Elastohydrodynamic Lubrication of - DiVA portalltu.diva-portal.org/smash/get/diva2:999453/FULLTEXT01.pdf · 2016. 9. 30. · Elastohydrodynamic Lubrication of Cam and Roller Follower

LICENTIATE T H E S I S

Department of Engineering Sciences and MathematicsDivision of Machine Elements

Elastohydrodynamic Lubrication of Cam and Roller Follower

Mohammad Shirzadegan

ISSN 1402-1757ISBN 978-91-7583-421-4 (print)ISBN 978-91-7583-422-1 (pdf)

Luleå University of Technology 2015

Moham

mad Shirzadegan E

lastohydrodynamic Lubrication of C

am and R

oller Follower

Elastohydrodynamic Lubrication of Cam

and Roller Follower

Mohammad Shirzadegan

Luleå University of TechnologyDepartment of Engineering Sciences and Mathematics

Division of Machine Elements

ISSN 1402-1757 ISBN 978-91-7583-421-4 (print)ISBN 978-91-7583-422-1 (pdf)

Luleå 2015

www.ltu.se


Copyright c© Mohammad Shirzadegan (2015).

This document is freely available at

www.ltu.se

This document may be freely distributed in its original form including the current author’sname. None of the content may be changed or excluded without permissions from theauthor. Printed by Luleå University of Technology, Graphic Production 2015

ISSN: 0000000ISBN: 000000000 (print)ISBN: 0000000000 (pdf)Luleå 2015www.ltu.se

This document was typeset in LATEX


Copyright c© Mohammad Shirzadegan (2015).

This document is freely available at

www.ltu.se

This document may be freely distributed in its original form including the current author’sname. None of the content may be changed or excluded without permissions from theauthor. Printed by Luleå University of Technology, Graphic Production 2015

ISSN: 0000000ISBN: 000000000 (print)ISBN: 0000000000 (pdf)Luleå 2015www.ltu.se

This document was typeset in LATEX

Preface

This work has been carried out at Luleå University of Technology at the Division ofMachine elements. I would like to express my special appreciation and thanks to mysupervisors Professor Roland Larsson and Associate Professor Andreas Almqvist for theirsupport, patient guidance and encouragement during the project. I have been very luckyto have supervisors who cared so much about my work.

I would like to thank Scania, Bosch and Bosch Rexroth Mellansel (Hägglunds Drives)for funding this work. Without their contribution, this work would have never been ac-complished. I also want to thank my industry supervisors, Dr. Hubert Herbst and Dr.Andrew Spencer at Scania, Dr. Johannes Muellers at Bosch and also Patrik Isaksson atBosch Rexroth Mellansel (Hägglunds Drives) for their valuable inputs and discussions.

I would also like to thank Dr. Marcus Björling at Luleå University of Technologyfor his collaboration which led to Paper B. I am thankful for his aspiring guidance andfriendly advice during the project work. Furthermore, I would also like to express mygratitude to all my friends and colleagues at Luleå University of Technology for providingan enjoyable place to work.

I am also grateful to Dr. Wassim Habchi for providing the research visit at LebaneseAmerican University, Byblos, and supporting my research while I was there.

A special thanks to my family, Amir, Jamileh, Shahin, Marjan and Rezvaneh for al-ways supporting me.

At the end I would like express appreciation to my beloved fiancée Anneli Mastanehwho I spent sleepless nights with and who always was my support in the moments whenthere was no one to answer my queries.

Mohammad Shirzadegan, Luleå, October 2015

i

AbstractModelling and simulation of friction is a research issue that still requires an extensiveamount of input from the scientific community. In a lubricated system, the dissipationof energy is connected to the direct contact between the surfaces, or more precisely thetribofilms, as well as of the shearing of the lubricant film. Elastohydrodynamic lubrica-tion (EHL) is a lubrication regime which is characteristic for contacts found in machinecomponents such as in roller bearings, gears and cam mechanisms. These contacts havein common that they carry load on a very small/concentrated area and exhibit elastic de-formations that are much greater than the thickness of the hydrodynamically formed film.There are a vast number of parameters that affect the friction in EHL contacts and it isa challenge to include other than the most basic ones in the model. The most advancedand sophisticated models are very complex with millions of degrees of freedom and are,therefore, not yet feasible to conduct parametric studies with. The extreme conditionsassociated with EHL, i.e., nm thin films, with phase transition from liquid to solid, GPapressure, temperature increase with considerable implications on lubricant flow and sur-face chemistry, etc., makes it even more difficult to model these systems. The shape of thecontacting parts further complicates modeling of EHL. More precisely, an EHL contactcan geometrically be either of a line, circular, elliptic or truncated contact type. Since theline contact appears between two cylindrical shaped bodies of infinite length, it permitsa 2D-model for the flow and there are analytical solutions, in the most elementary cases.The circular and the elliptic contacts more are complicated. The case when the surfacesare fully separated by the lubricant film has, however, been addressed by many research-ers, who also have presented numerical predictions validated by experimental data. Thefinite line contact appears to be the most challenging type, but it is also the only physic-ally reasonable model for EHL contacts where the edge effects cannot be neglected. Inthis work, both steady state and time dependent, fully deterministic models are utilizedand further developed to enable the study edge effects under variable operating conditionsin cam and roller follower systems. The numerical investigations were specified so thatgeneric knowledge about friction in these systems would be generated and also to providevalidation data for the development of a semi-analytical, low degree freedom model, forrapid estimation of friction. The main objective was to design such a low degree of free-dom model so that it can be employed in a multibody dynamic model, requiring frictionestimation in milliseconds. The semi-analytical low degree of freedom model developedin this work, takes thermal effects into account and is built on an advanced and well-characterized rheological model, including lubricant shear thinning, in order to estimatethe viscosity and volume of the lubricant. The model was utilized to perform friction pre-diction covering a range of operating conditions, which were also run in an experimentalinvestigation using a ball-on-disk test device. The results turned out to compare well,suggesting that it constitutes a suitable foundation for further developments.

iii

Contents

I Comprehensive Summary 5

1 Introduction 71.1 Hydrodynamic and Elastohydrodynamic lubrication . . . . . . . . . . . . 81.2 Lubrication regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Objective of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 EHL Theory 132.1 Reynolds Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Film Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Elastic Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Force Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5 Lubricant Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Rheology of lubricants 233.1 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1 Dowson & Higginson . . . . . . . . . . . . . . . . . . . . . . . 233.1.2 Tait equation of state . . . . . . . . . . . . . . . . . . . . . . . . 233.1.3 Murnaghan equation of state . . . . . . . . . . . . . . . . . . . . 24

3.2 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.1 Famous exponential expression . . . . . . . . . . . . . . . . . . 243.2.2 Roelands equation . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.3 Modified WLF viscosity equation . . . . . . . . . . . . . . . . . 253.2.4 Vogel-like viscosity equation . . . . . . . . . . . . . . . . . . . . 253.2.5 Carreau equation . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.6 Ree-Eyring expression . . . . . . . . . . . . . . . . . . . . . . . 26

4 Low degree of freedom approach 27

5 Results 295.1 Fully deterministic numerical simulation of cam and roller follower (Pa-

per A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.1.1 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . 305.1.2 Steady state and time dependent results for finite roller . . . . . . 31

5.2 Low degree of freedom approach for prediction of friction in elastohydro-dynamically lubricated contacts (Paper B) . . . . . . . . . . . . . . . . . 375.2.1 Model inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

v

5.2.2 Friction prediction . . . . . . . . . . . . . . . . . . . . . . . . . 38

6 Conclusions 45

7 Future work 47

II Appended Papers 49

A Fully deterministic numerical simulation of cam and roller follower 51A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54A.2 Elastohydrodynamic governing equations . . . . . . . . . . . . . . . . . 55

A.2.1 Lubricant properties . . . . . . . . . . . . . . . . . . . . . . . . 56A.2.2 Film thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . 56A.2.3 Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58A.2.4 Cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

A.3 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59A.3.1 Fully deterministic simulations . . . . . . . . . . . . . . . . . . . 60

A.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61A.4.1 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . 62A.4.2 Steady sate simulations . . . . . . . . . . . . . . . . . . . . . . . 64A.4.3 Time dependent simulations . . . . . . . . . . . . . . . . . . . . 66

A.5 In conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

B Low degree of freedom approach for prediction of friction in elastohydro-dynamically lubricated contacts 73B.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76B.2 THE MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

B.2.1 Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78B.2.2 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80B.2.3 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

B.3 FRICTION MEASUREMENT . . . . . . . . . . . . . . . . . . . . . . . 82B.4 FRICTION PREDICTION . . . . . . . . . . . . . . . . . . . . . . . . . 82B.5 RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . 84B.6 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

References 95

Appended Papers

[A] M. Shirzadegan, A. Almqvist and R. LarssonFully deterministic numerical simulation of cam and roller follower.

To be Submitted.

[B] M. Shirzadegan, M. Björling, A. Almqvist and R. LarssonLow degree of freedom approach for prediction of friction in elastohydro-dynamically lubricated contacts.

Accepted for publication, Tribology International.

vii

Nomenclature

α f ilm Equivalent pressure-viscosity coefficient [GPa −1]

βk Temperature coefficient of k0 [K−1]

∆ Dimensionless elastic deformation [-]

δ Elastic deformation [m]

γ Shear rate [s−1]

η Apparent viscosity [Pa· s]

η Viscosity of lubricant [Pa· s]

Λ Limiting stress pressure coefficient [-]

λR Relaxation time at TR and ambient pressure [s]

µ Limiting low-shear viscosity [Pa· s]

µg Viscosity value at glass transition temperature [Pa· s]

µR Low shear viscosity at TR and ambient pressure [Pa· s]

µr Viscosity value at reference state [Pa· s]

µinf Viscosity extrapolated to infinite temperature [Pa· s]

νs,eq Equivalent possion ratio [-]

φ Dimensionless viscosity scaling parameter [-]

φinf Viscosity scaling parameter for unbounded viscosity [-]

Ue Dimensionless entrainment velocity [-]

ℜ Surface feature [-]

ρ Density of lubricant [Kg/m3]

τ Shear stress [Pa]

η Dimensionless viscosity of lubricant [-]

1

ρ Dimensionless density of lubricant [-]

F Dimensionless load [-]

Ue Dimensionless entrainment velocity [-]

℘ Penalty term [-]

a Hertzian contact radius [m]

A1,A2 WLF parameters [-]

av Thermal expansivity ratio [K−1 ]

B1,B2 WLF parameters [-]

B f Fragility parameter in the new viscosity equation [-]

C1,C2 WLF parameters [-]

C f Coefficient of friction [-]

E ′ Equivalent Young’s modulus of elasticity [Pa]

Es,eq Equivalent elastic modules [-]

F Load [N]

G Dimensionless material parameter [-]

g Thermodynamic interaction parameter [-]

G(X ,Y ) Dimensionless Axial shape of roller [-]

g(x,y) Geometry profile [m]

H Dimensionless film thickness [-]

h Film thickness [m]

H0 Dimensionless mutual approach of two bodies [-]

k′0 Pressure rate of change of isothermal bulk modulus at p = 0 [-]

k0 Isothermal bulk modulus at p = 0 [Pa]

Kl Thermal conductivity of lubricant [W/mk]

k00 k0 at zero absolute temperature [Pa]

L Length of roller [m]

n Power law exponent [-]

P Dimensionless pressure [-]

p Pressure [Pa]

Ph Hertzian contact pressure [Pa]

R Reduced radius of curvature in line contact [m]

Ri Dimensionless radii of curvature [-]

Rx Reduced radius of curvature in x-direction [m]

Ry Reduced radius of curvature in y-direction [m]

T Dimensionless Time [-]

T Temperature [K]

t Time [s]

Tg Glass transition temperature [K]

U Surface velocity [m/s]

u Velocity component in x-direction [m/s]

u1 Velocity of roller [m/s]

u2 Velocity of cam [m/s]

UD Non-dimensional displacement in X-direction [-]

um Entrainment velocity [m/s]

V Volume [m3]

v Velocity component in y-direction [m/s]

V0 Volume at p=0 [m3]

VD Non-dimensional displacement in Y-direction [-]

VR Volume at reference state, TR, p=0 [m3]

W Dimensionless load parameter [-]

w Velocity component in z-direction [m/s]

WD Non-dimensional elastic deformation [-]

X ,Y,Z Dimensionless space coordinate [-]

x,y,z Space coordinate [m]

z Surface position [m]

EHL Elastohydrodynamic Lubrication

SRR Slide-to-roll ratio, SRR = (UA−UB)2 [-]

Part I

Comprehensive Summary

5

Chapter 1

Introduction

Obtaining a deep understanding of elastohydrodynamic lubrication (EHL) in machinecomponents is a significant issue in tribology and engineering. Elastohydrodynamic lub-rication is a lubrication regime where the hydrodynamically generated pressure inside theextremely thin lubricant film is large enough to generate substantial elastic deformation ofthe mating surfaces and a viscosity increase of several orders of magnitude. This lubric-ation regime is typical for non-conformal contacts, e.g. cam mechanisms, roller bearingsand gears. Cam and roller follower systems are frequently used in internal combustionengines (ICEs) to control the flow of the air and fuel intake and the exhaust gases. Amultibody dynamic simulation of ICE efficiency, requires many thousands of time-stepsto cover a complete engine cycle. For example, this implies that less than a millisecondof computation time can be spent for the simulation of the friction in the EHL contactbetween the cam and the roller follower in the cam mechanism controlling the fuel intake.This highlights the need for a method for EHL friction prediction that requires much lesscomputational time and resources than a fully deterministic one. However, developing ofmodels for rapid estimation, that obviously cannot have the accuracy of a fully determin-istic model, cannot be realized without having access to and being in full command overa fully deterministic model. The present work, aiming to present a semi-analytical lowdegree of freedom (LDOF) model for rapid estimation of friction in EHL contacts, havetherefore also resulted in the development and implementation of a fully deterministicmultiphysics model. The fully deterministic model considers the time dependent natureof the EHL contact between the cam and roller follower, with varying contact geometry,load, and relative speed of the contacting surfaces. It includes fluid structure interaction,in order to model the elastic deformation of the cam and roller follower caused by the im-mense hydrodynamic pressure that builds up under typical conditions associated with thisapplication. It also includes a facility for non-Newtonian lubricant behavior (rheology)and the possibility of simulating the EHL contact resembling cam and roller followersystems with rollers of finite width.

The following chapters of this thesis being with, a short background to EHL togetherwith a description of the state-of-the-art of the field. In chapters 2 and 3, the theoret-ical background for the fully deterministic model is presented. Chapter 2 presents thefull system of equations, which describe the EHL contact, except for the equations as-sociated with the rheology of the lubricant which have been given special attention inChapter 3. Chapter 4 gives a short summary of the semi-analytical low degree of freedom

7

8 Chapter 1. Introduction

(LDOF) approach developed in this work. In Chapter 5, the fully deterministic modelis first validated by comparison with existing results and then steady state and time de-pendent simulations of cam and roller follower systems with finite rollers are presented.The chapter then presents friction predictions using the LDOF approach. Chapter 6 listsconclusions. Chapter 7 describes possible future work. The appendices list the paperswhich this thesis is based upon, i.e., Paper A- "Fully deterministic numerical simulationof cam and roller follower" and Paper B- "Low degree of freedom approach for predictionof friction in elastohydrodynamically lubricated contacts".

Figure 1.1: Cam and roller: a typical machine component operating in the elastohydro-dynamic lubrication.

