/Ring)

75W9075W14080W90

(a) 150 rpm.

100 150 2000.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

Speed, [rpm]

Co

effi

cien

t o

f fr

icti

on

Coefficient of friction at 2800 Nm (Planet/Ring)

75W9075W14080W90

(b) 2800 Nm.

Figure 7.32.: Coe�cient of friction (Planet/Ring).

2000 2400 28000.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

Torque, [Nm]

Co

effi

cien

t o

f fr

icti

on

Coefficient of friction at 150 rpm (Sun/Planet)

75W9075W14080W90

(a) 150 rpm.

100 150 2000.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

Speed, [rpm]

Co

effi

cien

t o

f fr

icti

on

Coefficient of friction at 2800 Nm (Sun/Planet)

75W9075W14080W90

(b) 2800 Nm.

Figure 7.33.: Coe�cient of friction (Sun/Planet).

89


Tables 7.16 to 7.18 display the main results of the �nal tests with 5 di�erentoperating conditions for each oil.


oC] ∆ T [oC] Eff. [%] Λ(S/P ;P/R) µ(S/P ;P/R)

103.8 2709 29.44 419.0 50.2 25.0 98.58 0.171 0.261 0.074 0.057150.1 1936.3 30.44 413.9 54.1 25.8 98.64 0.207 0.316 0.065 0.050150.1 2323.1 36.52 511.1 56.8 29.4 98.60 0.186 0.284 0.067 0.052150.1 2708.6 42.58 624.2 60.3 33.3 98.53 0.162 0.248 0.069 0.054200.2 2708.4 56.79 837.5 71.9 45.3 98.53 0.142 0.2168 0.067 0.052



102.5 2709.4 29.09 388.4 53.2 22.9 98.66 0.282 0.431 0.069 0.053153.9 1935.4 31.21 466.9 56.1 26.5 98.50 0.357 0.545 0.059 0.046150.1 2323.3 36.52 503.8 57.2 27.3 98.62 0.332 0.507 0.062 0.048153.9 2709.2 43.68 595.3 61.0 30.5 98.64 0.297 0.453 0.064 0.049202.8 2709.6 57.55 783.9 72.6 41.1 98.64 0.259 0.397 0.062 0.048



102.6 2709.2 29.11 396.4 51.1 23.4 98.64 0.241 0.368 0.064 0.049152.9 1935.7 30.99 433.6 54.9 25.6 98.60 0.289 0.441 0.056 0.043152.9 2322.3 37.18 497.4 56.8 27.8 98.66 0.266 0.406 0.058 0.045152.6 2709.4 43.30 507.9 59.1 30.0 98.83 0.242 0.369 0.060 0.047204.2 2708.8 57.92 740.2 71.1 40.5 98.72 0.207 0.316 0.058 0.045

90

8. Conclusion and future works

8.1. Conclusion

8.1.1. Conclusions based on experimental evidence

1. For the most severe conditions, the 80W90 is the oil with the lowest powerlosses, followed by the 75W140 and the worst performing oil is the 75W90. So,in this case the mineral oil has better behaviour (in relation of the power loss)than the synthetic oils.

2. The power loss increases with both speed and torque, but does not alwaysincrease with input power.

3. The increase in the oil temperature is more sensitive to the speed than torque.

4. Analysing the stabilized temperatures it is veri�ed that the 75W90 reaches thehighest temperatures, generating 5oC more than the second worst oil (75W140).

5. The stabilized operating temperatures follow the same general trend of thepower loss.

6. For the speci�c �lm thickness, the 75W140 has the highest values and 75W90the lowest.

7. In relation to the churning loss the 75W140 promotes more losses (specially athigher speeds), because it has more viscosity than the other oils. The 75W90and 80W90 have a similar behaviour at 2800 Nm. The increase in speed is whatdrives the variation of the no-load loss in range 100/150 rpm (in this range thetemperatures are similar). For 150/200 rpm the temperature is what drives thevariation of the no-load loss, once that the viscosities decrease.

So, with these conclusions it is possible verify in terms of power loss and tem-perature operation, the best oil is the 80W90, but in some applications this maynot be the best choice. Despite the higher e�ciency 80W90 does not show the bestsurface distress probability characteristics, so in applications where the long life ofthe components is most important requirement the best solution would be the PAOoil (75W140).

The 75W90 can be the best solution, when loads are null or low and there areno requirements to limit temperatures, once the 80W90 reaches lower temperaturethan 75W90 and has the same behaviour without load.

91

8. Conclusion and future works

8.1.2. Conclusions based on numerical results

The experimental no-load measurements did not correlates well with the predic-tions. The measured no-load loss results were directly used in the power loss model.The oil parameter (XL) was adjusted based in the comparison between the exper-imental power loss and sum of the numerical load losses and experimental no-loadloss results.

1. The worst coe�cient of friction is the 75W90 and the lower is the 80W90. Thecoe�cient of friction makes the 75W90 has the highest power losses. To thethree tested oils, the coe�cient of friction reduce with increase of speed, whichsuggest that the e�ect of speed outweighs the viscosity decrease. When torqueincreases, at �xed speed, the coe�cient of friction increases too.

2. The oil parameter (XL) obtained in these tests for PAO oils was 1.15 and 1.1,for 75W90 and 75W140, respectively. For mineral oil (80W90) the XL factorwas 1.

3. The di�erence between numerical and experimental power loss is minimum,only one of the tests has a considerable di�erence between the simulations andthe results for all oils.

4. Like the experimental results demonstrated the predictions results show thatthe power loss is more sensitive to the speed than the torque.

5. In what regards to the surface distress probability, the 75W90 and 80W90shows dangerous values, at the most severe conditions can go beyond 80%.The 75W140 at the most severe conditions can go beyond 40%, but in theother conditions in planet/ring contact has a probability less than 5% and insun/planet contact has a probability between 5 and 40 %.

6. The overestimated component is the tapered roller bearings, in some points thisvalue exceeds the experimental no-load.

7. The higher contribute of the planetary gearbox is the friction losses, with arange between 65 and 90 %, the no-load losses can reach 29 %. The second mostimportant component in the gearbox are the tapered roller bearings, followingseals and lastly the deep groove ball bearing and needle roller bearings.

8. How demonstrated with experimental no-load results, the churning losses be-came impossible to calculate, but is possible observing that losses are low,because the experimental no-load losses are lower comparing with seals, needleand tapered roller bearings.

92

8.2. Future works

8.2. Future works

A new and more precise methodology for the no-load loss measurement shouldbe developed.

In order to improve the results, should be made more experimental tests andimplement other model with objective to obtain a better approximation of the XLfactor and thus obtain a better prediction of the power loss.

The tested planetary gearbox should be disassembled in order to check if oc-curred a problem with any component. As the results demonstrated the taperedroller bearings are very ine�cient, so to verify this fact should be changed this bear-ings for new bearings and perform load and no-load tests to compare with obtainedresults, having these results it could be con�rm that the tapered roller bearings wereine�cient.

The temperature in a machine, can be an indicator of some problems with acomponent. So, other technique to analyse the planetary gearbox would be verify theoperating conditions with a thermal process, using an infrared thermography, thus ifsome component has any problem with this technique it is possible prevent furtherdamage.

As the planetary gearbox should be disassembled, it would be interesting tomeasure the roughness of the planetary gears, because it changes with wear. Thespeci�c �lm thickness and the coe�cient of friction are parameters that vary withroughness and with a correct roughness, the speci�c �lm thickness and the surfacedistress probability would be more accurately evaluated.

93

Bibliography

[1] Anant S. Kolekar, Andrew V. Olver, Adam E. Sworski, and Frances E. Lockwood.The e�ciency of a hypoid axle�a thermally coupled lubrication model. TribologyInternational, 59(0):203 � 209, 2013.

[2] Fwd and rwd di�erentials. http://www.thecardr.com/di�erential.html. Web.17/03/2015.

[3] Karim Nice. How di�erentials work. http://auto.howstu�works.com/di�erential1.htm.Web. 17/03/2015.

[4] Karim Nice. How four-wheel drive works. http://auto.howstu�works.com/four-wheel-drive3.htm. Web. 17/03/2015.

[5] H. Heisler. Advanced Vehicle Technology. Number vol. 10 in Advanced vehicletechnology. Butterworth-Heinemann, 2002.

[6] API. Lubricant Service Designations for Automotive Manual Transmissions,Manual Transaxles and Axles. April 2013.

[7] Alexandre Sottomayor Jorge Seabra, Armando Campos. Lubri�cação Elast-ohridodinâmica. 2002.

[8] J. Frêne. Lubri�cation hydrodynamique: paliers et butées. Collection de la Dir-ection des études et recherches d'Electricité de France. Eyrolles, 1990.

[9] G. Stachowiak and A.W. Batchelor. Engineering Tribology. Elsevier Science,2013.

[10] Mobil Oil Portuguesa. Fundamentos da Lubri�cação. Mobil Oil Portuguesa,1980.

[11] ASTM International. Standard Pratice for Calculating Viscosity Index FromKinematic Viscosity at 40 and 100oC.

