research article engineering software solution for thermal...

Research ArticleEngineering Software Solution for Thermal ElastohydrodynamicLubrication Using Multiphysics Software

Thomas Lohner Andreas Ziegltrum Johann-Paul Stemplinger and Karsten Stahl

Gear Research Centre (FZG) Technical University of Munich (TUM) Boltzmannstraszlige 15 85748 Garching Germany

Correspondence should be addressed to Andreas Ziegltrum ziegltrumfzgmwtumde

Received 7 October 2015 Revised 8 December 2015 Accepted 17 December 2015

Academic Editor Michel Fillon

Copyright copy 2016 Thomas Lohner et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The complexity of thermal elastohydrodynamic lubrication (TEHL) problems has led to a variety of specialised numericalapproaches ranging from finite difference based direct and inverse iterative methods such as Multilevel Multi-Integration solversvia differential deflection methods to finite element based full-system approaches Hence not only knowledge of the physical andtechnical relationships but also knowledge of the numerical procedures and solvers is necessary to perform TEHL simulationsConsidering the state of the art of multiphysics software the authors note the absence of a commercial software package for solvingTEHLproblems embedded in largermultiphysics software By providing guidelines on how to implement a TEHL simulationmodelin commercial multiphysics software the authors want to stimulate the research in computational tribology so that hopefullythe research focus can be shifted even more on physical modelling instead of numerical modelling Validations as well as resultexamples of the suggested TEHL model by means of simulated coefficients of friction coated surfaces and nonsmooth surfaceshighlight the flexibility and simplicity of the presented approach

1 Introduction

Measurements numerical simulations and derived analyticalsolutions have for decades created detailed insights intothe behaviour of thermal elastohydrodynamically lubricated(TEHL) contacts Due to the large number of numericalinvestigations it is astonishing that so far no commercialsoftware package embedded in larger multiphysics softwarehas been available to solve TEHL problems

The simulation of TEHL contacts is mainly due toelastic deformations and the large pressure-induced increaseof lubricant viscosity highly nonlinear and has led tovarious specialised numerical simulation approaches Greatcomputational efforts and the tendency to instabilities athigh loads when using computational fluid dynamics (CFD)[1] make the use of simulations based on the Reynoldsequation necessary Hartinger et al [2] thereby compareTEHL solutions obtained with CFD with solutions based onthe Reynolds equation which show very good agreementLubrecht [3] and Venner [4] develop a direct iterative finitedifference based Multilevel Multi-Integration (MLMI) solverfor isothermal EHLproblemsHabchi [1] introduces the finite

element method (FEM) based full-system approach to solvethe fully coupled system of the highly nonlinear elastohy-drodynamic equations which are coupled with the energyequations by an iterative procedure Raisin et al [5] extendthe approach based on line contacts to transient operatingconditions Although both authors [5 6] mention the useof the commercial software COMSOL Multiphysics [7] tosolve the TEHL problem explicit implementation guidelinesare not given For the isothermal EHL line and point contactwith Newtonian fluid behaviour under transient and steady-state operating conditions Tan et al [8] describe someinformation for implementation inCOMSOLMultiphysics inmore detail

Considering the large number of self-developed approa-ches and solvers the modelling of TEHL contacts is hardlytransparent Hence the promotion of using commercialsoftware packages for solving TEHL problems is necessary tomake simulation techniques available to a broader audienceof the tribology society and to focus research more onphysical relationships rather than on numerical proceduresAccordingly the authors provide guidelines for the imple-mentation and solving of TEHL problems in the commercial

Hindawi Publishing CorporationAdvances in TribologyVolume 2016 Article ID 6507203 13 pageshttpdxdoiorg10115520166507203

2 Advances in Tribology

v

Gear contact

(2)

(1)

EA

Wheel (2)

Pinion (1)

A

Model contact Equivalent contact

1 12 2

ET2

T2

T1T1

FNleff

Rx

1 2Eeq eq

Figure 1 Line contact in a spur gear and its simplification to a model (disc) contact and to an equivalent contact

software COMSOL Multiphysics to stimulate and acceler-ate research in computational tribology The possibility ofdeveloping a complex TEHL model with moderate effort isdemonstrated in this paper