1.1 Hydrodynamic and Elastohydrodynamic lubrication

Hydrodynamic lubrication is normally found in conformal contacts, e.g. in journal andthrust bearings. The contact pressure is relatively low, usually less than 5 MPa, and theelastic deformation of the contacting bodies is small compared to the lubricant film thick-ness. Elastohydrodynamic lubrication (EHL) is the type of hydrodynamic lubricationfound for example in cam mechanism, gears and roller bearings. The contact betweenthe lubricated surfaces is in these situations usually non-conformal (concentrated) and theload is distributed over a very small area, typically a few mm2 in size. The maximumpressure in such EHL contacts (between two metal surfaces), is typically 0.5 to 4 GPa andthe minimum film thickness is commonly a few tens of nanometers thick. Hydrodynamicpressure within this range, is large enough to generate substantial elastic deformation ofthe mating surfaces. In other words, the conditions are severe and the fluid-structure inter-action (FSI) is strong. Moreover, in this range of hydrodynamic pressures, the viscositybecomes several orders of magnitude larger than under atmospheric conditions. Figure1.2 shows where typical conformal and non-conformal contacts can be found, by display-

1.2. Lubrication regimes 9

ing the schematics of a plain journal bearing and of a cam and roller follower system.

Figure 1.2: Schematics of a plain journal bearing (left) and a cam and roller followersystem (right).

1.2 Lubrication regimesElastohydrodynamic lubrication is considered as a fully lubricated regime. The film isoften thick enough to avoid contact between the mating surfaces. This means that ideal-istically there is no wear occurring on the surface contacts and the friction is determinedby shearing of the lubricant. In this case the friction coefficient is rather low. On theother hand, when the machine components operate under more severe conditions, e.g.high load or low speed, then the effects of the asperities should be taken into consider-ation. The lubrication regimes in elastohydrodynamic lubrication are divided into threeregimes, namely Boundary lubrication, Mixed lubrication, Full film lubrication.

1. Boundary lubrication where the lubricant does not have any role in the carryingload. The load is precisely carried by the asperities. In this regime, wear andfriction are not controlled by the bulk properties of flow film.

2. Mixed lubrication where the contact load is carried partially by the lubricant filmand the asperities interaction. It depends on operating conditions that how muchload is carried by the film and the asperities.

3. Full film lubrication where the load is almost carried by the flow film and the surfaceare totally separated by hydrodynamic actions. The friction is determined by liquidproperties and shearing of the lubricant.

The regimes of lubrication are usually shown by the Stribeck curve, see [66]. Figure1.3 shows a general Stribeck curve where the x-axis represents the Hersey number (H =ηw/p), where η, w and p correspond to absolute viscosity, rotational speed and pressurerespectively. The y-axis represents the friction coefficient.


Boundary Mixed

Hydrodynamic

Elastohydrodynamic

𝜂𝑤/𝑝

𝑓

Boundary filmBulk lubricant

Figure 1.3: A typical Stribeck curve showing different lubrication regimes.

1.3 Literature reviewBeauchamp Tower [67] was the first to observe a pressure variation in the lubricant film.This was investigated in an experimental work on a railroad bearing back in the 1880s.Later on, in the 1886 the theory of the pressure build up in a narrow gap was explainedby Osborn Reynolds [60]. He introduced a second order nonlinear differential equationwhich was extracted from the Navier-Stokes and the continuity equations. This equationhas been the foundation of lubrication analysis since then and is well known as "ReynoldsEquation". Martin [49] applied the Reynolds equation to a gear teeth contact and obtainedvery small values for the film thickness. Several research efforts were made to predictlubrication performance in the elastohydrodynamic lubrication. Elastic deformation andpressure-viscosity effects were taken into account in the Grubin and Vinogradova [21]study. They assumed that the elastic deformation in the highly loaded contact is the sameas that in the equivalent dry contact and the hydrodynamic pressure reaches infinity atthe boundary of the Hertzian contact zone. The pressure spike was first observed byPetrusevich [58] in 1951. A limited number of solutions for the EHL line contact problemwas presented and a high pressure spike at the outlet right before the film constriction wasshown in his work.

In 1959, Dowson and Higginson, which in [16] introduced the inverse solution forEHL line contacts, in which the pressure is assumed to be an independent variable andthe film thickness is considered as the dependent variable for which the Reynolds equa-tion is being solved for. Later on, Evans and Sindle [18] extended the inverse solutionto point contacts problem. Hamrock and Dowson [29] performed a series of EHL nu-merical analysis for elliptical contacts and covered a wide range of velocity, load andmaterial properties variations. Their regression analysis to numerical results is still themost popular and frequently used film thickness equation in the EHL community.

The Multigrid method was first implemented in the field of computational fluid dy-namic, see e.g Briggs et al. [13] and Brandt [12]. Lubrecht et al. [47], followed byVenner [68] implemented this numerical methodology in the elastohydrodynamic lubric-ation area. In this method, the standard iterative solver, e.g Gauss-Seidel and Jacobi, is

1.3. Literature review 11

used where its high frequencies error removes in the first few iterations. However, for thelow frequencies error a distributive relaxation scheme should be implemented to treat thelow frequencies errors in the same way as the high frequency one by reducing the griddensity. This technique has become popular among many researchers, see e.g. Larsson[42], Chevalier et al. [14] and Wijnant et al. [73].

The system of equations for the fully coupled EHL problem, was discretized by meansof the Galerkin least square finite element method, in order to stabilize the solutionprocedure, and the determining system was solved using the iterative Newton-Raphsonmethod.

Solving the EHL problem using finite element together with a Newton-Raphson iter-ative algorithm was introduced by Rohde and Oh in [62] firstly and this was followed byOkamura [54]. However, Results were presented only for the cases with low loaded in theEHL line contacts problem. Later on, Houpert and Hamrock [32] improved Okamura’swork and the solution covered the highly loaded contact up to 4 GPa Hertzian pressure.This method has the advantage of using a non-uniform mesh structure at the solution do-main. To start the iterative process, a proper initial guess is required. If it does converge,only few iterations are in practice needed to obtain the solution. All the unknown values,pressure and film thickness, are updated simultaneously. In each iteration the jacobianmatrices of the solution domain are inverted which results in a huge computation stor-age space. Therefore, at that time this method was not developed for the point contactcases. Habchi et al. [26] recently reported a comprehensive EHL analysis of line andpoint contacts in which the fully coupled elastohydrodynamic problem was solved usinga finite element discretization of the governing equations. In his work, the deformationwas obtained by using the classical linear elasticity equation, with the pressure from theReynolds equation being used as a boundary condition. In this approach, the degree offreedom is reduced by the use of a non-regular non-structure meshing system. A sta-bilization technique was implemented to deal with the undesired oscillation in the EHLsolution.

In the mentioned works, the contact geometries have been simplified to either a pointcontact or an infinite line contact. In practice, the machine components, e.g. cam mech-anisms, roller bearings and gears, have finite length and are slightly profiled towards theiredges to reduce the stress concentration. In 1963, experimental investigation of two testrigs, Ball-on-disk and roller-on-disk, were compared by Gohar and Cameron [20]. It wasreported that the minimum film thickness occurred at the sides of the contacts in roller-on-disk device, while in the ball-on-disk test the minimum film thickness happened at theboundary the Hertzian contact zone. This finding was followed by Wymer and Cameron[75]. More recently, several researchers have conducted numerical studies based on thesefindings. Mostofi and Gohar [51] calculated the EHL pressure distribution and film thick-ness for a profiled roller under low and moderate load conditions. Kuroda and Arai [39]simulated the contact between two rollers of finite width using a combination of the finitedifference and Newton-Raphson methods. However, their analysis was limited to a fewcases. Park and Kim [56] also conducted a numerical study on the EHL finite line con-tact. A good agreement between their obtained results and experimental investigation inRef. [20] was reported. Liu and Yang [44] solved the Reynolds and energy equation, bymeans of multigrid technique, simultaneously in order to analyze thermal effects in theEHL finite line contacts. It was reported that the magnitude of the pressure spike and film


thickness were relatively low in comparison to the isothermal case. Cam and flat followerwas modeled by Kushwaha and Rahnejat [40]. Their study accounts for transient effectse.g. variable load and velocity, and the minimum film thickness estimated over an en-gine cycle. Recently, Najjari and Guilbault [52] have carried out thermal finite line EHLsolution by means of multigrid method. They surveyed a range of rollers with profiledshape at the edges, for e.g logarithmic profiles, chamfer corners, and rounded edges andthe influence of profile shape on the pressure and film thickness were presented.

1.4 Objective of this thesisThe main objective of this thesis was to deliver a rapid and reliable model for frictionestimation in cam mechanisms. This means that this work has been concerning modelingand simulation of friction in the elastohydrodynamically lubricated contact between thecam and the roller follower.

To achieve the goal, a fully deterministic multiphysics model was developed in orderto study the EHL contact in detail to support the development of a semi-analytical lowdegree of freedom model for rapid estimation of friction.

Chapter 2

EHL Theory

This section describes equations, variables and parameters, which are used in the approachand the assumptions made to facilitate its analysis.

2.1 Reynolds EquationA small layer of lubricant in a narrow gap is considered. Both surfaces, z1(x,y,z) andz2(x,y,z), have a surface velocity of U1 = [u1,v1,w1] and U2 = [u2,v2,w2]. The velo-cities in the flow, based on the Newtonian lubricant assumption, can be obtained fromthe Navier-Stocks equation with proper boundary conditions. The compact form (vectorform) of the Navier-Stocks can be written as:

ρ

(∂U∂t

+U ·∇U)=−∇p+∇ ·

(η

(∇U +(∇U)T

))+∇

(−2η

3∇ ·U

)+ρg, (2.1)

where p represents the pressure, η represents the viscosity, U represents the velocity, ρ

represents the density and g represents the external force. The above equation expandedin the Cartesian coordinates as:

ρ

(∂u∂t

+u∂u∂x

+ v∂u∂y

+w∂u∂z

)=−∂p

∂x+

∂

∂x

(2η

∂u∂x− 2

3η∇ ·u

)+

∂

∂y

(η

(∂u∂y

+∂v∂x

))+

∂

∂z

(η

(∂u∂z

+∂w∂x

))+ρgx,

(2.2)

ρ

(∂v∂t

+u∂u∂x

+ v∂u∂y

+w∂u∂z

)=−∂p

∂y+

∂

∂x

(η

(∂v∂x

+∂v∂x

))+

∂

∂y

(2η

∂v∂y− 2

3η∇ ·u

)+

∂

∂z

(η

(∂v∂z

+∂w∂y

))+ρgy,

(2.3)

ρ

(∂w∂t

+u∂u∂x

+ v∂u∂y

+w∂u∂z

)=−∂p

∂z+

∂

∂x

(η

(∂w∂x

+∂u∂z

))+

∂

∂y

(η

(∂w∂y

+∂v∂z

))+

∂

∂z

(2η

∂w∂z− 2

3η∇ ·u

)+ρgz.

(2.4)

13

14 Chapter 2. EHL Theory

To take a simpler case, it is assumed that the body force and the mass inertia are negligible,the flow is laminar, the viscosity and density are constant across the film thickness. Then,the equations 2.2, 2.3 and 2.4 can be simplified in the Cartesian coordinate as:

−∂p∂x

=

∂

∂x

(43

η∂u∂x− 2

3η

(∂v∂y

+∂w∂z

))+

∂

∂y

(η

(∂u∂y

+∂v∂x

))+

∂

∂z

(η

(∂u∂z

+∂w∂x

)),

(2.5)

−∂p∂y

=

∂

∂x

(η

(∂v∂x

+∂u∂y

))+

∂

∂y

(43

η∂v∂y− 2

3η

(∂u∂x

+∂w∂z

))+

∂

∂z

(η

(∂v∂z

+∂w∂y

)),

(2.6)

−∂p∂z

=

∂

∂x

(η

(∂w∂x

+∂u∂z

))+

∂

∂y

(η

(∂w∂y

+∂v∂z

))+

∂

∂z

(43

η∂w∂z− 2

3η

(∂u∂x

+∂v∂y

)).

(2.7)

The distance between two moving surfaces is relatively narrow therefore all derivativeswith respect to z-direction will be much bigger than their equivalents with respect to xand y-directions. Hence, the above equations reduce to

−∂p∂x

=− ∂

∂z

(η

∂u∂z

), (2.8)

−∂p∂y

=− ∂

∂z

(η

∂v∂z

), (2.9)

−∂p∂y

= 0. (2.10)

The lower surface z1 is chosen as the reference point. It is assumed that there is no slipbetween the two surfaces and the following boundary conditions are implemented forabove equations as:

at z = z1 U =U1

at z = z2 U =U2.

Integration of the equations 2.8, 2.9 and 2.10 twice with the mentioned conditions, leadsto the equation for the velocity u and v in x and y direction, respectively. More precisely

u =1

2η

∂p∂x

(z2− (z2− z1)z− z2

1 + z1 (z2− z1))+

(u2−u1)

(z2− z1)(z− z1)+u1,

v =1

2η

∂p∂y

(z2− (z2− z1)z− z2

1 + z1 (z2− z1))+

(v2− v1)

(z2− z1)(z− z1)+ v1.

(2.11)

2.1. Reynolds Equation 15

Figure 2.1: Velocity profile of the fluid in a narrow gap.

Since z1 selected as the reference point then z1 =0 and 0≤ z≤ h where h = z2− z1. Thevelocity equations is rewritten as:

u =1

2η

∂p∂x

(z2−hz

)+

(u2−u1)

h(z)+u1,

v =1

2η

∂p∂y

(z2−hz

)+

(v2− v1)

h(z)+ v1.

(2.12)

The velocity profile consist of two components. The first is called "pressure term" wherehas a parabolic profile and the second is called "velocity term" where has a linear profile.The sum of these gives the actual velocity profile, see Fig.2.1. Now, the mass continuityequation for the volume V enclosed by the surface S entirely on the compressible flow isgiven by

ddt

∫V

ρ dV =−∫S

ρU ·n ds. (2.13)

Expanding the equation 2.13, the mass continuity expression becomes as:

∂

∂t(ρh)+

∂

∂x

(∫ h

0(ρu) dz

)+

∂

∂x

(∫ h

0(ρv) dz

)= 0. (2.14)

Substituting the velocities expression into the equation 2.14, the Reynolds equation readsas

∂

∂x

(ρh3

η

∂p∂x−ρh

(u1 +u2)

2

)+

∂

∂y

(ρh3

η

∂p∂y−ρh

(v1 + v2)

2

)− ∂(ρh)

∂t= 0, (2.15)

where um = (u1+u2)2 and v1 = v2 = 0. Hence, the Reynolds equation is rearranged as:

∂

∂x

((ρh3

η

∂p∂x

)+

∂

∂y

(ρh3

η

∂p∂y

))= 12um

∂(ρh)∂x

+12ρh∂um

∂x+12

∂(ρh)∂t

. (2.16)

The Reynolds equation describes the pressure distribution throughout the flow film betweenthe relatively narrow gaps. This equation consists of several specific terms. The two termsat the left side of Reynolds equation, Eq. 2.16, are known as "Poiseuille term" whichrepresent induced flows. The right hand side ones are known as "Couette term" whichconsists of the wedge effect, the stretch effect and the squeeze effect, respectively.