[12] P.W. Gold, A. Schmidt, H. Dicke, J. Loos, and C. Assmann. Viscosity-pressure-temperature behaviour of mineral and synthetic oils. pages 51�79, 2001.

[13] Raquel C. S. Dias. Torque loss in a planetary multiplier gearbox: In�uence ofoperating conditions and gear oil formulation, Porto, July 2014, Faculdade deEngenharia da Universidade do Porto.

[14] Bernd-Robert Höhn, Klaus Michaelis, and Thomas Vollmer. Thermal rating ofgear drives - balance between power loss and heat dissipation. October, 1996.

95

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[15] SKF. Rolling bearings. SKF, February, 2013.

[16] Pedro M.T. Marques, Carlos M.C.G. Fernandes, Ramiro C. Martins, andJorge H.O. Seabra. Power losses at low speed in a gearbox lubricated with windturbine gear oils with special focus on churning losses. Tribology International,62(0):186 � 197, 2013.

[17] A.J. Wimmer. Lastverluste von stirnradverzahnungen, Fakultät für Maschinen-wesen der Technischen Universität München, 2005.

[18] KISSsoft R©. Release 03/2013B. KISSsoft AG, Bubikon, Switzerland, 2013.

[19] L. Schlenk. Untersuchungen zur Freÿtragfähigkeit von Groÿzahnrädern. Lehr-stuhl für Maschinenelemente, Forschungsstelle für Zahnräder und Getriebebau.na, 1995.

[20] D. Dowson and G.R. Higginson. Elasto-hydrodynamic lubrication. Internationalseries on materials science and technology. Pergamon Press, 1977.

[21] Ecole nationale supérieure du pétrole et des moteurs (France) and Ayel J. LaLubri�cation industrielle, Introduction à la Lubri�cation des engrenages. Collec-tion Collegues et séminaires. Technip, 1984.

[22] R. Gohar. Elastohydrodynamics. Computing in Engineering. Imperial CollegePress, 2001.

[23] Ecole nationale supérieure du pétrole et des moteurs (France) and F. Roux. LaLubri�cation Industrielle, Notions de tribologie. Collection Collegues et sémin-aires. Technip, Paris, France, 1984.

[24] José A. Brandão, Mathilde Meheux, Fabrice Ville, Jorge H.O. Seabra, and JorgeCastro. Comparative overview of �ve gear oils in mixed and boundary �lmlubrication. Tribology International, 47(0):50 � 61, 2012.

[25] American Gear Manufactures Association. E�ect of Lubrication on Gear SurfaceDistress, March 13, 2003.

[26] Pedro M.T. Marques, Carlos M.C.G. Fernandes, Ramiro C. Martins, andJorge H.O. Seabra. E�ciency of a gearbox lubricated with wind turbine gearoils. Tribology International, 71(0):7 � 16, 2014.

96

Bibliography

97

Appendix

99

A. Tested oils

101

TOTAL

TRANSMISSION SYN FE 75W90Lubrificante sintético para transmissõesmecânicas

Este lubrificante utilizado de acordo com as nossas recomendações e paraa aplicação para o qual está prevista não apresenta riscos particulares.Uma ficha de dados de segurança conforme a legislação em vigor na UEencontra-se disponível junto do seu Gestor Comercial.

RV082012 - Anula e substitui todas as anterioresJAN2011

TOTAL TRANSMISSION SYN FE 75W90

CEPSA Portuguesa Petróleos, SARua General Firmino Miguel, Nº3 Torre 2 – 2º1600-100 LisboaTel: 00 351 217 217 600Fax: 00 351 217 230 838

Página 1 de 2

APLICAÇÕES

- O TOTAL TRANSMISSION SYN FE 75W90 é umlubrificante sintético de muito elevadaperformance desenvolvido para responder aoconceito T.D.L. (Total Drive Line) para caixas develocidades mecânicas sincronizadas ou nãosincronizadas, pontes e redutores.

- O TOTAL TRANSMISSION SYN FE 75W90 érecomendado para as caixas de velocidades ediferenciais que exijam um óleo do tipo API GL-4,API GL-5, API MT-1 ou SAE J2360.

- Particularmente adaptado para a lubrificaçãode eixos traseiros e reduções finais, de veículoscomerciais ligeiros das marcas MACK, MAN,DAF, IVECO, VOLVO, RENAULT TRUCKS,MERCEDES-BENZ, permitindo longos intervalosde manutenção que podem chegar aos540.000 km.

- Recomendado pela TOTAL para as seguintesaplicações: Arvin Meritor O76N, MAN 341-Z2,MAN 342-S1, VOLVO 97312, VOLVO 97316(GO101), FORD M2C-200C.

PROPRIEDADES

- Excelentes propriedades de Extrema Pressão (EP)e anti-desgaste, prevenindo o desgaste por atritoe o picado das engrenagens, que se encontramsujeitas a condições elevadas de carga.

- Grande resistência à oxidação e elevadaestabilidade a altas taxas de corte, permitindoalcançar longos intervalos de manutenção (até540.000 km), mantendo intacta a performance dolubrificante.

- Excelente filtrabilidade e bom comportamentoanti-espuma, permitindo uma excelente fluidez doóleo nas transmissões e nos diferenciais.

- Passagens de caixa facilitadas tanto a frio como aquente.

- Diminuição da fricção e das perdas porarrastamento mecânico, proporcionando umamelhoria na eficiência mecânica gerandoeconomias ao nível do consumo decombustível.

- O TOTAL TRANSMISSION SYN FE 75W90 é umproduto certificado pela TÜV Rheinland,distinção obtida após a realização de testesrealizados em Millbrook (UK) num camião EUROIV. Esta entidade confirma que a utilizaçãoconjunta, deste produto, com um óleo demotor e de caixa do tipo “Fuel Economy”permite diminuir, em média, o consumo decombustível em 3%.

CARACTERÍSTICAS

TOTAL TRANSMISSION SYN FE 75W90 Unidades Valores

Massa Volúmica a 15ºC kg/m3 866

Viscosidade a 40ºC mm2/s 101

Viscosidade a 100ºC mm2/s 15

Viscosidade a -40ºC mPa.s 66.000

Índice de Viscosidade - 157

Ponto de Fluxão ºC -51

Ponto de Inflamação Cleveland ºC 190Os valores típicos apresentados representam valores médios

TOTAL

TRANSMISSION SYN FE 75W90Lubrificante sintético para transmissõesmecânicas

Este lubrificante utilizado de acordo com as nossas recomendações e paraa aplicação para o qual está prevista não apresenta riscos particulares.Uma ficha de dados de segurança conforme a legislação em vigor na UEencontra-se disponível junto do seu Gestor Comercial.

RV082012 - Anula e substitui todas as anterioresJAN2011

TOTAL TRANSMISSION SYN FE 75W90

CEPSA Portuguesa Petróleos, SARua General Firmino Miguel, Nº3 Torre 2 – 2º1600-100 LisboaTel: 00 351 217 217 600Fax: 00 351 217 230 838

Página 2 de 2

ESPECIFICAÇÕES

API: MT-1 (Nível)API: GL-4 (Nível)API: GL-5 (Nível)MIL-PRF-2105E (Nível)

MERCEDES-BENZ: MB-Aprovação 235.8MACK GO-JMAN 3343 TYPE S (ex MAN 3343 TYPE SL)SCANIA: STO 1:0SAE J2360 (n.º PRI GL 0432)ZF TE-ML 02B, 05B, 07A, 12B, 16F, 17B, 19C, 21BDAF (longos intervalos de manutenção para diferencialtraseiro)

TOTAL TRANSMISSION SYN FE 75W140 Lubrificante Sintético para Transmissões Mecânicas

Este lubrificante utilizado de acordo com as nossas recomendações e para a aplicação para o qual está prevista não apresenta riscos particulares. Uma ficha de dados de segurança conforme a legislação em vigor na CE encontra-se disponível junto do vosso Gestor Comercial. TC022006 - Anula e substitui todas as anteriores JUL2005 TOTAL TRANSMISSION SYN FE 75W140

TOTAL PORTUGAL, Petróleos, SA Rua General Firmino Miguel, Nº3 Torre 2 – 2º 1600-100 Lisboa Tel: 00 351 21 723 0800 Fax: 00 351 21 723 0899

APLICAÇÕES

- TOTAL TRANSMISSION SYN FE 75W140 é um lubrificante sintético de muito elevada performance para pontes e redutores funcionando em regime de cargas muito elevadas.

- TOTAL TRANSMISSION SYN FE 75W140 é recomendado para transmissões com filtro no sistema de lubrificação.

- Recomendado para pontes que exijam um lubrificante do tipo API GL-5 e com este tipo de viscosidade

PROPRIEDADES

TOTAL TRANSMISSION SYN FE 75W140 apresenta o seguinte conjunto de propriedades: - Índice de Viscosidade muito elevado. - Propriedades de Extrema Pressão EP e anti-

desgaste reforçadas para uma lubrificação máxima de diferenciais hipóides

- Excelentes propriedades anti-espuma, anti-corrosão e anti-ferrugem.