2 Theory

In this section the theory and governing equations for sim-ulation of TEHL contacts are addressed For simplification atwo-dimensional line contact is considered

21 General Contact conditions of for example gears arecharacterised by varying load motion and geometry alongthe path of contact This transient contact can be interpretedas a transient model contact of two rolling elements withvarying load motion and geometry for each meshing pointIn TEHL simulations the transient contact is usually furthersimplified to an equivalent contact between a single (inelas-tic) roller and an elastic flat bodyThis is adopted in this studyand exemplarily shown for a line contact of a spur gear inFigure 1

Characteristic quantities of the equivalent contact aredescribed by the Hertzian contact parameters for example119901

119867 119887

119867 and 119877

119909[9] Poissonrsquos ratio ]eq and Youngrsquos Modulus

119864eq of the equivalent elastic flat body are calculated accordingto Habchi [1] The kinematics are characterised by the sumvelocity V

Σand the siding velocity V

119892 whereby for all analyses

in this study the velocity of the lower body is higher than thevelocity of the upper body V

1gt V

2

V119892= V

1minus V

2

VΣ= V

1+ V

2

(1)

The coefficient of friction 120583 of the TEHL contact is evaluatedby integrating the shear stress in the middle of the lubricantfilm (see also Habchi [1])

120583 =

int119909119890119909

119909119894119899

120591119911119909

100381610038161003816100381610038161003816119911=ℎ2

119889119909

119865119873119897eff

(2)

22 Generalised Reynolds Equation Derived from the two-dimensional compressible Navier-Stokes equations by con-sidering reasonable assumptions for TEHL contacts [1 10]and no-slip boundary conditions the shear rate 120574

119909and

velocity distribution V119909in the lubricant film can be expressed

by

120574119909 (119909 119911 119905) =

120597V119909

120597119911

=1

120578

120597119901

120597119909(119911 minus

intℎ

0(119911120578) 119889119911

intℎ

0(1120578) 119889119911

)

+V2minus V

1

120578 intℎ

0(1120578) 119889119911

(3)

V119909 (119909 119911 119905) =

120597119901

120597119909(int

119911

0

120578119889 minus int

119911

0

1

120578119889int

ℎ

0(119911120578) 119889119911

intℎ

0(1120578) 119889119911

)

+V2minus V

1

intℎ

0(1120578) 119889119911

int

119911

0

1

120578119889 + V

1

(4)

Inserting the velocity equation in the integrated transientcompressible continuity equation results in the generalisedReynolds equation according to Yang and Wen [11]

120597

120597119909[

[

120597119901

120597119909int

ℎ

0

120588(int

119911

0

120578119889 minus int

119911

0

1

120578119889int

ℎ

0(119911120578) 119889119911

intℎ

0(1120578) 119889119911

)119889119911]

]

+120597

120597119909[

[

V2minus V

1

intℎ

0(1120578) 119889119911

int

ℎ

0

120588(int

119911

0

1

120578119889) 119889119911

+ int

ℎ

0

120588 119889119911 sdot V1]

]

+120597

120597119905int

ℎ

0

120588 119889119911 = 0

(5)

In TEHL line contacts the generalised Reynolds equationdetermines the hydrodynamic pressure Due to the inte-grations when deriving the generalised Reynolds equation

Advances in Tribology 3

the pressure solution for line contacts is one-dimensionalThe integral terms in (3) to (5) can be understood asintegral viscosity anddensity termsAs a boundary condition119901(119909

119894119899) = 119901(119909

119890119909) = 0 has to be prescribed In addition the

cavitation model according to Wu [12] fulfils the Reynoldscavitation boundary condition [13]119901 ge 0 inΩ

119875and119901(119909cav) =

120597119901120597119909(119909cav) = 0

23 Contact Mechanics The calculation of the elastic defor-mation of the equivalent body is based on the finite elementmethod as introduced by Habchi [1] This has advantagesin terms of accuracy and resolution over the half-spaceassumption [10] The elastic deformation of the equivalentbody in gap height direction 120575 is calculated by applyingthe linear elasticity equation and neglecting the dynamicresponse of the solids [5]

nabla sdot 120590 = 0 with 120590 = 119862 sdot 120576 (119880)