Throughout this work, for Paper (A), the Reynolds equation is modified for the rollerwith finite length. The effects of stretch term in Eq. 2.16 is eliminated. For the sake ofsimplicity the following dimensionless parameters are defined and applied in the Reynoldsequation:

X =xa, Y =

yL, Z =

za,

P =p

Ph, H =

hRr

a2 , η =η

ηr,

ρ =ρ

ρr, F =

F(t)Fr

, Ue =um(t)

ur,

Ri =R(t)Rr

, T =tur

a,

(2.17)

where ηr and ρr represents the viscosity and the density at the reference temperature,respectively. a and Ph represent the Hertzian contact radius and the Hertzian pressurerespectively. These parameters are given as:

a =

√8FrRr

πE ′L, Ph =

2Fr

πaL, (2.18)

hence, the dimensionless time dependent Reynolds equation can be represented as:

∂

∂X

(ε

λ

∂P∂X

)+ϕ

2 ∂

∂Y

(ε

λ

∂P∂Y

)= Ue

∂(ρH)

∂X+

∂(ρH)

∂T, (2.19)

where

ε =ρH3

η, λ =

12urηrR3r

Pha3 , ϕ =aL.

2.2 Film ThicknessThe film thickness equation h(x,y), consists of mutual approach of two surfaces h0, geo-metry profile g(x,y) and elastic deformation of the mating surfaces δ(x,y). The contactgeometry can be estimated by parabolas for the undistorted contact. For the line contactcase the film thickness equation is obtained as

h(x) = h0 +x2

2R+δ(x), (2.20)

where R represents the reduced radius of curvature. In this case it is assumed that thecontacting bodies have an infinitely long length and the reduced radius of curvature isobtained as below, see Fig. 2.2.

1R=

1R1

+1

R2. (2.21)

For the point contact problem, the film thickness equation is obtained as

h(x,y) = h0 +x2

2Rx+

y2

2Ry+δ(x,y), (2.22)

2.2. Film Thickness 17

Figure 2.2: Equivalent contact for line contact geometry.

Figure 2.3: Equivalent contact for point contact geometry.


where the parabolically shaped surfaces have two local radii of curvature. The reductionof these types of contacts is depicted in Fig. 2.3, and the reduced radius of curvature in xand y-direction is obtained as

1Rx

=1

R1x+

1R2x

,

1Ry

=1

R1y+

1R2y

.(2.23)

Equation 2.21 is only valid for the infinitely long parabolic shape. In reality, the contact-ing bodies, i.e cylindrical rollers, often design with the rounded, crowned, chamfered orlogarithmic profile at the end. This should be considered in the film thickness expression.Hence, the parabola equation x2

2R should be replaced by the real geometry of the bodies.For example for a crowned roller with a crown radius of Ry, the film thickness equationreads as

Figure 2.4: Schematic view of a roller with crowned geometry.

h(x,y) = h0 +g(x,y)+δ(x,y),

g(x,y) = Rx−√

D2− x2,

D = Rx−Ry +√

R2y− y2.

(2.24)

Schematic view of the crowned roller is depicted in the figure 2.4. Description of theelastic deformation, δ(x,y), is given in the later section.

2.3 Elastic DeformationTo compute the elastic deformation of the mating surfaces, the linear elasticity equationsshould be employed for a three-dimensional elastic deformable cubic body, see Habchi[23]. These equations should be solved with appropriate boundary conditions. The fol-lowing sets of equations summarize the linear momentum equations for the cubic solidbody (under the static equilibrium). More precisely

2.3. Elastic Deformation 19

Figure 2.5: An elemental volume to define momentum equilibrium .

∂σxx

∂x+

∂τxy

∂y+

∂τxz

∂z+ρgx = 0 :x-component,

∂τyx

∂x+

∂σyy

∂y+

∂τyz

∂z+ρgy = 0 :y-component,

∂τzx

∂x+

∂τzy

∂y+

∂σzz

∂z+ρgz = 0 :z-component,

(2.25)

where σ and τ correspond to the stress tensor. ρgx, ρgy and ρgz correspond to body forcesin the x, y and z -directions, respectively. These forces are assumed to be zero and nothermal effects being included. According to the Hooke’s law the constitutive equationsfor a linear elastic isotropic material can be summarized as follow (Hooke’s law relates


the stress tensor (σ) of the cubic solid to the strain tensor (ε))

σxx =E

(1−ν)(1−2ν)[(1−ν)εxx +νεyy +νεzz] ,

σyy =E

(1−ν)(1−2ν)[νεxx +(1−ν)εyy +νεzz] ,

σzz =E

(1−ν)(1−2ν)[νεxx +νεyy +(1−ν)εzz] ,

τxy = τyx =E

1+νγxy,

τxz = τzx =E

1+νγxz,

τzy = τyz =E

1+νγzy.

(2.26)

To obtain the deformation in all directions, the kinematics equation which relates thestrain tensors to the displacement should be used. More precisely

εxx =∂uD

∂x,

εyy =∂vD

∂y,

εzz =∂wD

∂z,

γxy =12

(∂vD

∂x+

∂uD

∂y

),

γxz =12

(∂wD

∂x+

∂uD

∂z

),

γyz =12

(∂wD

∂y+

∂vD

∂z

).

(2.27)

inserting the equations 2.27 into the momentum equilibrium equations 2.25 leads to threesets of equations for the deformation of the contact bodies. To simplify the above equa-tions, Lamé parameters are introduced as

λs =νs,eqEs,eq(

1+νs,eq)(

1−2νs,eq) , µs =

Es,eq

2(1+νs,eq

) . (2.28)

Also, the following dimensionless parameters are needed to rewrite the linear elasticityequations, for the EHL finite line contact, in the dimensionless form.

UD = uDRr/a2, VD = vDRr/a2, WD = wDRr/a2, (2.29)

where UD, VD and WD correspond to the dimensionless deformation in X, Y and Z-direction, respectively. Therefore the system of equations for elastic deformation reads as

2.3. Elastic Deformation 21

follows; for more details see Habchi [24].

∂

∂X

[((λs +2µs)

∂UD

∂X+ϕλs

∂VD

∂Y+λs

∂WD

∂z

)]+ϕ

∂

∂Y

[µs

(ϕ

∂UD

∂Y+

∂VD

∂X

)]+

∂

∂Z

[µs

(∂UD

∂Z+

∂WD

∂X

)]= 0,

∂

∂X

[µs

(ϕ

∂UD

∂Y+

∂VD

∂X

)]+ϕ

∂

∂Y

[(λs

∂UD

∂X+ϕ(λs +2µs)

∂VD

∂Y+λs

∂WD

∂z

)]+

∂

∂Z

[µs

(∂VD

∂Z+ϕ

∂WD

∂Y

)]= 0,

∂

∂X

[µs

(∂UD

∂Z+

∂WD

∂X

)]+ϕ

∂

∂Y

[µs

(∂VD

∂Z+ϕ

∂WD

∂Y

)]+

∂

∂Z

[(λs

∂UD

∂X+ϕλs

∂VD

∂Y+(λs +2µs)

∂WD

∂z

)]= 0,

(2.30)

It is important to specify proper boundary conditions for these partial differential equa-tions. These boundary conditions should be either the displacement or the applied forceon every point of the boundaries. The computational domain (Ω) is shown in figure 2.6.

Figure 2.6: Geometry and boundaries of the contact

σn = σ ·n = σZZ = λs

∂UD∂X +ϕλs

∂VD∂Y +(λs +2µs)

∂WD∂z = −PPhR

a , on Ωc

UD =VD =WD, on ∂Ωb

VD = 0, on ∂Ωc

σn = 0, elsewhere

(2.31)


2.4 Force BalanceThe external load carried entirely by the flow film which is the double integration of thepressure distribution over the contact region. For the line and point contact situations theforce balance equations read as ∫

Ωc

p(x)dx = f ,

∫∫Ωc

p(x,y)dxdy = f .(2.32)

2.5 Lubricant PropertiesIn the EHL contacts the lubricant undergoes a rapid increase by the pressure which resultsin an increase in the viscosity. Compressibility of the lubricant also needs careful atten-tion. Hence, some suitable models are needed to approximate these behaviors. Severalempirical and physical base model have been proposed during previous decades. Some ofthem are described in the following chapters.

Chapter 3

Rheology of lubricants

Since the lubricant properties vary with e.g. pressure, temperature and shear rate, a rhe-ological model describing this variation is an essential part of the mathematical model forthe EHL contact. The main equations of particular interest are the viscosity and densityof the lubricants. Under elastohydrodynamic conditions, the viscosity of the lubricant in-creases within the contact by several orders of magnitude. Density of the lubricant varieswith pressure and temperature. Also, shear rate has a strong influence on the viscosityand in reality, most lubricants show a non-Newtonian behavior under EHL conditions.Rheology plays an important role for the friction estimation. The effect of rheology onfriction is described and discussed in Paper B. Here, in this section the lubricant transportproperties which are commonly used in the EHL modeling are described.

3.1 DensityThe variation of the density with the pressure and temperature can be defined by sev-eral equations. Here, the Dowson & Higginson [16], Tait equation of state [50] andMurnaghan equation [5] are explained briefly.

3.1.1 Dowson & HigginsonThis equation is the most frequently used density expression in the EHL community. Thecompressibility of the lubricant is estimated at a given pressure.

ρ(p) = ρ0

[1+

ap1+bp

], (3.1)

this equation is a curve fit to the experimental results. Both a = 0.6× 10−9 and b =1.7×10−9 are believed to be constant for all lubricants.

3.1.2 Tait equation of stateThis expression can estimate the density of a lubricant based on the volume V relative tothe volume at ambient pressure V0.

VV0

= 1− 11− k′0

ln[

1+pk0

(1+ k′0

)], (3.2)

23

24 Chapter 3. Rheology of lubricants

withk0 = k00 exp(−βkT ) , (3.3)

where k0 is the ambient pressure bulk modulus with an exponential relationship to thetemperature. Moreover, the relationship between the temperature and the ambient volumepressure is defined as

V0

VR= 1+av (T −TR) , (3.4)

therefore the density can be obtained from the Tait equation of state (EOS) as

ρ(p) = ρR

(VR

V

). (3.5)

All the required parameters in the Tait EOS can be obtained from experiments for eachlubricant, see e.g. Bair [5].

3.1.3 Murnaghan equation of state

The Murnaghan equation of state also has a quite good accuracy in order to predict thecompressibility of the lubricant. This model reads as

VVR

=1+av (T −TR)(

1+ k′0k0

) 1k′0

, (3.6)

The density of lubricant is defined as

ρ(p,T ) =ρ(R)

1+av (T −TR)×(

1+k′0k0

) 1k′0. (3.7)

3.2 ViscosityThere are plenty of models which describe the variation of viscosity with respect to thepressure and temperature. The following section introduces some of them in detail.

3.2.1 Famous exponential expression

Barus [9] presented a linear expression for the relationship between the pressure and vis-cosity based on experimental observations on the marine glue. Later on, an exponen-tial expression, Eq. 3.8, mistakenly attributed to Barus, has been extensively used in theEHL community. This expression is rarely applicable to viscosity prediction for pressuresabove a few 100 MPa.

µ(p) = µ0 exp(α f ilm p

), (3.8)

where µ0 represents the ambient pressure viscosity and α f ilm represents the pressure vis-cosity coefficient. For more details about α f ilm, see Vergne and Bair [72].

3.2. Viscosity 25

3.2.2 Roelands equationRoelands presented an exponential pressure-viscosity-temperature relationship which ismore accurate for slightly higher pressures, see [61]. Its simplicity has made it popular foruse in the EHL problems. The Roelands’ pressure-viscosity relationship is also obtainedbased on the empirical evidence.

µ(p,T ) = µ0 exp

[(ln(µ0)+9.67)

[−1+

(1+5.1×10−9 p

)z0

(T −138Tr−138

)−si]]

,

z0 =α f ilm

5.1−9 (ln(µ0)+9.67),

si =β(Tr−138)

ln(µ0)+9.67.

(3.9)

3.2.3 Modified WLF viscosity equationAnother alternative to the Eq. 3.9 is a viscosity model that is developed from the freevolume theory of viscous flow. Initially, the free volume equation was proposed byDoolitle [15]. Later on, William, Landel and Ferry (WLF) [19] modified this equationfor better accuracy. Yasutomi and co-workers [76] derived the modified model of WLF.This equation could estimate the viscosity of a lubricant in a broad pressure and temper-ature range with good accuracy.

µ = µg×10−C1(T−Tg(p))×F(p)C2(T−Tg(p))×F(p) ,

Tg(p) = Tg(0)+A1ln(1+A2 p) ,F(p) = 1−B1ln(1+B2 p) ,

(3.10)

where A1, A2, B1, B2, C1 and C2 are characteristic parameters for each lubricant and µgrepresents the viscosity at the glass transition temperature Tg. The function Tg(p) rep-resents the variation of the glass transition temperature with respect to pressure based onexperimental data whereas F(p) represents the variation of the thermal expansion coeffi-cient with pressure, see [23]. This equation is used in Paper A.

3.2.4 Vogel-like viscosity equationThis equation is extracted based on an accurate scaling function from the molecular dy-namic analysis.

µ(p,T ) = µ∞ exp(

BFφ∞

φ−φ∞

), (3.11)

where φ can be obtained from Ashurst-Hoover scaling rule, see Bair and Casselini [7].This expression is written as follows

φ =

(TTR

)(VVR

), (3.12)

the parameter φ∞ represents the viscosity scaling parameter for the unbounded viscosityand BF represents the fragility parameter.

26 Chapter 3. Rheology of lubricants

3.2.5 Carreau equationCarreau equation describes the apparent viscosity η as a function of shear rate γ, pressurep and temperature T .

η(p,T ) = µ(p,T )

[1+(

γλRµ(p,T )

µR

TR

TVVR

)2] n−1

2

, (3.13)

where the parameters λR, µR, TR, VR, V0 and n are lubricant specific parameters that arequantified by experiments, see [4]. This equation is also found in another form in theliterature.

η(p,T ) = µ(p,T )

[1+(

µ(p,T )Gr

γ

)2] n−1

2

, (3.14)

where Gr represents the critical stress.

3.2.6 Ree-Eyring expressionAnother alternative model to express the non-linear behavior of the viscosity is the Ree-Eyring equation. This equation is based on the sinh law and obtained from empiricalevidence.

η =τe

γsinh

(τ

τe

), (3.15)

where τe is the so-called "Eyring stress" which represent the stress level at which a specificlubricant changes from Newtonian to non-Newtonian behavior.

Chapter 4

Low degree of freedom approach

It is of interest to estimate the friction in the EHL contacts via a simplified model whichcontains only a few degrees of freedom. In Paper B, instead of solving the Reynoldsequation, Eq. 2.16, the Hertzian pressure solution for the circular contact was chosen.This equation can be obtained as

p = ph

√1− (r/a)2, (4.1)

wherea = 3

√2FRx/2E ′, ph = 3F/(2πa2).

Thermal effects are important for the friction estimation. Normally, in order to obtain atemperature distribution of a lubricant film the energy equation should be solved with theReynolds equation simultaneously. Here, however, an average temperature is obtainedacross the lubricant film over the whole contact area and used instead. This temperat-ure, Tavg, is obtained from the total value of the average lubricant bulk temperature, Tbavg ,the temperature rise due to the internal heating of the oil, ∆Tl , and the flash temperature,Tf , inside conjunction, see Archard [1] and Stachowiak and Batchelor [64]. More ex-planations about the method are mentioned in Paper B. The central film thickness, hc, isestimated by the formula, presented by Bair in [3], for EHL contacts. That is,

hNc

hc=

[1+0.79

((1+SRR)

(Ueη0)

(hNcG∗)

)(1/(1+0.2SRR))](3.6(1−n)1.7)

, (4.2)

where G∗ describes the critical stress and takes a value of 103−107 Pa. The Newtoniancentral film thickness, hNc, can be obtained from the Hamrock-Dowson formula [30]. Theviscosity and compressibility of the lubricant are obtained from Eq. 3.2.5 and Eq. 3.1.2,respectively. The average shear rate for the lubricant between the two surfaces can beobtained as

γ =UeSRR

hc. (4.3)

Finally, the shear stress in the fluid film is a function of viscosity and shear rate and thefriction force was determined by integrating the shear stress over the whole domain.