- Estabilidade térmica excepcional. - Muito baixa perda de viscosidade ao

cizalhamento.

- Excelente estabilidade em serviço. - Permite atingir os mais alongados intervalos de

manutenção. - Permite, quando associado à gama “ TOTAL

Eco Solution ”, contribuir para a economia de combustível

CARACTERÍSTICAS

TOTAL TRANSMISSION SYN FE 75W140 Unidades Grade SAE 75W140

Massa Volúmica a 15ºC Kg/m3 885 Viscosidade a 40ºC mm2/s 183 Viscosidade a 100ºC mm2/s 26 Viscosidade Brookfield a -40ºC mPa.s 130 000 Índice de Viscosidade - 178 Ponto de Congelação ºC -36 Os valores típicos apresentados representam valores médios

ESPECIFICAÇÕES

API GL-5 SCANIA STO 1:0

Units Grade SAE 80W-90

Volumetric mass at 15°C kg/m3 886

Viscosity at 40°C mm²/s 115

Viscosity at 100°C mm²/s 14,1

Viscosity Index - 123

Pour point °C - 33

TRANSMISSION RS FE 80W-90

SEMI-SYNTHETIC OIL FOR GEARBOXES AND AXLES

APPLICATIONS

- Semi-synthetic oil with a high viscosity index for the lubrication of gears under severe conditions of use.

- Designed to meet the challenge of the T.D.L. (Total Drive Line) concept: high performance lubrication of hypoid axles as well as synchronized gearboxes, to simplify maintenance without any compromise on components durability.

- Is recommended for use on manual gearboxes, rear axles, or any gear assembly requiring API GL-4, API GL-5, API MT-1 or SAE J2360 levels of performance

- Allows extended service intervals up to 160 000 km on ZF axles and gearboxes (without Intarder), and more generally on all commercial vehicles hypoid axles.

- Suitable for use on Scania gearboxes

PROPERTIES

- Very good low temperature fluidity due to a high viscosity index, generating benefits during cold starts and limiting drag losses and fuel consumption.

- Very high extreme-pressure performance for optimal protection of gears and bearings against scoring and scuffing.

CHARACTERISTICS

- Excellent antiwear, anticorrosion and antirust properties for the durability of components including gearbox synchronizers.

- Semi synthetic formulation, very resistant to oxidation and allowing extended drain intervals (up to 160 000 kms) compared to standard mineral.

TRANSMISSION RS FE

The typical characteristics mentioned represent mean values.

SPECIFICATIONS

MAN 341 TYPE E2, Z2 MAN 342 TYPE M2 ZF TE-ML 02B, 05A, 16B, 17B, 19B, 21A

Meets the performance requirements of the following international specifications : API GL-4 API GL-5 API MT-1 MIL-PRF-2105E / SAE J2360

TOTAL LUBRIFIANTS 16, rue de la République 92800 PUTEAUX

TRANSMISSION RS FE 80W-90 Updated : 05/2014

Référence étiquette : MPR/11/06

Ce lubrifiant utilisé selon nos recommandations et pour l’application pour laquelle il est prévu ne présente pas de risque particulier. Une fiche de données de sécurité conforme à la législation en vigueur dans la C.E. est disponible auprès de votre conseiller commercial.

B. Reports from the experimentaltests

B.1. Load Tests

B.1.1. Sixteen tests grid

Test number: 1 Date: 09/03/2015 By: David Costa

Oil 75W90 (PAO)

Imposed Conditions Units

nin 100 rpmTin 1600 NmTest duration 240 min

Actual Conditions

nin 102,4730 rpmnout 408,8815 rpmTin 1548,9 NmTout 382,6366 NmInput Power 16621 WOutput Power 16384 W

Temperature readings

Toil 41,2258 oCToil2 42,6396 oCTwall 41,0786 oCTamb 24,7552 oC

Additional information

E�ciency 98.5738 %Toil − Tamb 16.4706 oC

107

B. Reports from the experimental tests


Oil 75W90 (PAO)



Actual Conditions






108

B.1. Load Tests


Oil 75W90 (PAO)



Actual Conditions






109



Oil 75W90 (PAO)



Actual Conditions






110

B.1. Load Tests


Oil 75W90 (PAO)



Actual Conditions






111



Oil 75W90 (PAO)



Actual Conditions






112

B.1. Load Tests


Oil 75W90 (PAO)



Actual Conditions






113



Oil 75W90 (PAO)



Actual Conditions



Toil 67,8165 oCToil2 65.0967 oCTwall 63,3422 oCTamb 29,7526 oC



114

B.1. Load Tests


Oil 75W90 (PAO)



Actual Conditions





E�ciency 98,3936 %Toil − Tamb 30,6952 oC

115



Oil 75W90 (PAO)



Actual Conditions

nin 200.1236 rpmnout 798.4651 rpmTin 1935,3 NmTout 477,3941 NmInput Power 40558 WOutput Power 39917 W





116

B.1. Load Tests


Oil 75W90 (PAO)



Actual Conditions






117



Oil 75W90 (PAO)



Actual Conditions






118

B.1. Load Tests


Oil 75W90 (PAO)



Actual Conditions






119



Oil 75W90 (PAO)



Actual Conditions






120

B.1. Load Tests


Oil 75W90 (PAO)



Actual Conditions






121



Oil 75W90 (PAO)



Actual Conditions



Toil 95,7326 oCToil2 89.2645 oCTwall 87,9368 oCTamb 33,1260 oC



122

B.1. Load Tests

B.1.2. Five tests grid

75W90


Oil 75W90 (PAO)



Actual Conditions






123



Oil 75W90 (PAO)



Actual Conditions






124

B.1. Load Tests


Oil 75W90 (PAO)



Actual Conditions






125



Oil 75W90 (PAO)



Actual Conditions






126

B.1. Load Tests


Oil 75W90 (PAO)



Actual Conditions






127


75W140


Oil 75W140 (PAO)



Actual Conditions






128

B.1. Load Tests


Oil 75W140 (PAO)



Actual Conditions






129



Oil 75W140 (PAO)



Actual Conditions






130

B.1. Load Tests


Oil 75W140 (PAO)



Actual Conditions






131



Oil 75W140 (PAO)



Actual Conditions






132

B.1. Load Tests

80W90


Oil 80W90 (Mineral)



Actual Conditions






133



Oil 80W90 (Mineral)



Actual Conditions






134

B.1. Load Tests


Oil 80W90 (Mineral)



Actual Conditions






135



Oil 80W90 (Mineral)



Actual Conditions






136

B.1. Load Tests


Oil 80W90 (Mineral)



Actual Conditions






137


B.1.3. No-Load tests

80W90


Oil 80W90 (Mineral)


nin 100 rpmTin 2800 Nm

Actual Conditions

nin 102,5963 rpmTin 4,3774 NmT initial0 8,5 NmT final0 21 NmInput Power 47 WTest duration 60 min


Toil 52,5/50 oC

138

B.1. Load Tests


Oil 80W90 (Mineral)


nin 150 rpmTin 2000/2400/2800 Nm

Actual Conditions

nin 152,79 rpmTin 10,66/8,13/6,95 NmT initial0 18,5 NmT final0 22,5 NmInput Power 170,51/130,1/111,19 WTest duration 124 min


Toil 60/53,4 oC


Oil 80W90 (Mineral)



Actual Conditions

nin 204,1908 rpmTin 1,51 NmT initial0 18 NmT final0 23 NmInput Power 32,3 WTest duration 59 min


Toil 52,5/50 oC

139



Oil 80W90 (Mineral)



Actual Conditions

nin 102,5963 rpmTin 7,689 NmT initial0 7,5 NmT final0 14,5 NmInput Power 82,61 WTest duration 54 min


Toil 52,5/50 oC


Oil 80W90 (Mineral)


nin 150 rpmTin 2000/2400/2800 Nm

Actual Conditions

nin 152,79 rpmTin 7,35/6,62/7,05 NmT initial0 10,5 NmT final0 23,5 NmInput Power 117,68/105,56/112,76 WTest duration 124 min


Toil 60/53,4 oC

140

B.1. Load Tests


Oil 80W90 (Mineral)



Actual Conditions

nin 204,1908 rpmTin 2,62 NmT initial0 12,5 NmT final0 20 NmInput Power 55,94 WTest duration 58 min


Toil 52,5/50 oC


Oil 80W90 (Mineral)



Actual Conditions

nin 102,5963 rpmTin 6,22 NmT initial0 6 NmT final0 14,5 NmInput Power 66,85 WTest duration 57 min


Toil 52,5/50 oC

141



Oil 80W90 (Mineral)


nin 150 rpmTin 2000/2400/2800 Nm

Actual Conditions

nin 152,79 rpmTin 7,6/8,48/10,1 NmT initial0 3,5 NmT final0 19 NmInput Power 121,63/135,75/161,28 WTest duration 75 min


Toil 60/53,4 oC


Oil 80W90 (Mineral)