τ = ηγ,

Ff =∫ a

0τ 2πr dr.

(4.4)

27

28 Chapter 4. Low degree of freedom approach

Chapter 5

Results

This chapter contains a short summary of the results obtained from numerical simulationusing the fully deterministic EHL model, by means of COMSOL multiphysics software,as well as results from the semi-analytical low degree of freedom approach. Detailedresults and discussions are available in the appended Papers.

5.1 Fully deterministic numerical simulation of cam androller follower (Paper A)

In the work presented in Paper A, the model explained in the both chapters 2 and 3, wasapplied to simulate stationary operation of the EHL contact, assuming fully lubricatedconditions, in a cam and roller follower system with four different roller follower geo-metries, Case 1-4 presented in section 5.1.2 below. The time dependent case with a finitecrowned roller, see Fig. 2.4 for a schematic view of this type of roller, was also simu-lated by means of the fully deterministic model. For the stationary simulations, generalproperties of the lubricant are listed in Table 5.1 and for the time dependent case, with acrowned roller, the radii of curvature, load and velocity input parameters were specifiedby the operating conditions of a cam mechanism and they are depicted in Fig. 5.1. Inan effort to validate the model, two benchmark cases, i.e., Case A - the elliptical EHLproblem studied in [41] and [53] and Case B - the EHL line contact problem studied in[69], were chosen and the predicted results compared with the published work.

Table 5.1: Transport properties of lubricant for the stationary cam and roller analysis inCase 1-4.

WLF ParametersA1(

C) 19.17 C1 16.04A2 (GPa−1) 4.07 C2(

C) 18.18B1 0.23 Tg(0)(C) -73.86B2 (GPa−1) 24.9 µg (Pa·s) 1012

α f ilm (GPa−1) 23.6 ηr (Pa·s) 0.0282

29

30 Chapter 5. Results

Figure 5.1: Input parameters, velocity (red line), external load (black line) and radii ofcurvature (blue line) for the fully deterministic analysis.

Table 5.2: Input parameters used for the transient line contact analysis.

Ph 2 GPa Rx 0.0139 mu1 1.455 m/s u2 0.485 m/sµ0 0.04 Pa·s α 2.2 GPa−1

z 0.68 f 1.59× 106 NE ′ 2.26×1011 Pa ν 0.285

5.1.1 Model validation

The fully deterministic model, was used to simulate an elliptical EHL contact in two dif-ferent cases, Case A and B. Moes dimensionless load and dimensionless lubricant para-meters were specified as MA = 221.56, MB = 1000 and LA = 12.74, LB = 5, respectively.The entrainment velocity was set to Ue = 24.87 m/s and the ratio of reduced radii ofcurvature was D = Rx/Ry = 0.125. The obtained values for the central film thickness andthe minimum film thickness were compared with the results presented in the literature.The results are found in Table 5.3, showing that the film thicknesses predicted by ourmodel deviate from those reported in less than 5% in both cases.

The time dependent model was also validated to enable analysis of cam and rollerfollower systems where the load, geometry and speed input parameters vary over time.In [69], the passage of a surface feature, going from the inlet to the outlet of the contact,was investigated by means of numerical simulations. Figure A.3 shows the dimensionlesspressure and dimensionless film thickness obtained by means of the numerical simulation.These results agree well with the results presented in [69]. Some minor discrepanciesbetween the results presented here and the ones found in the literature, which may be dueto differences in the meshing and time steps was, however, observed.

5.1. Fully deterministic numerical simulation of cam and roller follower (Paper A) 31

Table 5.3: Minimum and central film thicknesses (in micrometers) obtained using ourmodel and those reported by [41] and [53]

Reference Case A Case Bhmin hcen hmin hcen

Nijenbanning [53] 1.257 1.430 0.246 0.285Kweh [41] 1.256 1.341 0.249 0.283Present model 1.294 1.495 0.237 0.287

Figure 5.2: Transient solution in the case of slip when the dent located at Xd = for−0.75,−0.5, 0, 0.5 for locations.

5.1.2 Steady state and time dependent results for finite roller

Results for four test cases are presented here. The first case considered is that of a crownedroller with Ry = 560 mm and Rx = 12.5 mm. The applied load is F = 8000 N, which cor-respond to the Hertzian pressures of Ph = 1.20 GPa. The entrainment velocity was setto Ue = 1 m/s. The dimensionless pressure distributions obtained using these inputs aredepicted in Figure 5.3. The pressure distribution and film thickness along the central linein the Y-direction are plotted in Fig. 5.4 where the predicted minimum film thickness islocated at the edge of the contacts. The second case considered is a roller with a log-arithmic profile, which smooths the pressures distribution at the edges. This is readilyapparent in Fig. 5.5, which shows the 3-D dimensionless pressure distribution. The cent-ral film thickness and pressure along the axial direction are shown in Fig. 5.6. The third


Figure 5.3: Dimensionless pressure distribution for a crowned roller (case 1).

Figure 5.4: Dimensionless pressure distribution and film thickness in the axial direction(left) and zoomed in plots of film thickness (right) for a crowned roller (case 1)

case also involves a roller with a logarithmic shape, but one with a different applicationthat is described by a different equation. The 3-D dimensionless pressure distribution forthis roller is plotted in the Fig. 5.7. Figure 5.8 shows the central film thickness and pres-sure distribution in the axial direction for the third case. The fourth case considered isa roller with a fillet radius at the edges. In this case the computational domain is set to−3≤ X ≤ 1.5 and −1.05≤ Y ≤ 0 and the lubricant properties are α f ilm = 18.05 GPa−1

and ηr = 0.201 Pa·s. For modified WLF viscosity constant parameters see Habchi andIssa [27]. In the test Case 4, the fillet radius, r, is varied form r/L = 25 to r/L = 7.45.The dimensionless pressure distribution and film thickness along the center line in the Y-direction are shown in Fig. 5.9. The maximum pressure and the minimum film thickness


Figure 5.5: Dimensionless pressure distribution for a logarithmic roller (case 2).

Figure 5.6: Dimensionless pressure distribution and film thickness in the axial direction(left) and zoomed in plots of film thickness (right) for a logarithmic roller (case 2).

belong to the roller with the lowest fillet radius. During the numerical simulation the filmthickness has a tendency to become zero or negative and it is difficult to see converge res-ults. Finally, results for the time dependent simulation of a cam and crowned roller overa complete engine cycle, where the radii of curvature, velocity and external load werevaried according to the values presented in Fig. 5.1, are found in Fig. 5.10. The variationin the central and minimum film thickness over the course of a complete engine cycle ispresented in Fig. 5.11.


Figure 5.7: Dimensionless pressure distribution for a logarithmic roller (case 3).

Figure 5.8: Dimensionless pressure distribution and film thickness in the axial direction(left) and zoomed in plots of film thickness (right) for a logarithmic roller (case 3).


Figure 5.9: Dimensionless pressure distribution (left) and film thickness (right) in theaxial direction for a roller with fillet at the edges (case 4).

Figure 5.10: Dimensionless pressure profiles for the time-dependent simulations at dif-ferent time steps for a crowned roller.


Figure 5.11: Dimensionless pressure and film thickness in the axial direction for differenttime steps for a crowned roller.

5.2. Low degree of freedom approach for prediction of friction inelastohydrodynamically lubricated contacts (Paper B) 37

5.2 Low degree of freedom approach for prediction offriction in elastohydrodynamically lubricated contacts(Paper B)

This section contains a short summary of the results obtained from the semi-analyticallow degree of freedom approach. Detailed results and discussions are available in theappended Papers.

5.2.1 Model inputsFor all of the results showed in this chapter, two types of lubricants, namely squalane andShell T9, are used. The transport properties of these two lubricant have been accuratelymeasured and presented in [4]. The transport properties of the squalane and Shell T9 canbe found in Table 5.4 and 5.5, respectively. The thermal conductivity parameters of the

Table 5.4: Transport properties of Squalane

k′0 11.74 av 8.36×10−4K−1

k00 8.658 GPa Λ 0.075βk 6.332×10−3 K−1 g 3.921φ∞ 0.11743 BF 24.50µ∞ 0.9506×10−4 P·s n 0.463G 6×106 Pa a 5

Table 5.5: Transport properties of lubricant Shell T9

k′0 10.545 av 7.734×10−4K−1

k00 9.234 GPa Λ 0.083βk 6.090×10−3 K−1 g 5.0348φ∞ 0.26844 BF 12.898µ∞ 1.489×10−4 P·s n 0.35G 7×106 Pa a 3

squalane and Shell T9 can be found in Table 5.6 and 5.7, respectively. The operating

Table 5.6: Thermal conductivity properties of Squalane

SqualaneA′ -0.115 Bk 0.0 W/mKCk 0.074 W/mK s 4.5

conditions and the material properties of the specimen where used in the simulations arelisted in the Table 5.8 and 5.9.


Table 5.7: Thermal conductivity properties of lubricant Shell T9

Shell T9A′ -0.101 Bk 0.053 W/mKCk 0.026 W/mK s 7.6

Table 5.8: Operating conditions and specimen material properties (Lubricant: Squalane)

Properties ValueInlet temperature 40 CEntrainment speed 1, 1.611, 3.145, 6.144 m/sSRR 0 to 1Load 80, 300 NBall Radius 10.3185 mmSteel Young’s modulus 210 GPaPoisson ratio 0.3Specific heat capacity 475 J/KgKThermal conductivity 46.6 W/mK

Table 5.9: Operating conditions and specimen material properties (Lubricant:Shell T9)

Properties ValueInlet temperature 30 CEntrainment speed 4 m/sSRR 0 to 0.5Load 50, 100, 200 NBall Radius 12.7 mmSteel Young’s modulus 210 GPaPoisson ratio 0.3Specific heat capacity 475 J/KgKThermal conductivity 50 W/mK

5.2.2 Friction prediction

The low degree of freedom model was used for friction prediction of squalane. The resultsare presented as friction coefficient versus slide-to-roll ratio (SRR) and compared with themeasured friction. The results are presented in Figs. 5.12 and 5.13 for the two loadingconditions; 80 N and 300 N. The friction curves are affected by several parameters. Atlow (SRR), where shear stress inside the lubricant is low, the friction variation is linearand the lubricant behavior is nearly Newtonian. At higher SRR, the shear thinning and/ortemperature rise influence the friction and the variation in friction coefficient is no longerlinear. Friction asymptotically reaches a plateau, and the prediction slightly deviate fromthe measured results. Finally, friction coefficient exhibit a decrease with increasing SRR,which indicates that both thermal dissipation and shear-thinning effects determine the


friction values. The variation of the average temperature versus slide-to-roll ratio are

Figure 5.12: Variation in the friction coefficient with slide-to-roll ratio (SRR) for the 80N loading condition. Results are for entrainment velocity 1 m/s, 1.61 m/s, 3.14 m/s and6.14 m/s.

Figure 5.13: Variation in the friction coefficient with slide-to-roll ratio (SRR) for the 300N loading condition. Results are for entrainment velocity 1 m/s, 1.61 m/s, 3.14 m/s and6.14 m/s.

depicted in Fig. 5.14. These results show that an increase in the slide-to-roll ration, theload and the velocity all lead to an increase in temperature.


Figure 5.14: Predicted average temperature for the 80 N and 300 N loading conditions.Results are for entrainment velocity 1 m/s, 1.61 m/s, 3.14 m/s and 6.14 m/s. Note that thehorizontal scale is logarithmic.

The effects of lubricant thermal conductivity on friction, for the F = 300 N, was alsoinvestigated. Three constant values for thermal conductivity, kl , were selected, i.e. 0.16W/mk, 0.36 W/mk and 0.66 W/mk. This investigation showed that the higher the lubricantthermal conductivity kl the higher friction coefficient becomes, see Fig. 5.15.

Figure 5.15: Comparison in the friction coefficient with different thermal conductivityvalues for the 300 N loading condition. Results are for entrainment velocity 3.14 m/s.Note that the horizontal scale is logarithmic.


Furthermore, a comparison between the results obtained using shear thinning filmthickness, Eq. 4.2, and the Hamrock-Dowson expression is depicted in Fig. 5.16. It wasobserved that the friction coefficient predicted by the LDOF model was underestimatedwhen the Hamrock-Dowson formula for the central film thickness was used instead of theformula in Eq. 4.2, especially at high slide to roll ratios.

Figure 5.16: Comparison of the friction coefficient with different film thickness equationsfor the 300 N loading condition. Results are for entrainment velocity 1.61 m/s. Note thatthe horizontal scale is logarithmic.

To study the effect of the limiting shear viscosity, the Vogel-like equation (Eq. B.9)appearing in the Carreau equation (Eq. 3.13) was replaced by four other alternative equa-tions. These four alternative equations are the exponential expression in Eq. 27-A andthree different variations/types of the Roelands equation, i.e., equations 27 B-D. For moredetails see Paper B, Table B.3. The rest of the parameters were kept the same as before.The results obtained for all operating conditions are shown in figures 5.17 and 5.18. Theresults using the temperature dependent type of Roelands equation in Eq. 27 - C, showedthe closest match to the measured values. This suggests that Eq. 27 - C could be usedinstead of the Vogel-like equation, Eq. B.9. However, further analysis using data for otherlubricants should be carried out in order to ascertain which expression is most suitable.

To further examine the reliability of the LDOF approach, the squalane model lubricantwas replaced by the Shell T9. Again the variation in the friction coefficient with the slide-to-roll ratio was assessed. The resulting prediction from the LDOF approach, for threeloading conditions, i.e., 50 N, 100 N and 300 N, is depicted in Fig. 5.19. The predictionsfrom a fully deterministic model presented by Habchi and co-workers in [25], are also de-picted in this figure. In the results presented in [25], the inlet temperature was specified tobe 303 K and the entrainment velocity to 4 m/s. Moreover, they employed the Murnaghanequation of state (Eq. 3.6) and therefore the Tait equation of state (Eq. B.6) used in theLDOF for the previously described analyses was replaced to enable comparison.

Finally, the Carreau expression (Eq. 3.13) was replaced by the Ree-Eyring equa-tion (Eq. 3.15) in order to investigate which one is most suitable to represent the non-


Figure 5.17: Coefficients of friction obtained with different limiting shear viscosity equa-tions for the 80 N loading condition.


Newtonian effects. The Eyring shear stress was set to 5.5 MPa, the load F = 80 N andF = 300 N and the entertainment velocity Ue = 3.14 m/s and Ue = 6.14 m/s. The predic-tions calculated by means of the LDOF approach are depicted together with the frictionmeasured in a ball-on-disc apparatus in Fig. 5.20.





Chapter 6

Conclusions

In this thesis work, the long term objective is to develop a semi-analytical low degree offreedom model for rapid estimation of friction in the EHL contact between a cam and aroller follower, found in e.g. the cam mechanisms controlling the fuel intake in an in-ternal combustion engine. The resulting LDOF approach turned out to give very accuratepredictions of friction in EHL contacts between a ball and a plate. However, to be able totest the accuracy and to give more insight in to the physics of the cam and roller followercontact, a fully deterministic multiphysics model was developed. This highly complexmodel was developed in order to systematically analyze effects associated with lubricantrheology, contact geometry as well as the operating conditions in terms of varying loadsand speed, etc. The development of these models has been progressing in ways making itpossible to subsequently add effects to the LDOF model and then systematically validatethe friction estimation by means of the progressively more sophisticated fully determin-istic model. Below, the main conclusions of this thesis work are listed in an order relatedto the overall progress of the thesis work as follows:

• A fully coupled isothermal numerical model for simulating the EHL finite line con-tacts was developed. Finite element discretization by means of Newton-Raphsonwas used in order to deal with the Reynolds equation and the elastic deformationequation. Undesired oscillations and computational instability were mitigated usingthe Galerkin least squares finite element method.

• In addition, this model was proven feasible for time-dependent analysis of finiteEHL line contacts found in real applications such as the one typically found in acam mechanism. The model was shown to be suitable for investigating the effectsof varying contact geometry, load, and relative speed of the contacting surfaces onthe lubricant film thickness and pressure distribution.

• The proposed low degree of freedom model (LDOF) can be used to predict the fric-tion in an application operating at high slide-to-roll ratios. At higher SRR values,the use of the thermal algorithm was also found to be important for the accurateestimation of friction.

• Shear thinning were found to play a decisive role in friction estimation and byincluding a rheology considering shear thinning the friction predicted by the modelcame closer to the values measured in the ball-on-disc experiment.

45

46 Chapter 6. Conclusions

• The proposed isothermal model predicted the coefficient of friction with an accept-able accuracy for low slide-to-roll ratios (SRR <0.12) suggesting that it can be usedto predict friction in applications operating under low slide-to-roll ratios. Remarkthat it is not recommended to employ the exponential expression 3.8 or Roelandstemperature independent equations together with a limiting shear stress model inthe present LDOF approach, as this could lead to considerable deviations fromexperiments dependent on the operating conditions. In addition, the use of well-characterized lubricant parameter values, based on the real physical behavior ofthe lubricant, render more accurate estimations of friction compared to estimationsbased on the frequently used Eyring model.

Chapter 7

Future work

The long term goal is to develop the semi-analytical low degree of freedom model sothat friction in cam and roller follower systems with finite width rollers can be estimated.As was shown in Section 5.1.2. by using the fully deterministic model to analyze EHLcontacts resembling a cam mechanism, with the cam being wider than the roller follower,the edge effects affected the minimum film thickness and pressure distribution. The casewhen roller follower is wider than the cam, which means that the contact patch will takethe shape of a truncated ellipse is also an extension that deserves to be addressed in thefuture. The challenge in trying to include either of these cases, i.e., wider cam or widerroller, in the LDOF approach is to establish general curve-fit expressions such as theHamrock-Dowson film thickness formula, that also considers the effects associated withthe non-elliptic contact geometries.

The LDOF approach was originally designed for predictions of friction in smoothEHL contacts operating in the full film lubrication regime. Another future work objectiveis therefore to develop the facility of the model considering surface roughness of thecontacting surfaces. An important boundary condition for the inclusion of these facilitiesis again the computation time, which needs to be of the order of microseconds in order tobe feasible to be implemented as a contact element in the multi body dynamic simulationsoftware.

47

48 Chapter 7. Future work

Part II

Appended Papers

49

Paper A

Fully deterministic numericalsimulation of cam and roller follower

51

53

To be Submitted

Fully deterministic numerical simulation of cam and rollerfollower

M. Shirzadegan, A. Almqvist and R. LarssonDivision of Machine Elements, Luleå University of Technology,

Luleå, SE-971 87 Sweden

AbstractThis paper presents a fully deterministic multiphysics model developed for numerical sim-ulation of the elastohydrodynamically lubricated contact between a cam and roller fol-lower, typically found in cam mechanisms controlling the fuel injection in an internalcombustion engine. Special attention was focused in order to enable systematic ana-lyses of effects associated with, roller crowning and edge geometries, lubricant rheologyas well as typical cam mechanism operating conditions, in terms of varying loads, camspeed and geometry. The model is built on the assumption that the contact operates in thefull film EHL regime and the Reynolds equation was adopted as the equation governingthe fluid flow. The elastic deformation was modelled by the full set of classical equationsfor linear elasticity. The system of equations for the fully coupled EHL problem, wasdiscretized by means of the Galerkin least square finite element method, in order to sta-bilize the solution procedure, and the determining system was solved using the iterativeNewton-Raphson method. The model was also validated against some particular modelproblems, found in the literature.

Keywords: Elastohydrodynamic, cam mechanisms, numerical simulation, finite ele-ment method

54 Paper A. Fully deterministic numerical simulation of cam and roller follower

A.1 Introduction

From the moment an engine starts to run, hydrodynamic forces are transferred to thetribological components, causing significant elastic deformations, especially at non- con-formal contacts. Cam mechanisms represent a major class of systems that interact in thisway, i.e., under elastohydrodynamic lubrication (EHL) conditions. The numerical invest-igation of such systems presents several complexities. The discretization of a physicallyrepresentative model for the lubricant behavior and the implementation of an appropri-ate solution procedure are two of them. Another one, is related to the modeling and theinclusion of the elastic deformation of the mating surfaces. Moreover, since interactionbetween the cam and roller follower is a time-dependent process, it is necessary to takethe operating conditions into account, e.g. variable load, velocity and radii of curvature.It has become commonplace to use the Reynolds equation in studies of EHL problem.However, the Reynolds equation is a complex nonlinear partial differential equation, andcare is required to obtain smooth and valid solutions. There are two important issues toconsider. The first one is the elastic deformation of the contact bodies. This is since thatin non- conformal contacts, such as those between a cam and a roller follower, gears, orrolling bearings, the load carrying area is small and the pressure generated becomes highenough to cause significant deformation. The other one is related to the piezo-viscous be-havior of the lubricant that, due to the high pressure exerted on the thin lubricant film, thatdiminishes the influence of the term describing the pressure driven flow in the Reynoldsequation. Mathematically, this means that the equation goes from elliptic to hyperbolicwhich requires completely different methodology when discretized.

Several methods for dealing with these problems have been proposed. For example,Dowson and Higginson, which in [16] introduced the inverse solution for EHL line con-tacts, in which the pressure is assumed to be an independent variable and the film thick-ness is considered as the dependent variable for which the Reynolds equation is beingsolved for. Later, in [18], this method was extended to the point contact problem byEvans and Sindle . Venner and Lubrecht [71], proposed the multi-level method, in whichsophisticated relaxation techniques were used to stabilize the solution procedure. Theelastic deformation was estimated using the multi-level multi-summation, see [46]. Hab-chi et al., see [26], recently presented EHL analysis of line and point contacts in which thefully coupled elastohydrodynamic problem was numerically, by the application of usinga finite element discretization. The deformation was obtained using the classical linearelasticity equations and the coupling between the equations is that the pressure from theReynolds equation is being used as a boundary condition. In their approach, the degreeof freedom is reduced by the use of a non-regular non-structure meshing system.

Cam mechanisms have a wide range of industrial applications. A great variety of rollerdesigns exist because optimizing the roller’s geometry can greatly reduce surface stressesand damages. In 1963, Gohar and Cameron [20] compared experimental results obtainedusing ball-on-disc and roller-on-disc test devices, and they showed that the minimumfilm thickness occurred at the sides of the contact zone in the latter one, whereas in theformer the minimum film thickness occurred at the boundary of the Hertzian contactarea. This finding was subsequently confirmed experimentally by Wymer and Cameron in[75]. After this, researchers have conducted numerical simulations of their experiments.Mostofi and Gohar, [51], calculated the pressure distribution and film thickness for a

A.2. Elastohydrodynamic governing equations 55

profiled roller under low and moderate load conditions. Kuroda and Arai, [39], simulatedthe contact between two rollers of finite width using a combination of the finite differenceand Newton-Raphson methods. However, only a few rather limited cases were examined.In [56], Park and Kim also conducted a numerical study on the systems investigated byGohar and Cameron, obtaining good agreement with the experimental results. Liu andYang, [44], used multi-level techniques to analyze TEHL finite contacts and found thataccounting for thermal effects reduced the predicted film thickness and the magnitudeof the pressure spikes relative to the isothermal case. Kushwaha and Rahnejat, see [40],adopted an approach that accounts for transient effects and used it to model a cam andflat follower over an engine cycle. Recently, Najjari and Guilbault [52], employed multi-level method to study the steady state TEHL contact between several rollers with finitelength. A range of rollers with different crownings, logarithmic profiles, chamfering, androunded edges were numerically simulated and the influence of the edge properties on thepressure and film thickness throughly discussed.

This paper presents models for fully deterministic steady state and time dependentsimulation of finite EHL contacts. The model, which is similar to the one presented in[26], is used to perform numerical simulation of cam and roller follower contacts, overa complete engine cycle with varying cam operating conditions in terms of load, speedand cam geometry. The elastic deformation of the cam and roller are obtained using amodified classical linear elasticity equation. The stabilized Galerkin least squares finiteelement method was implemented to deal with the severe instabilities encountered underconditions involving high loads. The contact geometry considered is the one between aroller with a finite length, exhibiting crowning, fillet radius at the ends and logarithmicprofile shape. The results of the steady state analyses conducted by means of the presentmodel are compared to those obtained using the numerical approach reported by Kweh[41] and Nijenbanning [53] and results from the time dependent case study are comparedto published reference data presented by Venner and Lubrecht [69].

A.2 Elastohydrodynamic governing equations

The pressure distribution inside the lubricant film is obtained by solving the time-dependentReynolds equation. It is assumed that the mating surfaces have unidirectional velocitiesu1 and u2, parallel to the x-axis and that the velocity in the y-direction is zero. It is also as-sumed that the inlet of the contact is fully flooded and that the effect of surface roughnessis small enough to be disregarded. The following set of scaling parameters are definedand applied to transform the Reynolds equation in dimensionless form:

X =xa, Y =

yL, Z =

za,

P =p

Ph, H =

hRr

a2 , η =η

ηr,

ρ =ρ

ρr, F =

F(t)Fr

, Ue =ue(t)

ur,

Ri =R(t)Rr

, T =tur

a,

(A.1)


where ηr and ρr represents the viscosity and density at the reference temperature, respect-ively and a, Ph represents the Hertzian contact radius and Hertzian pressure, respectively.For line contacts, these parameters are defined as follows:

a =

√8FrRr

πE ′L, Ph =

2Fr

πaL. (A.2)

Hence, the dimensionless time dependent Reynolds equation can be represented as:

∂

∂X

(ε

λ

∂P∂X

)+ϕ

2 ∂

∂Y

(ε

λ

∂P∂Y

)= Ue

∂(ρH)

∂X+

∂(ρH)

∂T, (A.3)

where

ε =ρH3

η, λ =

12urηrR3r

Pha3 , ϕ =aL.

A.2.1 Lubricant propertiesThe relationship between density and pressure was taken to be that proposed by Dowsonand Higginson [16]. This expression accounts for the compressibility of the lubricant toa certain extend.

ρ = ρr

[1+

0.6×10−9 p1+1.7×10−9 p

], (A.4)

Under the elastohydrodynamic lubrication regime, The viscosity of the lubricant increasesby several orders of magnitude. To describe the dependence of lubricant viscosity onpressure, the modified WLF equation is employed. According to [76] modified WLFreads as follow:

η = ηg×10−C1(T−Tg(p))×F(p)C2(T−Tg(p))×F(p) ,

Tg(p) = Tg(0)+A1ln(1+A2 p) ,F(p) = 1−B1ln(1+B2 p) ,

(A.5)

where A1, A2, B1, B2, C1 and C2 are characteristic parameters for the lubricant and ηg rep-resents the viscosity at the glass transition temperature Tg. The function Tg(p) representsthe variation of the glass transition temperature with respect to pressure based on experi-mental data while F(p) represents the variation of the thermal expansion coefficient withpressure, see e.g. Habchi et al. [26].

A.2.2 Film thicknessThe key variables in the EHL film thickness equation are usually the mutual separationof the two contact surfaces, h0, geometry of the contact, g, and the elastic deformationof the two surfaces, w. Cylindrical rollers typically have logarithmically profiled edgeswith crowning, chamfering, and corners to reduce the stress concentration and establish asmooth pressure distribution this must also be accounted for in the film thickness equation.In dimensionless form, the film thickness equation reads

H(X ,Y,T ) = H0 (T )+G(X ,Y,T )+W (X ,Y,T ) , (A.6)

A.2. Elastohydrodynamic governing equations 57

Figure A.1: Schematic view of a crowned roller. Roller radius, Rx and crowning radiusRy.

where the mutual separation H0 (T ) is determined by a force balance equation, G(X ,Y,T )correspond to the dimensionless contact geometry and W (X ,Y,T ) correspond to the di-mensionless elastic deformation.

A great diversity of roller shapes exists due to the diverse applications of rollers inmachines. Rollers with axial crowning as shown in Fig. A.1 are quite common; thegeometry of such rollers is described by the following equations:

g(x,y) = Rx−√

d2− x2,

d = Rx−Ry +√

R2y− y2,

(A.7)

where Rx and Ry correspond to the roller and the crowning radius, respectively.

The elastic deformation of the cam and roller follower is estimated by applying athree-dimensional linear elasticity equation to the contacting structure. The following di-mensionless parameters are used to rewrite the linear elasticity equation in dimensionlessform.

U = uRr/a2, V = vRr/a2, W = wRr/a2, (A.8)

here, U , V , W correspond to the dimensionless deformation in the X , Y , Z direction,respectively. The system of equations for elastic deformation is thus as shown below; for


more details see the work of Habchi [24].

∂

∂X

[((λs +2µs)

∂U∂X

+ϕλs∂V∂Y

+λs∂W∂z

)]+ϕ

∂

∂Y

[µs

(ϕ

∂U∂Y

+∂V∂X

)]+

∂

∂Z

[µs

(∂U∂Z

+∂W∂X

)]= 0,

∂

∂X

[µs

(ϕ

∂U∂Y

+∂V∂X

)]+ϕ

∂

∂Y

[(λs

∂U∂X

+ϕ(λs +2µs)∂V∂Y

+λs∂W∂z

)]+

∂

∂Z

[µs

(∂V∂Z

+ϕ∂W∂Y

)]= 0,

∂

∂X

[µs

(∂U∂Z

+∂W∂X

)]+ϕ

∂

∂Y

[µs

(∂V∂Z

+ϕ∂W∂Y

)]+

∂

∂Z

[(λs

∂U∂X

+ϕλs∂V∂Y

+(λs +2µs)∂W∂z

)]= 0,

(A.9)

where λs and µs are the Lamé parameters

λs =νs,eqEs,eq(

1+νs,eq)(

1−2νs,eq) , µs =

Es,eq

2(1+νs,eq

) , (A.10)

in which νs,eq and Es,eq represents the equivalent possion ratio and equivalent elastic mod-ules for the contacting bodies, respectively. The equivalent elastic modulus is used inthese equations for the sake of simplicity. This means that instead of solving the equa-tions for two domains, only one domain is needed to calculate the total elastic deforma-tion of the mating surfaces. The following relation is used to define the equivalent elasticmodulus:

Es,eq =E2

1 E2 (1+ν2)2 +E2

2 E1 (1+ν1)2

(E1 (1+ν2)+E2 (1+ν1))2 × a

PhRr,

νs,eq =E1ν2 (1+ν2)+E2ν1 (1+ν1)

(E1 (1+ν2)+E2 (1+ν1)).