Actual Conditions



Toil 52,5/50 oC

142

B.1. Load Tests

75W140


Oil 75W140 (PAO)



Actual Conditions

nin 102,5196 rpmTin 5,98 NmT initial0 -14,5 NmT final0 8 NmInput Power 64,15 WTest duration 55 min


Toil 54,2/52 oC

143



Oil 75W140 (PAO)


nin 150 rpmTin 2000/2400/2800 Nm

Actual Conditions

nin 153,98/150,11/153,92 rpmTin 8,36/6,55/7,69 NmT initial0 4,5 NmT final0 10 NmInput Power 134,85/102,96/124,04 WTest duration 81 min


Toil 62/55 oC


Oil 75W140 (PAO)



Actual Conditions

nin 202,81 rpmTin 6,03 NmT initial0 -9,5 NmT final0 7,5 NmInput Power 128 WTest duration 57 min


Toil 73,6/71,5 oC

144

B.1. Load Tests


Oil 75W140 (PAO)



Actual Conditions

nin 102,5196 rpmTin 8,92 NmT initial0 -9,5 NmT final0 11,5 NmInput Power 95,72 WTest duration 54 min


Toil 54,2/52 oC


Oil 75W140 (PAO)


nin 150 rpmTin 2000/2400/2800 Nm

Actual Conditions

nin 153,98/150,11/153,92 rpmTin 7,38/8,19/8,81 NmT initial0 8,5 NmT final0 19 NmInput Power 119,039/128,8/141,95 WTest duration 98 min


Toil 62/55 oC

145



Oil 75W140 (PAO)



Actual Conditions



Toil 73,6/71,5 oC


Oil 75W140 (PAO)



Actual Conditions

nin 102,5196 rpmTin 8,59 NmT initial0 -6 NmT final0 7 NmInput Power 92,32 WTest duration 50 min


Toil 54,2/52 oC

146

B.1. Load Tests


Oil 75W140 (PAO)


nin 150 rpmTin 2000/2400/2800 Nm

Actual Conditions

nin 153,98/150,11/153,92 rpmTin 7,57/7,83/6,95 NmT initial0 6 NmT final0 11 NmInput Power 122,11/123,02/111,99 WTest duration 95 min


Toil 62/55 oC


Oil 75W140 (PAO)



Actual Conditions



Toil 73,6/71,5 oC

147


75W90


Oil 75W90 (PAO)



Actual Conditions



Toil 54/52 oC

148

B.1. Load Tests


Oil 75W90 (PAO)


nin 150 rpmTin 2000/2400/2800 Nm

Actual Conditions

nin 150,135/150,131/150,1071 rpmTin 6,19/6,76/8,13 NmT initial0 6 NmT final0 11 NmInput Power 97,43/106,35/127,74 WTest duration 89 min


Toil 64,8/55,5 oC


Oil 75W90 (PAO)



Actual Conditions



Toil 78,8/76,8 oC

149



Oil 75W90 (PAO)



Actual Conditions



Toil 54/52 oC


Oil 75W90 (PAO)


nin 150 rpmTin 2000/2400/2800 Nm

Actual Conditions

nin 150,135/150,131/150,1071 rpmTin 7,08/7,23/8,69 NmT initial0 -8 NmT final0 6 NmInput Power 111,39/113,64/136,63 WTest duration 87 min


Toil 64,8/55,5 oC

150

B.1. Load Tests


Oil 75W90 (PAO)



Actual Conditions

nin 200,2288 rpmTin 2,57 NmT initial0 -11 NmT final0 5,5 NmInput Power 53,92 WTest duration 67 min


Toil 78,8/76,8 oC


Oil 75W90 (PAO)



Actual Conditions

nin 103,7788 rpmTin 5,89 NmT initial0 -9,5 NmT final0 1,5 NmInput Power 63,96 WTest duration 48 min


Toil 54/52 oC

151



Oil 75W90 (PAO)


nin 150 rpmTin 2000/2400/2800 Nm

Actual Conditions

nin 150,135/150,131/150,1071 rpmTin 5,82/5,97/6,73 NmT initial0 -7,5 NmT final0 9,5 NmInput Power 91,42/93,87/105,81 WTest duration 102 min


Toil 64,8/55,5 oC


Oil 75W90 (PAO)



Actual Conditions

nin 200,2288 rpmTin 2,45 NmT initial0 -4 NmT final0 3,5 NmInput Power 51,43 WTest duration 65 min


Toil 78,8/76,8 oC

152

C. FTIR analysis

153

C. FTIR analysis

154

D. Kissoft data

155

1/10

KISSsoft Release 03/2014 F KISSsoft University license - Universidade do Porto File Name : UnnamedChanged by: up201306632 on: 20.03.2015 at: 16:09:45

CALCULATION OF A HELICAL PLANETARY GEAR

Drawing or article number:Gear 1: 0.000.0Gear 2: 0.000.0Gear 3: 0.000.0

Calculation method ISO 6336:2006 Method B

------- Gear 1 --------- Gear 2 --------- Gear 3 ---Number of planets [p] (1) 3 (1)

Power (kW) [P] 65.45Speed (1/min) [n] 1000.0 0.0Speed difference for planet bearing calculation (1/min) [n2] 750.0Speed planet carrier (1/min) [nSteg] 250.0

Torque (Nm) [T] 625.0 0.0 1875.0Torque Pl.-Carrier (Nm) [TSteg] 2500.000

Application factor [KA] 1.25Power distribution factor [Kgam] 1.00Required service life [H] 20000.00Gear driving (+) / driven (-) + -/+ -

1. TOOTH GEOMETRY AND MATERIAL

(geometry calculation according to DIN 3960:1987) ------- Gear 1 ------------ Gear 2 ------------ Gear 3 ---Center distance (mm) [a] 73.035Centre distance tolerance ISO 286:2010 Measure js7

Normal module (mm) [mn] 2.0000Pressure angle at normal section (°) [alfn] 20.0000Helix angle at reference circle (°) [beta] 10.0000Number of teeth [z] 36 36 -108Facewidth (mm) [b] 42.00 42.00 42.00Hand of gear right left left

Planetary axles can be placed in regular pitch.: 120°

Accuracy grade [Q-ISO1328:1995] 6 6 6Inner diameter (mm) [di] 0.00 0.00External diameter (mm) [di] 0.00Inner diameter of gear rim (mm) [dbi] 0.00 0.00Outer diameter of gear rim (mm) [dbi] 0.00

2/10

MaterialGear 1: 18CrNiMo7-6, Case-carburized steel, case-hardened ISO 6336-5 Figure 9/10 (MQ), core strength >=25HRC Jominy J=12mm<HRC28Gear 2: 18CrNiMo7-6, Case-carburized steel, case-hardened ISO 6336-5 Figure 9/10 (MQ), core strength >=25HRC Jominy J=12mm<HRC28Gear 3: 18CrNiMo7-6, Case-carburized steel, case-hardened ISO 6336-5 Figure 9/10 (MQ), core strength >=25HRC Jominy J=12mm<HRC28 ------- Gear 1 ------------ Gear 2 ------------ Gear 3 ---Surface hardness HRC 61 HRC 61 HRC 61Material quality according to ISO 6336:2006 Normal (Life factors ZNT and YNT >=0.85)Fatigue strength. tooth root stress (N/mm²) [sigFlim] 430.00 430.00 430.00Fatigue strength for Hertzian pressure (N/mm²) [sigHlim] 1500.00 1500.00 1500.00Tensile strength (N/mm²) [Rm] 1200.00 1200.00 1200.00Yield point (N/mm²) [sigs] 850.00 850.00 850.00Young's modulus (N/mm²) [E] 206000 206000 206000Poisson's ratio [ny] 0.300 0.300 0.300Mean roughness, Ra, tooth flank (µm) [RAH] 0.60 0.60 0.60Mean roughness height, Rz, flank (µm) [RZH] 4.80 4.80 4.80Mean roughness height, Rz, root (µm) [RZF] 20.00 20.00 20.00

Gear reference profile 1 :Reference profile 1.25 / 0.25 / 1.0 ISO 53.2:1997 Profil CDedendum coefficient [hfP*] 1.250Root radius factor [rhofP*] 0.250Addendum coefficient [haP*] 1.000Tip radius factor [rhoaP*] 0.000Protuberance height factor [hprP*] 0.000Protuberance angle [alfprP] 0.000Tip form height coefficient [hFaP*] 0.000Ramp angle [alfKP] 0.000 not topping



3/10

Summary of reference profile gears:Dedendum reference profile [hfP*] 1.250 1.250 1.250Tooth root radius Reference profile [rofP*] 0.250 0.250 0.250Addendum Reference profile [haP*] 1.000 1.000 1.000Protuberance height factor [hprP*] 0.000 0.000 0.000Protuberance angle (°) [alfprP] 0.000 0.000 0.000Tip form height coefficient [hFaP*] 0.000 0.000 0.000Ramp angle (°) [alfKP] 0.000 0.000 0.000