(A.11)

It is important that the above equations are applied to the appropriate domains. Thus,the following boundary conditions are defined for the total computational domain Ω, in-cluding the EHL domain, Ωc, the symmetry boundary, ∂Ωc and the bottom boundary,∂Ωc, as shown in Figure A.2.

σn = σ ·n = σZZ = λs∂U∂X +ϕλs

∂V∂Y +(λs +2µs)

∂W∂z = −PPhRr

a , on Ωc

U =V =W, on ∂Ωb

V = 0, on ∂Ωc

σn = 0, elsewhere.

(A.12)

A.2.3 LoadThe load equilibrium over the entire solution domain is completely satisfied by the filmflow. Consequently, summing the pressures within the contact domain balances the ex-

A.3. Numerical analysis 59

Figure A.2: Computational domain

ternal load: ∫∫Ωc

P(X ,Y,T ) dXdY =πF(t)2Fr

. (A.13)

A.2.4 CavitationTo deal with the free boundary problem (and cavitation in particular) at the outlet of theEHL contact, the penalty method is applied to the Reynolds equation [74]. This methodintroduces an auxiliary term to the Reynolds equation which deals with the negative partof the pressure calculation. It has no influence on the results as long as the obtainedpressure is positive. The compact form of the steady state Reynolds equation ( Eq.A.3) ismodified as follows by the incorporation of the cavitation term:

∇ · (εP)− ∂(ρH)

∂X−℘·P− = 0, (A.14)

where ℘ is an arbitrary large positive number and P− = min(P,0).

A.3 Numerical analysisThe Reynolds equation (A.3), film thickness equation (A.6), linear elasticity equation(A.9) and force balance equation (A.13) must be solved simultaneously to obtain convin-cing pressure distribution and film thickness. A finite element approach was used to solvethese non-linear differential equations, bring in mind the fact that their solution are likelyto exhibit undesired oscillations, particularly under high load conditions, see e.g. [26].This instability was managed by modifying equation (A.3) with the Galerkin least square


finite element method (GLS), presented in [33] and the isotropic diffusion technique [77].The weak formulation of the steady state Reynolds equation is∫

Ωc

ε∇Ph ·∇W hp dΩ+

∫Ωc

(β ·∇Ph−Q

)∇W h

p dΩ+∫Ωc

℘·P−Wp dΩ+

nce

∑e=1

∫Ωce

R(

Ph)

τ

(β ·∇W h

p −∇ ·(

ε∇W hp

))dΩ+

nce

∑e=1

∫Ωce

ρidhe|β|2l′

∇Ph ·∇W hp dΩ = 0,

(A.15)

where the first two terms represent the application of Galerkin formulation to theReynolds equation, the third describe the cavitation, the fourth represent the stabilizingGLS term, and the final terms represents the application of the isotropic diffusion tech-nique. W h

p represents a weighting function, the subscript p denotes the (unknown) pres-sure and Ωce is the set of elements belonging to the calculating domain Ωc. Some of theimportant parameters in equation (A.15) are obtained using the following expressions:

R(P) =−∇ · (ε∇P)+β ·∇P−Q,

τ =heξ(Pe)

2|β|l′,

Pe =|β|he

2εl′, ξ(Pe) = coth(Pe)−

1Pe,

β = H∂ρ

∂P, Q =−ρ

∂H∂X

,

(A.16)

where ρid represents the relative amount of Isotropic diffusion with respect to the originalmethod. he and Pe are respectively the characteristic length and the local Peclet numberof the element e.

The discrete form of the linear elasticity equation is∫Ω

−C+ εs(Uh) · εs(W hU)dΩ+

∫Ωc

−Ph ·nW hU dΩ = 0, (A.17)

where U represents the displacement vector, εs represents the strain tensor and W hU rep-

resents the weighting functions and subscript, U , denotes the displacement unknown.Finally, the discrete form of the force balance equation reads as follow:∫

Ωc

PWH0 dΩ− π

4WH0 = 0, (A.18)

where WH0 represents the weighting functions.

A.3.1 Fully deterministic simulationsThe model described in the preceding section was implemented in a multi-physics mod-eling package and used to simulate the contact between a cam and a roller follower. The

A.4. Results 61

parameters used to define the rheology of the lubricant are listed in Table 1. Both steadystate and time-dependent simulations were conducted. Four profiled rollers with differentgeometries were considered in the steady state simulations, with a constant entrainmentvelocity, load, and radius of curvature. The Hertzian pressure solution was taken as aninitial guess; see equation (A.19).

P(X ,Y ) =

√1−X2 +Y 2, if

√X2 +Y 2 ≤ 1

0, otherwise(A.19)

The initial value for H0 needs to be chosen with care in order to minimize the timerequired to obtain the final solution. Then the Reynolds equation (A.3) is coupled with thelinear elasticity equation (A.9) and the Newton Raphson method is employed to solve thedetermining system of equations. The force balance equation (A.13) is used to determinethe updating of the H0 value until the pressure solution, of the Reynolds equation (A.3)converges.

The time dependent simulations were focused on a cam mechanisms with a crownedroller and simulations for a complete engine cycle, with varying cam geometry, load,and relative speed of the contacting surfaces, were carried out. The Hertzian pressuresolution and the initial mutual separation value were used as described above, and theresults obtained for the steady state case were taken as the first iteration of the time-dependent solution process. To validate the steady-state approach presented herein, it wasused to simulate the same systems as considered in two previously published studies onthe elliptical EHL problem, i.e., see [41] and [53]. In the former one of these two studies,the film thickness was obtained by means of multi-level techniques and in the latter onean inverse approach was employed for this purpose. A range of operating conditions andaspect ratios of the contact ellipse were considered, see Section A.4.1 and for the resultsof comparison. The time-dependent approach was validated against the results presentedin [69], by Venner and Lubrecht, where the passage of a surface feature was studied bymeans of numerical simulations.

Table A.1: Transport properties of lubricant

WLF ParametersA1(

C) 19.17 C1 16.04A2(GPa−1) 4.07 C2(

C) 18.18B1 0.23 T g(0)(C) -73.86B2(GPa−1) 24.9 µg(Pa.s) 1012

α f ilm(GPa−1) 23.6 ηr(Pa.s) 0.0282

A.4 Results

This section contains the summary of the results obtained from numerical simulationusing the fully deterministic EHL model.


A.4.1 Model validationAs mentioned in the preceding section A.3.1, two test cases, Case A and B, that weredescribed in publications by [41] and [53], respectively, were selected to be used forvalidation of the present steady state model. The two test cases are defined by Moes di-mensionless load parameters (MA = 221.56 and MB = 1000). The corresponding Moesdimensionless lubricant parameters are LA = 12.74 and LB = 5, respectively. The entrain-ment velocity was set to Ue = 24.87 m/s and the ratio of reduced radii of curvature wasD = Rx/Ry = 0.125. The central and minimum film thicknesses reported in the two refer-ence studies were predicted by means of the present model and the results are presentedin Table A.2. The film thicknesses predicted in our model, yields results deviate fromthose reported by less than 5%. This shows that the simultaneous solution of the linearelasticity equation and Reynolds equation as used in our model yields results that agreewell with conventional approaches.

Table A.2: Minimum and central film thicknesses (in micrometers) obtained using ourmodel and those reported by Kweh [41] and Nijenbanning [53]

Reference Case A Case Bhmin hcen hmin hcen

Nijenbanning [53] 1.257 1.430 0.246 0.285Kweh [41] 1.256 1.341 0.249 0.283Present model 1.294 1.495 0.237 0.287

To validate our time-dependent approach, a test case also found in the work [69],by Venner and Lubrecht, that analyzed the passage of a surface feature, i.e. an indent-ation, located on the slower moving surface, is considered here. It is assumed that theupper body has a smooth surface, and the contact is fully lubricated. Thermal effects arenegligible throughout the simulation. Table A.3 shows the input parameters for these sim-ulations, and equation (A.20) describes the behavior of the dent. Figure A.3 compares theoutput of our model to the published data [69]. There is a minor discrepancy between thetwo sets of results, which may be due to differences in meshing and time steps, the latterof which were selected after optimization by the multi-physics software package used inthis work.

ℜ(X ,T ) =−0.11−10(X−Xd)2cos(2π(X−Xd)) ,Xd =−2.5+2

U2

UsT. (A.20)

Table A.3: Input parameters used for the transient line contact analysis.

Ph 2(GPa) Rx 0.0139(m)u1 1.455 (m/s) u2 0.485(m/s)µ0 0.04 (Pa.s) α 2.2 GPa−1

z 0.68 f 1.59×106(N)E ′ 2.26×1011(Pa) ν 0.285

A.4. Results 63

Figure A.3: Transient slippage solutions for a dent located Xd = located at four locations−0.75,−0.5,0,0.5.


A.4.2 Steady sate simulationsAs mentioned above, cylindrical bodies are profiled at their edges to reduce stress con-centrations and smoothen the pressure distribution. This section describes the steady-statemodeling of four different rollers with industrial applications, using the lubricant prop-erties presented in Table A.1. The first case considered is that of a crowned roller withRy = 560 mm and Rx = 12.5 mm. The applied load is 8000 N, which correspond to aPh value of 1.20 GPa. The entrainment velocity was specified as Ue = 1 m/s. Due to thesystem’s symmetry, only half of the computational domain is modeled: −3 ≤ X ≤ 1.5and 0 ≤ Y ≤ 0.5. The chosen boundary conditions were P ≥ 0 on Ωc, P = 0 on ∂Ωcand P = ∇P ·nc = 0 on the cavitation boundary. The dimensionless pressure distributionsobtained using these inputs are shown in Fig. A.4. The pressure distribution and filmthickness along the central line in the Y-direction are shown in Fig. A.5. The minimumfilm thickness is located at the edge of the roller.

Figure A.4: Dimensionless pressure distribution for a crowned roller (case 1).

The second case considered is a roller with a logarithmic profile. This is readilyapparent in Fig. A.6, which shows the 3-D dimensionless pressure distribution. For thiscase the values for α f ilm and ηr are chosen from Table A.1. The central film thicknessand pressure along the axial direction are shown in Fig. A.7.

The third case also involves a roller with a logarithmic shape, but one with a differentapplication that is described by a different equation. For this case the values for α f ilmand ηr are set to 18.05 GPa−1 and 0.2021 Pa·s, respectively. For modified WLF viscosityconstant parameters see Habchi and Issa [27]. The 3-D dimensionless pressure distribu-tion for this roller is shown in Fig. A.8. Figure A.9 shows the central film thickness andpressure distribution in the axial direction for the third case.

The fourth case considered is a roller with a fillet radius at the edges. In this casethe computational domain is set to -3 ≤ X ≤ 1.5 and -1.05 ≤ Y ≤ 0 and the lubricant

A.4. Results 65

Figure A.5: Dimensionless pressure distribution and film thickness in the axial direction(left) and zoomed in plots of film thickness (right) for a crowned roller (case 1).

Figure A.6: Dimensionless pressure distribution for a logarithmic roller (case 2).

properties are α f ilm =18.05 GPa−1 and ηr = 0.2021 Pa·s. The fillet radius, r is variedform r/L = 25 to r/L = 7.45. The dimensionless pressure distribution and film thicknessa long the center line in Y-direction are shown in Fig. A.10. The maximum pressureand the minimum film thickness belongs to the roller with the smallest fillet radius. Themaximum pressure and the minimum film thickness belong to the roller with the lowestfillet radius. During the numerical simulation the film thickness has tendency to becomezero or negative and it is difficult to obtain converged results.


Figure A.7: Dimensionless pressure distribution and film thickness in the axial direction(left) and zoomed in plots of film thickness (right) for a logarithmic roller (case 2).

Figure A.8: Dimensionless pressure distribution for a logarithmic roller (case 3).

A.4.3 Time dependent simulations

Time-dependent simulations were performed to study the variation in the pressure dis-tribution of the cam and roller over a complete engine cycle, during which the radii ofcurvature, entrainment velocity, and applied load all varied. The dimensionless inputparameters used in these simulations are presented in Figure A.11. The lubricant proper-ties are α f ilm =18.05 GPa−1 and ηr = 0.2021 Pa·s. During the early stages of the cycle,the load and entrainment velocity increase while the radii of curvature are constant. Thecomponents then enter into a highly loaded state that persists for some time. During thisphase, the entrainment velocity increases linearly while the radii remain constant. To-wards the end of the engine cycle, the load and radii reach their minima while the velocityincreases modestly. At the very end of the cycle, the velocity and radii rise to their peakvalues because the system is operating in a low-load state. Dimensionless pressure distri-

A.4. Results 67

Figure A.9: Dimensionless pressure distribution and film thickness in the axial direction(left) and zoomed in plots of film thickness (right) for a logarithmic roller (case 3).

Figure A.10: Dimensionless pressure distribution (left) and film thickness (right) in theaxial direction for a roller with fillet at the edges (case 4).

butions for this system at different time steps are plotted in Figure A.12, and the variationin the central and minimum film thickness over the course of a complete engine cycle isshown in Figure A.13. Dimensionless central film thickness and minimum film thicknessover an engine cycle are depicted in Fig. A.14.


Figure A.11: Input parameters, external load (black line), velocity (red line), and radii ofcurvature (blue line) for the fully deterministic analysis.

Figure A.12: Dimensionless pressure profiles for the time-dependent simulations at dif-ferent time steps.

A.4. Results 69

Figure A.13: Dimensionless pressure and film thickness in the axial direction for differenttime steps

Figure A.14: Dimensionless central film thickness (dash red line) and minimum filmthickness (solid blue line) over an engine cycle.


A.5. In conclusion 71

A.5 In conclusionA fully coupled isothermal numerical model for simulating the EHL finite line contactswas developed. Finite element discretization by means of Newton-Raphson was used inorder to deal with the Reynolds equation and the elastic deformation equation. Undesiredoscillations and computational instability were mitigated using the Galerkin least squaresfinite element method.

In addition, this model was proven feasible for time-dependent analysis of finite EHLline contacts found in real applications such as the one typically found in a cam mechan-ism. The model was shown to be suitable for investigating the effects of varying contactgeometry, load, and relative speed of the contacting surfaces on the lubricant film thick-ness and pressure distribution.

The contact between a cam and four different finite length roller followers exhibitingdifferent profiles in the axial direction were analyzed by means of the proposed model.In general, the edge effects associated with the profile, reduced stress concentrations andsmoothed the pressure distributions in the film. It was demonstrated that the present modelcan be used effectively to investigate the effects of roller geometry on pressure peaks andfilm thickness.

The new models were validated by comparing their output to representative numericalresults from the literature; good agreement with the reference data was achieved in allcases.

ACKNOWLEDGEMENTSThe authors wish to thank Scania, Bosch and Bosch Rexroth Mellansel (Hägglunds Drives)for financial support. Without their contribution, this work would have never been accom-plished. The authors wish to thank Dr. Wassim Habchi for providing the research visit atLebanese American University in Beirut.

Paper B

Low degree of freedom approach forprediction of friction inelastohydrodynamically lubricatedcontacts

73

75

Accepted for publication, Tribology International

Low degree of freedom approach for prediction of friction inelastohydrodynamically lubricated contacts

M. Shirzadegan, M. Björling, A. Almqvist and R. LarssonDivision of Machine Elements, Luleå University of Technology,

Luleå, SE-971 87 Sweden

AbstractA low degree of freedom, semi-analytical model for rapid estimation of the friction coeffi-cient in elastohydrodynamically lubricated contacts was developed and tested. Its estim-ates are based on the shear rate dependent Carreau equation for the apparent viscosity,together with the hydrodynamic pressure and the temperature of the lubricant. To valid-ate the approach, the model’s predictions were compared to experimental coefficient offriction measurements acquired using a ball-on-disc test device at various applied loads,entrainment velocities, and slide to roll ratios. The model’s predictions were in goodagreement with the experimental results, showing that it is suitable for use in multibodydynamics analyses where rapid computation of elastohydrodynamic friction is required tominimize computing time and resource consumption.

Keywords: Elastohydrodynamic, coefficient of friction, lubricant rheology, low degreeof freedom model

76Paper B. Low degree of freedom approach for prediction of friction in

elastohydrodynamically lubricated contacts

B.1 INTRODUCTION

The continuing development of engineering machines has provided many challenges inthe field of tribology. Many machine elements require lubrication to reduce friction andwear. Consequently, there is great interest in studying the static, dynamic, and vibrationalbehavior of lubricants and components in order to improve their function and efficiency.Friction and the risk of failure due to various contact types have been investigated in arange of applications such as the contacts between the roller and the raceway in rollingelement bearings, between gear teeth in gear transmissions, and between the follower andthe camshaft in cam mechanisms. All of these systems operate in the elastohydrodynamiclubrication (EHL) regime, where the lubricant pressure is high enough to significantlychange the geometry of the lubricated conjunction through elastic deflection of the mat-ing surfaces. EHL has been studied for several decades and many models of varyingcomplexity have been developed to predict variables such as the minimum film thickness,maximum pressure and friction. However, accurate modeling of the lubricant’s rheolo-gical behavior in systems operating under EHL regimes continues to be challenging dueto the thinness of the oil film and the relatively large elastic deformation of the matingsurfaces, see e.g. [48]. Under EHL conditions, the relationship between shear stress andshear rate is not always linear. In many studies, this relationship has been modeled usingthe Eyring sinh law. One early example is the work [37] of Tevaarwerk and Johnson,who used this model to estimate friction in EHL contacts. Evans and Johnson [17] alsoapplied the Eyring sinh law and used it to identify four traction regimes with differentnonlinear behaviors. A more recent example of its use was reported by Jacod et al. [34],who proposed a generalized traction curve for isothermal EHL contact. Although Eyringfriction models remain popular, multiple studies have shown that they are inadequate fordescribing shear thinning behavior, see e.g. [36] and [2].

Recently, Liu et al. [45] implemented a finite difference discretization of the Reynoldsequation, which was coupled with a rheological model that was not based on the Eyringsinh law. Instead, the rheology was described using the free volume model for compress-ibility together with the Carreau expression for the shear thinning response. This approachyielded a traction curve for low sliding velocities and relatively low loads. Subsequently,Habchi and co-workers [28] successfully predicted film thicknesses and pressure distri-butions at higher loads using a finite element model. This approach has also been used byHabchi et al. [25] with a non-Newtonian rheology model, which takes the limiting shearstress into account, based on lubricant transport properties to assess EHL friction. In thesame paper, they defined four traction regimes that may be encountered in EHL contacts(linear, non-linear, plateau and thermoviscous), and linked them to four dimensionlessnumbers. In their approach, the Murnaghan equation of state and a Vogel-like equa-tion were employed to model the lubricant’s rheological behavior. These models make itpossible to perform detailed studies on the effect of shear thinning and piezoviscous para-meters. The output of their model was shown to be in good agreement with empirical datafrom friction measurements, presented in the paper by Björling et al. [10], acquired usinga ball-on-disc machine. Since the lubricant’s properties vary with e.g. pressure, tem-perature and shear rate, a rheological model describing this variation is essential for themathematical modelling of EHL contact. Barus [9] presented a linear expression for therelationship between pressure and viscosity based on experimental observations on mar-

B.1. INTRODUCTION 77

ine glue. Later on, an exponential expression (Eq. 27-A), mistakenly attributed to Barus,has been extensively used in the EHL community. The exponential expression is rarelyaccurate at pressures up to a few 100 MPa. Roelands [61] also presented an exponentialpressure-viscosity relationship that for some fluids provides reasonably good results evenat somewhat higher pressures, but it is generally not applicable when the pressure exceeds1 GPa. The Roelands’ pressure-viscosity relation has become widely used in numericalsimulations of EHL contact, see e.g. [42], [70] and [68].

Friction is determined mainly by the viscosity in the central part of the contact, wherethe pressure is high, and it has been observed that using the exponential expression or Roe-lands’ pressure-viscosity relationship leads to unrealistically high friction coefficients, seee.g. Habchi [23]. The film thickness is, however, governed by the conditions in the inletof the EHL contact, where the pressure is relatively low. Rheological characterization byBair [4] of a variety of lubricants, has highlighted the importance of selecting an appro-priate model when simulating highly loaded EHL contacts. To reliably capture both thebehavior of the viscosity and the density over the range of pressures typically found inEHL contacts, the rheological model must include a realistic representation of the lubric-ant’s compressibility.

Fully deterministic models such as those discussed above, i.e. [25], [38] and [45],can predict the film’s thickness and pressure distributions with good accuracy and in highdetail. However, they are not fast enough to be used to compute the film thickness andpressure over a whole engine cycle in the dynamically loaded EHL contact between thecam and the follower. A fast and reliable tool for estimating friction in EHL contactswould therefore be extremely useful in various industrial applications, particularly if itwere compatible with Multibody Dynamic (MBD) software such as BEAST [65], a threedimensional MBD software package with a contact model for EHL line and point contactsthat is based on numerical solution of the Reynolds equation, see e.g. Venner [68]. In anMBD software package, the contact behavior must be evaluated (in terms of stiffnessand damping) repeatedly, so it is very important to be able to obtain acceptably accurateestimates at the lowest possible computational cost.

This paper presents a method for evaluating contact behavior under EHL conditionsthat requires much less computational time and resources than a fully deterministic modelbut retains sufficient accuracy to be used for determining the friction in bearings, gears,cam mechanisms and other typical tribological components. The methodology developedin this work is relatively similar to that presented by Poll and Wang [59], which is basedon the Maxwell model. The present approach is based on the Tait equation of state (in-stead than the Maxwell model) and on a so-called shifted Carreau model that includes thecompressibility of the lubricant. A simplified temperature distribution algorithm is im-plemented in the present approach, in order to consider the thermal effects. The model’sfeasibility was evaluated by comparing its predictions to the results of friction measure-ments, conducted using a ball-on-disc device at a range of loads, entrainment velocitiesand SRRs. The comparison showed that the model’s predictions were in good agreementwith the experimental data over a wide range of slide-to-roll ratios.



B.2 THE MODELThis section describes the equations, variables and parameters used in our approach andthe assumptions made to facilitate analysis.

B.2.1 Rheology

A low degree of freedom (LDOF) approach for engineering applications was developed toobtain rapid estimates of friction in EHL contacts. This approach is based on the shiftedCarreau equation, which describes the apparent viscosity η as a function of shear rate γ,p and temperature T . More precisely:

η(γ, p,T ) = µ(p,T )

[1+(

γλRµ(p,T )

µR

TR

TVVR

)2] n−1

2

, (B.1)

where the parameters λR, µR, TR, VR, V0 and n are lubricant-specific parameters that werequantified by experiments performed by Bair and co-workers [4], [8]. The parameters forsqualane oil are listed in Table B.2. The temperature dependent dynamic viscosity, µ(p,T )is defined below. Since Couette flow dominates under EHL conditions, it is assumed thatthe average shear rate for a lubricant between two surfaces can be approximated as

γ =UeSRR

hc, (B.2)

where SRR is the slide-to-roll ratio, Ue represents the entrainment velocity, and hc isthe central film thickness. Any influence of surface roughness is neglected here. Thefilm thickness is established at the inlet of the EHL contact and in a fully deterministicapproach it is calculated by simultaneously solving the Reynolds equation and the filmthickness equation, including the elastic deformation and the load balance equation. Inthe present approach, however, the central film thickness (hc) is estimated by the formula,presented by Bair in [3], for EHL contacts:

hNc

hc=

[1+0.79

((1+SRR)

(Ueη0)

(hNcG∗)

)(1/(1+0.2SRR))](3.6(1−n)1.7)

, (B.3)

where G∗ describes the critical stress and takes a value of 103 to 107 Pa. The Newtoniancentral film thickness, hNc, can be obtained from the Hamrock-Dowson formula.

hNc = 2.69Rx(G0.53)(U0.67)(W−0.067)(1−0.61e−0.73k), (B.4)

G = α f ilmE, U =µ0Ue

ERxand W =

FR2

x,

see Hamrock and Dowson [29].The low shear viscosity at the ambient pressure and the viscosity pressure coefficient

are represented by µ0 and α f ilm, respectively. The latter parameter can be obtained by

B.2. THE MODEL 79

measurement or using the expression proposed in [4]. In [22], Gupta et al. presented anempirical thermal factor, Ct to account for the inlet shear heating, i.e.

Ct =1−13.2(Ph/E)L0.42

1+0.213(1+2.23SRR0.83)L0.64 (B.5)

where L≈(βµ0U2

e)/Kl represents the thermal loading factor, β represents the temperature-

viscosity coefficient and Kl represents the thermal conductivity of the lubricant. In [57]Pensado et al. derived β-values for squalane and two pentaerythritol ester lubricants.

The lubricant undergoes a rapid increase in viscosity at high pressure. Consequently, itbecomes very difficult to squeeze the lubricant out from this contact region. A model thataccounts for the compressibility of the lubricant was developed by Bair and co-workers[8]. Their model and the one presented here are both based on the Tait equation of state,which is used to calculate the ratio of the apparent volume, V , to the volume at ambientpressure, V0, i.e.

VV0

= 1− 11− k′0

ln[

1+pk0

(1+ k′0

)], (B.6)

withk0 = k00 exp(−βkT ) (B.7)

where k0 is the ambient pressure bulk modulus, which has an exponential relationshipto the temperature. Moreover, the relationship between temperature and ambient volumepressure is defined as follows

V0

VR= 1+av (T −TR) , (B.8)

where av is the thermal expansivity and TR is the reference temperature. For more detailedinformation about the Tait equation of state, see Millat and Castro [50]. Also, in thefollowing Vogel-like viscosity-temperature relationship

µ(p,T ) = µ∞ exp(

BFφ∞

φ−φ∞

), (B.9)

the dimensionless viscosity scaling parameter, φ, is a function of the volume relative tothe ambient pressure volume according to

φ =

(TTR

)(VVR

), (B.10)

where BF represents the fragility parameter and g is the thermodynamic interaction para-meter. The shear stress in the fluid film is a function of viscosity and shear rate, i.e.

τ = ηγ (B.11)

It is assumed that the shear stress, τ, is restricted by the limiting shear stress, τl , i.e.,

τ =

ηγ τ < τl

τl τ≥ τl.(B.12)

The limiting shear stress is assumed to vary linearly with pressure according to τ = Λp,where Λ is the limiting shear stress-pressure coefficient.



B.2.2 PressureThe Hertzian pressure solution was considered to be a reasonably good approximationof the EHL pressure distribution [31]. The contact region is discretized into a one-dimensional radial grid whose origin lies at the contact center. Expression (B.12) wasevaluated along the radial coordinate and the friction force was determined by integratingthe shear stress over the whole domain, i.e.

Ff =∫ a

0τ 2πr dr (B.13)

The contact radius and maximum Hertzian pressure were obtained using the followingequations

a = 3√

2FRx/2E ′, (B.14)

ph = 3F/(2πa2). (B.15)

B.2.3 TemperatureThe temperature has a substantial influence on the lubricant’s properties in rolling/slidingcontacts as shown in previous section, more specifically in equations (B.6-B.10). In thiswork, instead of solving the energy equation, an average temperature is calculated acrossthe film’s thickness over the whole contact area. This temperature, Tavg, is obtained fromthe total value of the average lubricant bulk temperature Tbavg , the temperature rise due tointernal heating of the oil ∆Tl and the flash temperature (Tf ) between conjunctions. Moreprecisely

Tavg = Tbavg +∆Tl +Tf . (B.16)

This method for calculating the average temperature is mainly based on the moving hot-spot theory, which was introduced by several authors such as Blok [?], Jaeger [35] andArchard [1]. Notably, the latter of these authors proposed a specific solution for circularcontacts and provided several equations for estimating the flash temperature for differentvelocity ranges and contact geometries. Our approach makes several assumptions whencalculating the temperature rise. That is, it is assumed that the heat source is uniform, thatthe thermal properties of the upper and lower surfaces do not change with temperature andthat all of the heat is transferred to the contact surfaces via conduction. Consequently, theeffect of convective cooling by the lubricant is neglected. The average bulk temperaturecan be estimated with adequate precision by the following equation:

Tbavg =TbA +TbB

2, (B.17)

where TbA and TbB correspond to the surface temperatures of the two contacting solids. IfTbA 6= TbB then equation (B.17) should be modified as follows [64]:

Tbavg =TbA +TbB

2+

TbA−TbB

2

(ψ−1ψ+1

), (B.18)

where

ψ =

(UAρAcAKA

UBρBcBKB

)

B.2. THE MODEL 81

is a constant. In this expression, ρ denotes density, c specific heat and K thermal conduct-ivity. The material properties used in the thermal algorithm are listed in Table B.1. Thetemperature rise due to internal heating of the oil (∆Tl) can be calculated as

∆Tl =C f hcF∆U

8πa2Kl(B.19)

where C f represents the coefficient of friction, ∆U represents the sliding velocity, hc rep-resents the central film thickness and Kl represents the thermal conductivity of the oil,see [1]. Equation (B.19) is based on the assumption that viscosity is constant and theheat is uniformly distributed across the film. However, it is well-known that under elast-ohydrodynamic conditions the viscosity varies with the pressure, temperature and shearrate. According to Eq. (B.19), the thermal conductivity of the lubricant has a strong influ-ence on the temperature rise. It was reported by Larsson and Andersson [43] that thermalconductivity varies with the pressure and temperature. Moreover, numerical studies byKumar and Khonsari [38] showed that doubling the Kl value would increase the friction.Equations (B.20) and (B.21), which were proposed by Bair [6], are used to determinethe variation of Kl with pressure and temperature. The average value of Kl is used inEq. (B.19). The constants (A′, q, Ck and s) in Eqs. (B.20) and (B.21) are obtained frommeasurements of the specific lubricant under consideration. The thermal conductivity ofthe oil is modeled as

Kl =Ck(k′)−s, (B.20)

where

k′ =VVR

[1+A′

(TTR

)(VVR

)]. (B.21)

The Peclet number reflects the rate at which the surface temperature of the contact spreadsinto the two contacting surfaces and it is defined as

PecA,B =UA,Ba2 χA,B

(B.22)

where U represent the velocity of the upper and lower surfaces, a represents the contactradius, and χ = K/ρc represents the thermal diffusivity. To estimate the flash temper-ature rise in a point contact, three different scenarios based on the Peclet number wereintroduced in [1]. More precisely

Tf A,B =0.25 FC f ∆U

Ka, PecA,B < 0.1 (B.23)

Tf A,B =(0.215−0.0255 PecA,B) FC f ∆U

Ka, 0.1≤ PecA,B ≤ 5 (B.24)

Tf A,B =0.218 FC f ∆U

Ka(PecA,B)0.5 , 0.1≤ PecA,B ≤ 5 (B.25)

All of the generated heat is partitioned between the two surfaces. Here, it is assumed thatthe flash temperature can be obtained from the following equation

Tf =Tf ATf B

Tf A +Tf B(B.26)

Also, it is assumed that surface A moves faster than B so all of the heat moves to thissurface via conduction. For more general cases, see [55] and [64].