Type of profile modification: none (only running-in)Tip relief (µm) [Ca] 2.00 2.00 2.00

Lubrication type oil bath lubricationType of oil Oil: ISO-VG 220Lubricant base Mineral-oil baseKinem. viscosity oil at 40 °C (mm²/s) [nu40] 220.00Kinem. viscosity oil at 100 °C (mm²/s) [nu100] 17.50FZG test A/8.3/90 ( ISO 14635-1:2006) [FZGtestA] 12Specific density at 15 °C (kg/dm³) [roOil] 0.895Oil temperature (°C) [TS] 70.000

------- Gear 1 ------------ Gear 2 ------------ Gear 3 ---Overall transmission ratio [itot] 4.000Gear ratio [u] 1.000 -3.000Transverse module (mm) [mt] 2.031Pressure angle at pitch circle (°) [alft] 20.284Working transverse pressure angle (°) [alfwt] 20.122 20.122 [alfwt.e/i] 20.154 / 20.090 20.090 / 20.154Working pressure angle at normal section (°) [alfwn] 19.841 19.841Helix angle at operating pitch circle (°) [betaw] 9.990 9.990Base helix angle (°) [betab] 9.391Reference centre distance (mm) [ad] 73.111 -73.111Sum of profile shift coefficients [Summexi] -0.0378 0.0378Profile shift coefficient [x] -0.0189 -0.0189 0.0567Tooth thickness (Arc) (module) (module) [sn*] 1.5570 1.5570 1.6121

Tip alteration (mm) [k*mn] 0.000 0.000 0.000Reference diameter (mm) [d] 73.111 73.111 -219.332Base diameter (mm) [db] 68.577 68.577 -205.731Tip diameter (mm) [da] 77.035 77.035 -215.105 (mm) [da.e/i] 77.035 / 77.025 77.035 / 77.025 -215.105 / -215.115Tip diameter allowances (mm) [Ada.e/i] 0.000 / -0.010 0.000 / -0.010 0.000 / -0.010Tip form diameter (mm) [dFa] 77.035 77.035 -215.105 (mm) [dFa.e/i] 77.035 / 77.025 77.035 / 77.025 -215.105 / -215.115Active tip diameter (mm) [dNa.e/i] 77.035 / 77.025 77.035 / 77.025 -215.105 / -215.115Operating pitch diameter (mm) [dw] 73.035 73.035 / 73.035 -219.105 (mm) [dw.e] 73.050 73.050 / 73.020 -219.060 (mm) [dw.i] 73.020 73.020 / 73.050 -219.150Root diameter (mm) [df] 68.035 68.035 -224.105Generating Profile shift coefficient [xE.e/i] -0.0670 / -0.0944 -0.0670 / -0.0945 -0.0086 / -0.0429Manufactured root diameter with xE (mm) [df.e] 67.843 67.843 -224.366 (mm) [df.i] 67.733 67.733 -224.504Theoretical tip clearance (mm) [c] 0.500 0.500/ 0.500 0.500Tip clearance upper allowance (mm) [c.e] 0.671 0.671/ 0.719 0.671Tip clearance lower allowance (mm) [c.i] 0.581 0.581/ 0.616 0.581Active root diameter (mm) [dNf] 70.232 70.232/ 69.718 -222.731 (mm) [dNf.e] 70.256 70.256/ 69.740 -222.689

4/10

(mm) [dNf.i] 70.213 70.213/ 69.702 -222.764Root form diameter (mm) [dFf] 69.725 69.725 -223.078 (mm) [dFf.e/i] 69.627 / 69.573 69.627 / 69.573 -223.373 / -223.526Internal toothing: Calculation dFf with pinion type cutter (z0= 35, x0= 0.000)Reserve (dNf-dFf)/2 (mm) [cF.e/i] 0.341 / 0.293 0.083 / 0.037 0.419 / 0.305Addendum (mm) [ha = mn * (haP*+x)] 1.962 1.962 2.113 (mm) [ha.e/i] 1.962 / 1.957 1.962 / 1.957 2.113 / 2.108Dedendum (mm) [hf = mn * (hfP*-x)] 2.538 2.538 2.387 (mm) [hf.e/i] 2.634 / 2.689 2.634 / 2.689 2.517 / 2.586Roll angle at dFa (°) [xsi_dFa.e/i] 29.321 / 29.303 29.321 / 29.303 17.492 / 17.502Roll angle to dNf (°) [xsi_dNf.e/i] 12.754 / 12.590 12.754 / 12.590 [xsi_dNf.e/i] 10.595 / 10.420 23.738 / 23.793Roll angle at dFf (°) [xsi_dFf.e/i] 10.065 / 9.801 10.065 / 9.800 24.232 / 24.341Tooth height (mm) [H] 4.500 4.500 4.500Virtual gear no. of teeth [zn] 37.555 37.555 -112.666Normal tooth thickness at tip cylinder (mm) [san] 1.520 1.520 1.769 (mm) [san.e/i] 1.452 / 1.405 1.452 / 1.405 1.679 / 1.626Normal spacewidth at root cylinder (mm) [efn] 0.000 0.000 1.218 (mm) [efn.e/i] 0.000 / 0.000 0.000 / 0.000 1.205 / 1.198Max. sliding velocity at tip (m/s) [vga] 0.783 0.783/ 0.261 0.329Specific sliding at the tip [zetaa] 0.568 0.568/ 0.189 0.400Specific sliding at the root [zetaf] -1.315 -1.315/ -0.667 -0.234Sliding factor on tip [Kga] 0.273 0.273/ 0.091 0.115Sliding factor on root [Kgf] -0.273 -0.273/ -0.115 -0.091Pitch on reference circle (mm) [pt] 6.380Base pitch (mm) [pbt] 5.984Transverse pitch on contact-path (mm) [pet] 5.984Lead height (mm) [pz] 1302.603 1302.603 3907.810Axial pitch (mm) [px] 36.183 36.183 36.183Length of path of contact (mm) [ga] 9.969 11.268 (mm) [ga.e/i] 10.013 / 9.903 11.311 / 11.196Length T1-A (mm) [T1A] 7.578 17.547/ 6.280 -31.405Length T1-B (mm) [T1B] 11.563 13.563/ 11.563 -36.688Length T1-C (mm) [T1C] 12.563 12.563/ 12.563 -37.688Length T1-D (mm) [T1D] 13.563 11.563/ 12.264 -37.389Length T1-E (mm) [T1E] 17.547 7.578/ 17.547 -42.672Diameter of single contact point B (mm) [d-B] 72.371 73.747/ 72.371 -218.425(mm) [d-B.e] 72.371 73.715/ 72.371 -218.454(mm) [d-B.i] 72.364 73.787/ 72.364 -218.388Diameter of single contact point D (mm) [d-D] 73.747 72.371/ 72.832 -218.900 (mm) [d-D.e] 73.715 72.371/ 72.802 -218.900(mm) [d-D.i] 73.787 72.364/ 72.873 -218.911

Transverse contact ratio [eps_a] 1.666 1.883Transverse contact ratio with allowances [eps_a.e/i] 1.673 / 1.655 1.890 / 1.871Overlap ratio [eps_b] 1.161 1.161Total contact ratio [eps_g] 2.827 3.044Total contact ratio with allowances [eps_g.e/i] 2.834 / 2.816 3.051 / 3.032

2. FACTORS OF GENERAL INFLUENCE

5/10

------- Gear 1 ------------ Gear 2 ------------ Gear 3 ---Nominal circum. force at pitch circle (N) [Ft] 5699.119 5699.119Axial force (N) [Fa] 1004.9 1004.9 1004.9Axial force (total) (N) [Fatot=Fa* 3] 3014.7 3014.7Radial force (N) [Fr] 2106.309 2106.309Normal force (N) [Fnorm] 6158.4 6158.4 6158.4Tangent.load at p.c.d.per mm (N/mm) (N/mm) [w] 135.69 135.69Only as information: Forces at operating pitch circle:Nominal circumferential force (N) [Ftw] 5705.040 5705.040Axial force (N) [Fa] 1004.9 1004.9/ 1004.9 1004.9Axial force (total) (N) [Fatot=Fa* 3] 3014.7 3014.7Radial force (N) [Fr] 2090.219 2090.219Circumferential speed reference circle (m/s) [v] 2.87Circumferential speed operating pitch circle (m/s) [v(dw)] 2.87

Running-in value (µm) [yp] 0.525 0.600Running-in value (µm) [yf] 0.487 0.563Gear body coefficient [CR] 1.000 1.000Correction coefficient [CM] 0.800 0.800Reference profile coefficient [CBS] 0.975 0.975Material coefficient [E/Est] 1.000 1.000Singular tooth stiffness (N/mm/µm) [c'] 13.113 14.931Meshing stiffness (N/mm/µm) [cgalf] 19.661 24.816Meshing stiffness (N/mm/µm) [cgbet] 16.712 21.094Reduced mass (kg/mm) [mRed] 0.0045 0.0181Resonance speed (min-1) [nE1] 17485 9822Resonance ratio (-) [N] 0.043 0.076Running-in value (µm) [ya] 0.525 0.600Planets are supported by fixed restraint boltslpa (mm) = 54.60 b (mm) = 42.00 dsh (mm) = 36.56Tooth trace deviation (active) (µm) [Fby] 4.25 5.87from deformation of shaft (µm) [fsh*B1] 4.10 0.45Tooth trace 0 0(0:without, 1:crowned, 2:Tip relief, 3:full modification)from production tolerances (µm) [fma*B2] 14.14 14.14Running-in value y.b (µm) [yb] 0.75 1.04