B.3 FRICTION MEASUREMENTFriction measurements were conducted using a ball-on-disc test device-specifically, a We-deven Associates Machine (WAM11) as shown in Fig. B.1. The ball samples were madefrom AISI 52100 bearing steel and were of grade 5 with a 13/16 in. (20.637 mm) outerdiameter and a hardness of about 60 HRC. The discs had a 4 in. (101.6 mm) outer dia-meter, a polished surface and were through hardened to about 60 HRC. The RMS (rootmean square) surface roughness of the balls and disc were 20 nm and 35 nm, respect-ively. Surface topographies were characterized at 10x magnification (with a 1x field ofview lens) using a Wyko NT1100 optical profilometer (Veeco). The experiments wereperformed with squalane oil, a commercially available low molecular weight branchedalkane (2, 6, 10, 15, 19, 23-hexamethyltetracosane) that has been characterized by Bairand co-workers [3, 7]. Its viscosity at ambient pressure and pressure-viscosity coefficientwere determined to be 15 mPa and 18 GPa−1, respectively, at 40 C (see Table B.1). Thelubricant was applied at the center of the disc via an oil dispenser, which circulated theoil across the disc surface. All tests were performed at 40 C, and to ensure temperaturestability, the lubricant was circulated over the ball and disc with the desired entrainmentspeed until a steady state temperature was reached. In each test, the entrainment speed andcontact pressure were held constant while the slide to roll ratio was varied from 0.0002 to1.05. Two loads were used in the test, 80 and 300 N, corresponding to a maximum Hert-zian pressure of 1.25 and 1.94 GPa respectively. The entrainment speed was varied from1.0 to 6.14 m/s. A more detailed description of the test set-up and the experimental pro-cess is given in the paper by Björling et al. [11]. The material properties of the specimensare listed in the Table B.1:

Table B.1: Properties of the steel surfaces used in experiments and simulations.

Material 52100 (AISI)Young modulus (GPa) 210Poisson’s coefficient 0.3Specific heat capacity (J/KgK) 475Thermal conductivity (W/mK) 46.6Thermal diffusivity (m2/s) 13.6×10−6

Density (Kg/m3) 7850

B.4 FRICTION PREDICTIONThe model described in the preceding sections was used to simulate the contact betweenthe ball and the disc in the experiments. The parameters used to characterize the rheologyof the lubricant are those determined experimentally by Bair and co-workers [4] and [8]for squalane oil (see Table B.2). In the simulations, the SRR was varied from 0.0002 to1.05 while keeping the entrainment speed and load constant. The Hertzian pressure solu-tion was considered to be an acceptable approximation of the EHL pressure distribution.The bulk temperature was determined using Eq. (B.17), and the Newtonian and shear

B.4. FRICTION PREDICTION 83

Figure B.1: Photograph of the ball-on-disc test apparatus used to obtain the friction meas-urements.

Figure B.2: A flowchart outlining the approach used to predict friction in our model.

thinning film thicknesses were determined using Eqs. (B.3) and (B.4). The initial guessfor the average temperature is preferably set to a value close to the reference temperature.The values of α f ilm and µ0 for squalane were specified as 18.18 GPa−1 and 0.0157 Pa·srespectively. The friction force was estimated using Eqs. (12) and (13). The averagetemperature, described in Eq. B.16, is a function of friction coefficient, which in turn istemperature dependent. Hence, an iterative process is needed to obtain the friction andthe average temperature. The relative error of less than 10−7 was used as a terminatingcondition and it takes roughly 4 to 12 iterations to get the converged results. The flowchartshown in Fig. B.2 outlines each step of the friction calculation process.



Table B.2: Important physical properties of squalane oil.

k′0 11.74 av 8.36×10−4K−1

k00 8.658 GPa Λ 0.075βk 6.332×10−3 K−1 g 3.921φ∞ 0.11743 BF 24.50µ∞ 0.9506×10−4 P·s n 0.463G 6×106 Pa a 5A′ -0.115 Ck 0.074 W/mKs 4.5

B.5 RESULTS AND DISCUSSION

The friction coefficients predicted using the LDOF model were consistently in reasonablygood agreement with the measured values. Figures B.3 and B.4 plot the coefficient offriction ( f ) against the slide-to-roll ratio (SRR) for loads of 80 N and 300 N, respectively,which correspond to Hertzian pressures of 1.25 GPa and 1.94 GPa, respectively. Each fig-ure shows the measured and predicted friction coefficients obtained at four entrainmentvelocities: 1.0, 1.61, 3.14 and 6.14 m/s. In addition, Figure B.5 plots the variation of thefriction with the SRR (on a logarithmic scale) for two loads, 80 N and 300 N, with anentrainment velocity of 3.14 m/s. The friction curves appear to be linear in their initialsections (i.e. when the shear stress and SRR are low), which implies that the lubricant isbehaving in a Newtonian fashion. The model assumes that elastic effects can be neglected,which may be partly responsible for the small observed discrepancies between the pre-dictions and measurements at low SRR values. At higher SRR values, the shear thinningand/or temperature rise influence the friction and the curve deviates from linearity. Thefriction asymptotes towards a plateau, and the experimental results deviate from the pre-dicted values somewhat more than was the case in the linear region. Finally, as the SRRincreases still further, the friction decreases, reflecting the combined influence of thermaldissipation and shear-thinning effects. The predicted average temperatures for these casesare plotted in Fig B.6. It is clear that the load, entrainment velocity and SRR all havedirect effects on the predicted temperature; as the values of these parameters increase, thetemperature rises.

To study thermal effects on the friction coefficients, the model’s temperature algorithmwas disabled; the results obtained under these conditions are plotted in Figures B.7 andB.8. Figure B.7 shows the results obtained with a low entrainment velocity at low SRRvalues; under these conditions, there is reasonably good agreement between the meas-ured and predicted friction coefficients at low slide-to-roll ratios (SRR), although somedeviation is apparent at higher SRR values under the 300 N loading condition. FigureB.8 shows the results obtained over a wider range of SRR values and with a higher en-trainment velocity of 6.14 m/s. Under these conditions the agreement between the ex-perimental measurements and the predictions made by the model with the temperaturealgorithm disabled becomes quite poor as the SRR increases. This may be due to theassumption of isothermal contact, which necessarily means that potential thermal effectsare ignored and friction response is governed by limiting shear stress or shear thinning ef-

B.5. RESULTS AND DISCUSSION 85

Figure B.3: Variation in the friction coefficient with the slide-to-roll ratio (SRR) for the80 N loading condition. Results are shown for entrainment velocities of 1, 1.61, 3.14 and6.14 m/s.

fects. It is also notable that in both load cases, the deviation between the predicted valuesand the measurements increases with the entrainment velocity.

It was previously noted that the average value for lubricant’s thermal conductivity,Kl , have taken from equation (B.21). To illustrate the thermal conductivity’s impact onthe friction curve, Figure B.9 plots the model’s output assuming three different constantvalues for Kl: 0.16 W/mk, 0.36 W/mk and 0.66 W/mk. The highest Kl value yields thegreatest friction, which means that the heat is transferred more rapidly to the surfaces.This behavior is in agreement with the findings of Larsson [43] and Kumar [38].

Two methods for estimating the film thickness are implemented in the LDOF model:the Newtonian method of Hamrock-Dowson and Bair’s shear-thinning method. To visu-alize the effect of varying the method used to predict film thickness on the estimatedfriction, the results obtained using these two methods are shown in Fig. B.10, based on anentrainment velocity and load of Ue = 1.611 m/s and F = 300 N. The Newtonian methodclearly leads to an underestimation of the friction.

As mentioned earlier, it is essential to use an appropriate rheological model whenmodelling EHL contacts. Equation (B.9), which is known as the Vogel-like equation,was used to define the limiting low shear viscosity and applied to the Carreau equationin order to estimate the viscosity of the lubricant. To determine the impact of using theVogel-like equation, the limiting shear viscosity was also computed using the exponentialequation (Eq. 27-A) and three variants of the Roelands equation (Eq. 27-B-D); see TableB.3. Equations (27-A) and (27-B) are temperature independent whereas equations (27-C)and (27-D) describe the limiting low shear viscosity as being temperature-dependent. Thelubricant parameters used in the calculations with these equations are listed in Table B.3;all other parameters were assigned the same values as in the previous computations. Asshown in Figures B.11 and B.12 (which present results obtained for loads of 80 and 300



Figure B.4: Variation in the friction coefficient with the slide-to-roll ratio (SRR) for the300 N loading condition. Results are shown for entrainment velocities of 1, 1.61, 3.14and 6.14 m/s.

Figure B.5: Variation in the friction coefficient with the slide-to-roll ratio (SRR) for the300 N and 80 N loading conditions with an entrainment velocity of 3.14 m/s. Note thatthe horizontal scale is logarithmic.


Figure B.6: Predicted average temperature for the 80 N and 300 N loading conditionswith entrainment velocities of 1, 1.61, 3.14, and 6.14 m/s. Note that the horizontal scaleis logarithmic.

Figure B.7: Variation in the isothermal friction coefficient at low slide-to-roll ratios (SRR)for the 80 N and 300 N loading conditions with an entrainment velocity of 1.16 m/s. Notethat the horizontal scale is logarithmic.

N, respectively), the use of Eq. (27-A) leads to a rapid increase in the predicted shearstress to the limiting value. Therefore the predicted coefficient of friction is constantover most of the SRR range and is insensitive to thermal effects. It is also apparent from



Figure B.8: Variation in the isothermal friction coefficient with the slide-to-roll ratio(SRR) for the 80 N and 300N loading conditions with an entrainment velocity of 6.144m/s. Note that the SRR scales are logarithmic.

Figure B.9: Friction coefficients predicted for different thermal conductivity values underthe 300 N loading condition with an entrainment velocity of 3.14 m/s. Note that thehorizontal scale is logarithmic.

both figures that combining the Eq. (27-A) and shifted Carreau equations does not yieldreasonable predictions of the viscosity and friction behavior. Eq. (27-B) shows similarresults as Eq. (27-A). At the low SRR there is an agreement between prediction and


Figure B.10: Friction coefficients obtained using different film thickness equations for the300 N loading condition with an entrainment velocity of 1.61 m/s. Note that the horizontalscale is logarithmic.

experiment. However, when the SRR increases the predicted friction starts to deviate fromthe experiment. This is due to the fact that Eq. (27-B) is not temperature dependent. Eq.(27-C) has better results on friction prediction than the rest of the mentioned equations. At80 N and low entrainment velocity (1 m/s, 1.61 m/s) there is a good agreement betweenthe friction prediction and measurement. When the entrainment velocity increases thepredicted friction slightly over estimated the experimentally measured friction. But, thecurve still follows a similar trend. The same behavior is also observed at the higher loadcondition (300 N), see Fig. B.12. Equation (27-D) is not a suitable choice for the lowlimiting shear viscosity. It is shown in Fig. B.11 and Fig. B.12 that the predicted frictionis overestimated in all operating conditions

To further validate the LDOF approach, studies were performed using Shell T9 inplace of squalane oil. The coefficient of friction ( f ) determined for this oil are plottedagainst the slide-to-roll ratio (SRR) in Figure B.13, and compared to the results of a fullydeterministic simulation performed by Habchi and co-workers [25]. Three loading con-ditions are considered: 50 N, 100 N and 300 N; these correspond to Hertzian pressures of0.93 GPa, 1.17 GPa and 1.47 GPa, respectively. The inlet temperature was set to 303 Kand the entrainment velocity to 4 m/s. All of the required parameters and input data are asreported by Habchi et al. [25]. The volume equation is replaced by the Murnaghan equa-tion of state. The obtained coefficient of friction from LDOF model was compared withthe "full potential" results in Ref [25], meaning that thermal effects, shear thinning andthe limiting shear stress were all taken into account. Under these conditions, remarkablygood agreement was obtained with the results reported by Habchi et al. [25].

As mentioned in section B.1, the Ree-Eyring expression is quite widely used by tri-bologists to estimate friction in EHL contacts. To compare its performance to that of theLDOF model, Ree-Eyring calculations were performed using equations (5) and (15) fromreference [63]. The equation 27-A was used to predict the viscosity behavior of the oil



because this is commonly done in Ree-Eyring calculations [63]. The low shear viscosityand viscosity-pressure coefficient values for squalane oil were taken from Table 3. TheEyring shear stress was set to 5.5 MPa, while the load and entertainment velocity wereassigned the same values as before (F = 80 and 300 N, Ue = 3.14 and 6.14 m/s), andthe temperature algorithm outlined in section B.2.3 was applied. The friction generatedduring ball-on-disc contact was estimated and compared to the results obtained using theLDOF approach and the experimental data, as shown in Fig .B.14. The Ree-Eyring ap-proach yields results that are in relatively good agreement with experiment for a load of80 N. However, there is some deviation at higher loads and velocities. This may be dueto the use of the exponential expression to describe the oil’s pressure-viscosity behavior.

B.6 CONCLUSIONSA model with a low degree of freedom was developed to predict friction in EHL contactsusing a rigorously scrutinized set of rheological equations. The relative density of thelubricant is estimated using the Tait equation of state, and the shear dependence of theviscosity is defined by the Carreau equation. Due to the importance of thermal effects inEHL friction, an iterative procedure for estimating the average temperature distributionfor a circular contact was implemented. The new method’s performance was evaluated

B.6. CONCLUSIONS 91

Figure B.11: Coefficients of friction obtained with different limiting shear viscosity equa-tions for the 80 N loading condition.

Figure B.12: Coefficients of friction obtained with different limiting shear viscosity equa-tions for the 300 N loading condition.

by comparing its predictions to the results of friction measurements acquired using a ball-and-disc device at two different loads with two different entrainment speeds over a rangeof slide-to-roll ratios.

Thermal effects become very important and the thermal model proposed herein yieldsacceptably accurate predictions. The use of the shear thinning film thickness was alsofound to be important for the accurate estimation of friction: calculations based on Newto-



Figure B.13: Coefficients of friction predicted using the LDOF approach and the fullydeterministic simulations described by Habchi et al. [25].

Figure B.14: Coefficients of friction from the experimental measurements, dash line,compared to results obtained using the LDOF (Carreau), blue line, and LDOF (Eyring),black line, approaches (Lubricant: squalane).

nian film thicknesses tended to underestimate the coefficients of friction whereas the shearthinning approach yielded results that agreed more closely with experiment. Moreover, itwas shown that it is important to account for variation in the lubricant’s thermal conduct-ivity, which depends on the pressure and temperature. Results from calculations madewith constant thermal conductivity suggest that even small changes in this parameter willaffect the friction prediction. The proposed isothermal model predicted the coefficient offriction with an acceptable accuracy for low slide-to-roll ratios (SRR ≤ 0.12) and maytherefore be useful for predicting friction in applications where low slide-to-roll ratios areto be expected.

Finally, we wish to note that it is not advisable to employ the equation 27-A or Roe-lands temperature independent viscosity equations together with a limiting shear stressmodel in this LDOF approach as this could lead to substantial deviations from experi-

B.6. CONCLUSIONS 93

ment. In addition, the use of quantitative [15] values based on the lubricant’s physicalbehavior will generally yield more reliable estimates of the coefficient of friction than canbe obtained using the popular Eyring model.

ACKNOWLEDGEMENTSThe authors wish to thank Scania, Bosch and Bosch Rexroth Mellansel (Hägglunds Drives)for financial support. Without their contributions, this work would have never been ac-complished. The authors also thank Dr. Wassim Habchi at the Lebanese American Uni-versity for providing the numerical results.

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