Dynamic factor [KV=max(KV12,KV23)] 1.03 [KV12,KV23] 1.01 1.03

Face load factor - flank [KHb] 1.20 1.35 - Tooth root [KFb] 1.18 1.31 - Scuffing [KBb] 1.20 1.35

Transverse load factor - flank [KHa] 1.18 1.26 - Tooth root [KFa] 1.18 1.26 - Scuffing [KBa] 1.18 1.26

Helical load factor scuffing [Kbg] 1.27 1.29

Number of load cycles (in mio.) [NL] 2700.0 900.0 900.0

3. TOOTH ROOT STRENGTH

6/10

Calculation of Tooth form coefficients according method: BTooth form factors calculated with manufacturing profile shift xE.eInternal toothing: Calculation of roF and sFn according to ISO 6336-3:2007-04-01Internal toothing: Calculation of YF, YS with pinion type cutter (z0= 35, x0= 0.000, rofP*= 0.250) ------- Gear 1 ------------ Gear 2 ------------ Gear 3 ---Tooth form factor [YF] 1.46 1.46/ 1.16 0.95Stress correction factor [YS] 2.04 2.04/ 2.22 2.73Bending lever arm (mm) [hF] 2.11 2.11/ 1.67 2.28Working angle (°) [alfFen] 18.88 18.88/ 16.75 20.15Tooth thickness at root (mm) [sFn] 4.19 4.19/ 4.19 5.37Tooth root radius (mm) [roF] 0.94 0.94/ 0.94 0.73(sFn* = 2.094/ 2.094/ 2.094/ 2.684 roF* = 0.470/ 0.470/ 0.470/ 0.366 dsFn = 68.495/ 68.495/ 68.495/ -224.179 alfsFn = 30.0/ 30.0/ 30.0/ 60.0)

Helix angle factor [Ybet] 0.92 0.92Deep tooth factor [YDT] 1.00 1.00Gear rim factor [YB] 1.00 1.00 1.00Effective facewidth (mm) [beff] 42.00 42.00/ 42.00 42.00Nominal stress at tooth root (N/mm²) [sigF0] 185.03 185.03/ 160.98 161.09Tooth root stress (N/mm²) [sigF] 330.33 330.33/ 342.13 342.36Permissible bending stress at root of Test-gearSupport factor [YdrelT] 0.997 0.997/ 0.997 1.010Surface factor [YRrelT] 0.957 0.957 0.957size factor (Tooth root) [YX] 1.000 1.000 1.000Finite life factor [YNT] 0.873 0.892 0.892Alternating bending coefficient [YM] 1.000 0.700 1.000Stress correction factor [Yst] 2.00Yst*sigFlim (N/mm²) [sigFE] 860.00 860.00 860.00Permissible tooth root stress (N/mm²) [sigFP=sigFG/SFmin] 511.48 366.00/ 366.00 529.55Limit strength tooth root (N/mm²) [sigFG] 716.07 512.41/ 512.41 741.37Required safety [SFmin] 1.40 1.40/ 1.40 1.40Safety for Tooth root stress [SF=sigFG/sigF] 2.17 1.55/ 1.50 2.17Transmittable power (kW) [kWRating] 101.34 72.52/ 70.02 101.23

4. SAFETY AGAINST PITTING (TOOTH FLANK)

------- Gear 1 ------------ Gear 2 ------------ Gear 3 ---Zone factor [ZH] 2.47 2.47Elasticity coefficient (N^.5/mm) [ZE] 189.81 189.81Contact ratio factor [Zeps] 0.775 0.729Helix angle factor [Zbet] 1.008 1.008Effective facewidth (mm) [beff] 42.00 42.00Nominal flank pressure (N/mm²) [sigH0] 706.42 383.63Surface pressure at operating pitch circle (N/mm²) [sigHw] 953.41 568.66Single tooth contact factor [ZB,ZD] 1.00 1.00/ 1.00 1.00Flank pressure (N/mm²) [sigHB, sigHD] 953.41 953.41/ 568.66 568.66

Lubrication coefficient at NL [ZL] 1.020 1.020/ 1.020 1.020Speed coefficient at NL [ZV] 0.971 0.971/ 0.971 0.971Roughness coefficient at NL [ZR] 0.951 0.951/ 0.980 0.980Material pairing coefficient at NL [ZW] 1.000 1.000/ 1.000 1.000Finite life factor [ZNT] 0.885 0.915 0.915Small no. of pittings permissible: no

7/10

Size factor (flank) [ZX] 1.000 1.000 1.000Permissible surface pressure (N/mm²) [sigHP=sigHG/SHmin] 1249.78 1292.61/ 1331.04 1331.04Limit strength pitting (N/mm²) [sigHG] 1249.78 1292.61/ 1331.04 1331.04

Required safety [SHmin] 1.00 1.00/ 1.00 1.00Safety for surface pressure at operating pitch circle [SHw] 1.31 1.36/ 2.34 2.34Safety for stress at single tooth contact [SHBD=sigHG/sigHBD] 1.31 1.36/ 2.34 2.34(Safety regarding nominal torque) [(SHBD)^2] 1.72 1.84/ 5.48 5.48Transmittable power (kW) [kWRating] 112.46 120.31/ 358.58 358.58

4b. MICROPITTING ACCORDING TO ISO TR 15144-1:2010

Pairing Gear 1- 2:Calculation did not run. (Lubricant: Load stage micropitting test is unknown.)

Pairing Gear 2- 3:Calculation did not run. (Lubricant: Load stage micropitting test is unknown.)

5. STRENGTH AGAINST SCUFFING

Calculation method according to ISO TR 13989:2000

Lubrication coefficient (for lubrication type) [XS] 1.000Multiple meshing factor [Xmp] 2.0 2.0Relative structure coefficient (Scuffing) [XWrelT] 1.000 1.000Thermal contact factor (N/mm/s^.5/K) [BM] 13.780 13.780 13.780Relevant tip relief (µm) [Ca] 2.00 2.00 2.00Optimal tip relief (µm) [Ceff] 8.63 6.83Ca taken as optimal in the calculation (0=no, 1=yes) 0 0/ 0 0Effective facewidth (mm) [beff] 42.000 42.000Applicable circumferential force/facewidth (N/mm) [wBt] 247.172 298.155((1)Kbg = 1.268, wBt*Kbg = 313.430)((2)Kbg = 1.286, wBt*Kbg = 383.360)Angle factor [Xalfbet] 0.976 0.976

Flash temperature-criteriaLubricant factor [XL] 0.830 0.830Tooth mass temperature (°C) [theMi] 83.42 73.09theM = theoil + XS*0.47*Xmp*theflm [theflm] 14.28 3.28Scuffing temperature (°C) [theS] 348.80 348.80Coordinate gamma (point of highest temp.) [Gamma] -0.397 -0.500(1) [Gamma.A]= -0.397 [Gamma.E]= 0.397(2) [Gamma.A]= -0.500 [Gamma.E]= 0.397Highest contact temp. (°C) [theB] 117.90 83.18Flash factor (°K*N^-.75*s^.5*m^-.5*mm) [XM] 50.058 50.058Approach factor [XJ] 1.017 1.017Load sharing factor [XGam] 0.780 0.690Dynamic viscosity (mPa*s) [etaM] 41.90 41.90Coefficient of friction [mym] 0.070 0.058Required safety [SBmin] 2.000

8/10

Safety factor for scuffing (flash-temp) [SB] 5.820 21.158

Integral temperature-criteriaLubricant factor [XL] 1.000Tooth mass temperature (°C) [theM-C] 86.47 72.16theM-C = theoil + XS*0.70*theflaint [theflaint] 11.77 1.54Integral scuffing temperature (°C) [theSint] 360.78 360.78Flash factor (°K*N^-.75*s^.5*m^-.5*mm) [XM] 50.058 50.058Running-in factor (well run in) [XE] 1.000 1.000Contact ratio factor [Xeps] 0.255 0.271Dynamic viscosity (mPa*s) [etaOil] 41.90 41.90Averaged coefficient of friction [mym] 0.087 0.055Geometry factor [XBE] 0.305 0.058Meshing factor [XQ] 1.000 1.000Tip relief factor [XCa] 1.210 1.363Integral tooth flank temperature (°C) [theint] 104.12 74.48Required safety [SSmin] 1.800Safety factor for scuffing (intg.-temp.) [SSint] 3.46 4.84Safety referring to transferred torque [SSL] 8.52 64.95

6. MEASUREMENTS FOR TOOTH THICKNESS

------- Gear 1 ------------ Gear 2 ------------ Gear 3 ---Tooth thickness deviation DIN 3967 cd25 DIN 3967 cd25 DIN 3967 cd25Tooth thickness allowance (normal section) (mm) [As.e/i] -0.070/ -0.110 -0.070/ -0.110 -0.095/ -0.145

Number of teeth spanned [k] 5.000 5.000 0.000(Internal toothing: k = (Measurement gap number)Base tangent length (no backlash) (mm) [Wk] 27.597 27.597 0.000Actual base tangent length ('span') (mm) [Wk.e/i] 27.531/ 27.493 27.531/ 27.493 0.000/ 0.000Diameter of contact point (mm) [dMWk.m] 73.753 73.753 0.000

Theoretical diameter of ball/pin (mm) [DM] 3.389 3.389 3.320Eff. Diameter of ball/pin (mm) [DMeff] 3.500 3.500 3.500Theor. dim. centre to ball (mm) [MrK] 39.020 39.020 -106.971Actual dimension centre to ball (mm) [MrK.e/i] 38.935/ 38.885 38.935/ 38.885 -107.110/ -107.183Diameter of contact point (mm) [dMMr.m] 73.059 73.059 -218.950Diametral measurement over two balls without clearance (mm) [MdK] 78.041 78.041 -213.942Actual dimension over balls (mm) [MdK.e/i] 77.870/ 77.771 77.870/ 77.771 -214.220/ -214.365Actual dimension over rolls (mm) [MdR.e/i] 77.870/ 77.771 77.870/ 77.771 0.000/ 0.000Actual dimensions over 3 rolls (mm) [Md3R.e/i] 0.000/ 0.000 0.000/ 0.000 0.000/ 0.000Note: Internal gears with helical teeth cannot be measured with rollers.

Tooth thickness (chordal) in pitch diameter (mm) ['sn] 3.113 3.113 3.224 (mm) ['sn.e/i] 3.043/ 3.003 3.043/ 3.003 3.129/ 3.079Reference chordal height from da.m (mm) [ha] 1.992 1.992 2.099Tooth thickness (Arc) (mm) [sn] 3.114 3.114 3.224 (mm) [sn.e/i] 3.044/ 3.004 3.044/ 3.004 3.129/ 3.079

Backlash free center distance (mm) [aControl.e/i] 72.839/ 72.725 -73.261/ -73.382Backlash free center distance, allowances (mm) [jta] -0.196/ -0.309 -0.000/ -0.000dNf.i with aControl (mm) [dNf0.i] 69.860 69.388 -223.498Reserve (dNf0.i-dFf.e)/2 (mm) [cF0.i] 0.116 -0.120 -0.062Centre distance allowances (mm) [Aa.e/i] 0.015/ -0.015 0.015/ -0.015

9/10

Circumferential backlash from Aa (mm) [jtw_Aa.e/i] 0.011/ -0.011 0.011/ -0.011Radial clearance (mm) [jrw] 0.324/ 0.181 0.362/ 0.211Circumferential backlash (transverse section) (mm) [jtw] 0.234/ 0.131 0.270/ 0.156Normal backlash (mm) [jnw] 0.217/ 0.121 0.250/ 0.145

Entire torsional angle (°) [j.tSys] 0.1890/ 0.1214(j.tSys: Torsional angle of planet carrier for blocked shaft)

7. GEAR ACCURACY

------- Gear 1 ------------ Gear 2 ------------ Gear 3 ---According to ISO 1328:1995Accuracy grade [Q-ISO1328] 6 6 6Single pitch deviation (µm) [fptT] 7.50 7.50 8.50Base circle pitch deviation (µm) [fpbT] 7.00 7.00 8.00Sector pitch deviation over k/8 pitches (µm) [Fpk/8T] 12.00 12.00 16.00Profile deviation (µm) [ffaT] 6.50 6.50 7.50Profile slope deviation (µm) [fHaT] 5.50 5.50 6.00Total profile deviation (µm) [FaT] 8.50 8.50 10.00Helix form deviation (µm) [ffbT] 10.00 10.00 10.00Helix slope deviation (µm) [fHbT] 10.00 10.00 10.00Total helix deviation (µm) [FbT] 14.00 14.00 15.00Total cumulative pitch deviation (µm) [FpT] 26.00 26.00 35.00Runout (µm) [FrT] 21.00 21.00 28.00Single flank composite, total (µm) [FisT] 37.00 36.00 46.00Single flank composite, tooth-to-tooth (µm) [fisT] 11.00 10.00 12.00Radial composite, total (µm) [FidT] 31.00 31.00 37.00Radial composite, tooth-to-tooth (µm) [fidT] 9.50 9.50 9.50

Axis alignment tolerances (recommendation acc. ISO TR 10064:1992, Quality 6)Maximum value for deviation error of axis (µm) [fSigbet] 9.10 9.10Maximum value for inclination error of axes (µm) [fSigdel] 18.20 18.20

8. ADDITIONAL DATA

Mean coeff. of friction (acc. Niemann) [mum] 0.073 0.059Wear sliding coef. by Niemann [zetw] 0.946 0.578

Meshpower (kW) 49.087 49.087Power loss from gear load (kW) 0.153 0.052Total power loss (kW) 0.615Total efficiency 0.991Weight - calculated with da (kg) [Mass] 1.533 1.533 2.236Total weight (kg) [Mass] 8.367

Moment of inertia (System referenced to wheel 1): calculation without consideration of the exact tooth shapesingle gears ((da+df)/2...di) (kg*m²) [TraeghMom] 0.0008875 0.0008875 0.02501System ((da+df)/2...di) (kg*m²) [TraeghMom] 0.002385

Indications for the manufacturing by wire cutting:Deviation from theoretical tooth trace (µm) [WireErr] 187.4 187.4 62.5Permissible deviation (µm) [Fb/2] 7.0 7.0 7.5

10/10

9. DETERMINATION OF TOOTHFORM

Data for the tooth form calculation :Data not available.

REMARKS:- Specifications with [.e/i] imply: Maximum [e] and Minimal value [i] with consideration of all tolerances Specifications with [.m] imply: Mean value within tolerance- For the backlash tolerance, the center distance tolerances and the tooth thicknessdeviation are taken into account. Shown is the maximal and the minimal backlash corresponding the largest resp. the smallest allowances The calculation is done for the Operating pitch circle..- Calculation of Zbet according Corrigendum 1 % ISO 6336-2:2008 with Zbet = 1/(COS(beta)^0.5)- Details of calculation method: cg according to method B KV according to method B KHb, KFb according method C fma following equation (64), Fbx following (52/53/56) fsh calculated by exactly following the method in Annex D, ISO 6336-1:2006 Literature: Journal "Antriebstechnik", 6/2007, p.64. KHa, KFa according to method B End of Report lines: 531

E. Detailed results of theimplemented numericalapproach, (Program printouts)

E.1. Results before optimization [16 test grid]

The operating conditions of these results are shown in table 7.1.

nu_vogel [cSt] =

109.4674 76.0597 48.6130 38.4821101.6477 65.5402 46.3010 31.429986.9330 56.7009 32.3625 24.783972.2821 39.3956 23.9062 19.5493

dens [kg/cm3] =

0.8536 0.8484 0.8409 0.83620.8526 0.8461 0.8399 0.83170.8504 0.8436 0.8324 0.82580.8476 0.8367 0.8248 0.8190

eta0 [Pa · s] =

93.4393 64.5313 40.8767 32.179486.6643 55.4523 38.8898 26.141173.9288 47.8351 26.9388 20.466461.2694 32.9627 19.7183 16.0108

XL =

1

mu_mz_sp =

0.0560 0.0527 0.0511 0.04940.0589 0.0555 0.0536 0.05220.0615 0.0579 0.0566 0.0548

167

E. Detailed results of the implemented numerical approach, (Program printouts)

0.0641 0.0609 0.0593 0.0573

mu_mz_pr =

0.0436 0.0410 0.0397 0.03840.0458 0.0431 0.0416 0.04060.0478 0.0450 0.0440 0.04260.0498 0.0473 0.0461 0.0445

P_VZP [W] =

150.8764 211.3609 268.7949 325.4323196.9809 280.1807 351.7748 429.5113247.4603 351.2479 445.3209 540.8872299.1523 430.5499 544.4779 658.8029

M_DGB [N · mm] =

218.0120 213.3816 185.4900 178.4122230.9348 217.4044 199.4107 174.6623229.6408 216.8825 176.0733 167.4344221.0173 189.9237 167.2494 165.1970

P_VL_DGB [W] =

9.3348 13.6051 15.5221 18.67519.8400 13.9832 16.6737 18.28209.7993 13.9499 14.7141 17.52349.3810 12.2171 13.9789 17.2870

M_TRB11 [N · mm] =

1.0e+03 *

3.6149 3.5954 3.4830 3.46303.6381 3.5933 3.5471 3.49993.6506 3.6265 3.5661 3.54893.6929 3.6591 3.6032 3.5954

M_TRB12 [N · mm] =

1.0e+03 *

3.0005 2.9770 2.8068 2.77152.8464 2.7741 2.6822 2.56082.6377 2.5798 2.3804 2.3322

168

E.1. Results before optimization [16 test grid]

2.4171 2.2664 2.1146 2.0992

M_TRB [N · mm] =

1.0e+03 *

6.6154 6.5724 6.2898 6.23456.4845 6.3675 6.2293 6.06066.2884 6.2063 5.9465 5.88116.1101 5.9254 5.7178 5.6946

P_VL0_TRB [W] =

71.3066 105.4837 132.4119 164.163669.7218 103.3105 131.3415 159.817667.8885 100.9304 125.4121 155.287665.7561 96.4718 120.7354 150.5487

P_VL_TRB [W] =

70.9894 105.0189 131.8903 163.536269.2492 102.6393 130.5474 158.965067.2659 100.0455 124.5232 154.247665.0044 95.5305 119.7564 149.3496

TVL0_NRB [N · m] =

0.0266 0.0272 0.0242 0.02410.0253 0.0248 0.0235 0.02100.0228 0.0225 0.0185 0.01800.0201 0.0177 0.0151 0.0153

P_VL0_NRB [W] =

5.1452 7.8371 9.1469 11.36774.8578 7.2010 8.8461 9.93144.3890 6.5386 6.9580 8.47613.8459 5.1304 5.6877 7.2355

TVL1_NRB [N · m] =

0.0192 0.0192 0.0192 0.01920.0239 0.0239 0.0239 0.02390.0287 0.0287 0.0287 0.02870.0335 0.0335 0.0335 0.0335

169


TVL_NRB [N · m] =

0.0458 0.0464 0.0434 0.04320.0492 0.0488 0.0474 0.04500.0515 0.0513 0.0472 0.04670.0536 0.0512 0.0486 0.0488

P_VL_NRB [W] =

8.8459 13.3480 16.3788 20.41589.4577 14.1474 17.8767 21.23489.9192 14.8738 17.7779 22.035310.2607 14.8551 18.3168 23.0548

P_VD_IN [W] =

9.5350 14.1980 18.6319 23.30759.4890 14.3228 18.6213 23.30599.5048 14.3235 18.6067 23.30489.4532 14.3253 18.6102 23.3037

P_VD_OUT [W] =

22.7176 33.8282 44.3982 55.535822.6070 34.1251 44.3629 55.534022.6403 34.1256 44.3379 55.527822.5194 34.1292 44.3450 55.5206

P_VD [W] =

32.2526 48.0262 63.0301 78.843332.0960 48.4480 62.9842 78.839932.1451 48.4492 62.9446 78.832531.9726 48.4546 62.9552 78.8243

lambda_SP =

0.2744 0.2703 0.2263 0.21870.2518 0.2353 0.2127 0.18070.2176 0.2050 0.1547 0.14540.1831 0.1488 0.1181 0.1172

lambda_PR =

0.4191 0.4128 0.3456 0.33410.3846 0.3594 0.3249 0.27600.3324 0.3131 0.2362 0.2220

170

E.2. Results after optimization [5 test grid]

0.2797 0.2273 0.1803 0.1790


The operating conditions of these results are shown in table 7.3.

E.2.1. 75W90

nu_vogel [cSt] =

65.370057.541251.649244.663829.6889

dens [kg/cm3] =

0.84600.84390.84200.83920.8304

eta0 [Pa · s] =

55.305648.558743.487437.483624.6529

XL =

1.1500

mu_mz_sp =

0.07380.06450.06730.06990.0674

171


mu_mz_pr =

0.05740.05020.05230.05430.0524

P_VZP [W] =

351.9602318.0738397.9789482.0231619.7843

M_DGB [N · mm] =

210.8167198.5741202.6661200.7532182.4246

P_VL_DGB [W] =

9.142412.456012.712912.591815.2628

M_TRB11 [N · mm] =1.0e+03 *

3.68313.54553.60553.67213.6466

M_TRB12 [N · mm] =

1.0e+03 *

2.36872.6733

172


2.51032.32362.2259

M_TRB [N · mm] =

1.0e+03 *

6.05196.21886.11585.99575.8725

P_VL0_TRB [W] =

66.494998.361796.947895.2381124.2913

P_VL_TRB [W] =

65.769797.772396.151494.2473123.1349

TVL0_NRB [N · m] =

0.01900.02240.02080.01890.0174

P_VL0_NRB [W] =

3.72646.33365.89325.34776.5837

173


TVL1_NRB [N · m] =

0.03350.02400.02870.03350.0335

TVL_NRB [N · m] =

0.05260.04630.04960.05240.0509

P_VL_NRB [W] =

10.281713.111714.025414.827719.2285

P_VD_IN [W] =

9.656513.969913.969513.967318.6311

P_VD_OUT [W] =

23.008533.280533.281333.278244.3900

P_VD [W] =

32.665047.250447.250847.2455

174


63.0211

Power_Loss_no_LOAD [W] =

61.2990100.0790104.6170123.392047.2740

Lambda_SP =

0.17100.20720.18610.16240.1419

Lambda_PR =

0.26110.31640.28420.24800.2168

E.2.2. 75W140

nu_vogel [cSt] =

119.3111106.7735102.240088.989359.7590

dens [kg/cm3] =

0.86330.86160.86100.85880.8521

eta0 [Pa · s] =

175


102.999091.999188.025576.424850.9226

XL =

1.1000

mu_mz_sp =

0.06860.05950.06210.06420.0620

mu_mz_pr =

0.05330.04620.04830.04990.0482

P_VZP [W] =

323.1997300.5738367.4606454.1454577.8600

M_DGB [N · mm] =

209.2327202.8109212.7347214.1315196.8390

P_VL_DGB [W] =

8.962713.048813.3415

176


13.774516.6798

M_TRB11 [N · mm] =

1.0e+03 *

3.68173.55473.62013.68513.6672

M_TRB12 [N · mm] =

1.0e+03 *

2.36092.69562.55872.38892.3070

M_TRB [N · mm] =

1.0e+03 *

6.04266.25036.17886.07405.9742

P_VL0_TRB [W] =

65.5821101.405197.969199.0289128.1928

P_VL_TRB [W] =

64.8724100.785697.125497.9244

177


126.8803

TVL0_NRB [N · m] =

0.02820.03440.03280.03040.0280

P_VL0_NRB [W] =

5.45339.97599.28848.832110.7218

TVL1_NRB [N · m] =

0.03350.02390.02870.03350.0335

TVL_NRB [N · m] =

0.06170.05830.06160.06390.0616

P_VL_NRB [W] =

11.929816.924817.420018.557323.5349

P_VD_IN [W] =

9.539314.327913.9672

178


14.325118.8710

P_VD_OUT [W] =

22.727134.136333.273734.129544.9588

P_VD [W] =

32.266548.464147.240948.454663.8298


94.0170125.3350118.2610125.9930122.6960

Lambda_SP =

0.28230.35660.33200.29690.2598

Lambda_PR =

0.43120.54460.50700.45340.3968

179


E.2.3. 80W90

nu_vogel [cSt] =

74.344563.015158.280353.105934.3201

dens [kg/cm3] =

0.86250.86000.85880.85730.8495

eta0 [Pa · s] =

64.123454.194250.050445.526629.1536XL =

1

mu_mz_sp =

0.06390.05560.05790.06000.0579

mu_mz_pr =

0.04960.04320.04500.04660.0450

P_VZP [W] =

180


300.9734279.0283348.5100420.7864542.9610

M_DGB [N · mm] =219.2929207.3974216.4569221.4728202.1983

P_VL_DGB [W] =

9.399113.248713.825814.123817.2516

M_TRB11 [N · mm] =

1.0e+03 *

3.69143.56563.62583.69233.6726

M_TRB12 [N · mm] =

1.0e+03 *

2.40862.71932.57762.42352.3349

M_TRB [N · mm] =

1.0e+03 *

6.10006.2849

181


6.20346.11586.0075

P_VL0_TRB [W] =

66.2907101.2474100.185898.8847129.8263

P_VL_TRB [W] =

65.5381100.616299.309297.7438128.4567

TVL0_NRB [N · m] =

0.02060.02410.02280.02140.0195

P_VL0_NRB [W] =

3.98336.93536.58316.17037.4924

TVL1_NRB [N · m] =

0.03350.02390.02870.03350.0335

TVL_NRB [N · m] =

0.0541

182


0.04800.05160.05500.0530

P_VL_NRB [W] =

10.464313.835414.860915.811720.3890

P_VD_IN [W] =

9.546514.225114.224714.201118.9997

P_VD_OUT [W] =

22.740433.892733.888633.835045.2676

P_VD [W] =

32.286848.117848.113348.036164.2674


65.4960119.6510123.9000111.977044.7040

Lambda_SP =

183


0.24080.28880.26590.24220.2071

Lambda_PR =

0.36780.44100.40620.36990.3163

184

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