120575 (119909 119905) =1003816100381610038161003816119880V (119909 119905)

1003816100381610038161003816

with the displacement vector 119880 = (119880

119906

119880V)

(6)

Boundary conditions of the linear elasticity equation are zerodisplacement on the bottom of the calculation domain Ω

120575

and the hydrodynamic pressure 119901 between 119909119894119899and 119909

119890119909on

the top of the calculation domain Ω120575 which is applied as a

normal stress (Figure 3 top left) A free boundary conditionassuming zero normal and tangential stress is applied to theremaining boundaries of the computational domain Ω

120575

The film thickness equation describes the height of theseparating lubricant film and consists of the constant param-eter ℎ

0 the undeformed geometry the elastic deformation

of the equivalent body 120575(119909 119905) and the deviation from thesmooth profileR

ℎ (119909 119905) = ℎ0+119909

2

2119877119909

+ 120575 minusR (7)

Hydrodynamic pressure and the applied load have to be inbalance This is ensured by the load balance equation whichdetermines the constant parameter ℎ

0in (7)

intΩ119875

119901 (119909 119905) 119889Ω119875= (

119865119873

119897eff) (119905) (8)

24 Energy Conservation Similarly to the generalisedReynolds equation the transient energy equations aresimplified with reasonable assumptions typically used inTEHL calculations [1 10] The resulting energy equation ofthe lubricant can be written as

120588119888119901sdot (120597119879

120597119905+ V

119909

120597119879

120597119909) minus (

120597

120597119909(120582

120597119879

120597119909) +

120597

120597119911(120582

120597119879

120597119911))

= minus119879

120588

120597120588

120597119879(120597119901

120597119905+ V

119909

120597119901

120597119909) + 120578(

120597V119909

120597119911)

2

(9)

The transient energy equations of the solids (1 2) are shownin (10) The transitions between the solids and the lubricant

are described by a temperature and conductive heat fluxcontinuity condition

120588119894119888119901119894sdot (120597119879

120597119905+ V

119894

120597119879

120597119909)

minus (120597

120597119909(120582

119894

120597119879

120597119909) +

120597

120597119911(120582

119894

120597119879

120597119911)) = 0

with 119894 = 1 2

(10)

Thus based on heat sources (shearing and compression ofthe lubricant) heat sinks (expansion of the lubricant) andheat transfer due to convection and conduction the temper-ature distribution in lubricant and solids can be calculatedBoundary conditions ensure that lubricant entering the com-putational domain Ω

119879(V

119909(119909

119894119899 119911 119905) ge 0 and V

119909(119909

119890119909 119911 119905) le

0) has bulk temperature 119879119872

and for lubricant leaving Ω119879

(V119909(119909

119894119899 119911 119905) le 0 and V

119909(119909

119890119909 119911 119905) ge 0) the conductive heat

flux is zero (minus120582 sdot 120597119879120597119909 = 0) Similarly solids entering thecomputational domains Ω

1198791and Ω

1198792(V

119894(119909

119894119899 119905) ge 0 and

V119894(119909

119890119909 119905) le 0) have bulk temperature 119879

119872and for solids

leaving Ω1198791

and Ω1198792

(V119894(119909

119894119899 119905) le 0 and V

119894(119909

119890119909 119905) ge 0) the

conductive heat flux is zero (minus120582119894sdot 120597119879120597119909 = 0) The height

of solids Ω1198791

and Ω1198792

is with 119889 = 315 sdot 119887119867[14] sufficient so

that the conductive heat flux into the solids becomes zero andbulk temperature 119879

119872can be assumed at 119889

25 Lubricant Properties Pressure temperature and shearrate distributions in TEHL contacts have a significant influ-ence on lubricant properties The accompanying changes inviscosity have the largest influence on the TEHL contactitself The models for pressure and temperature dependencyof the viscosity as suggested in [15] are used The dynamicviscosity for a given temperature and ambient pressure 120578(119879)is described by the Vogel model [16]

120578 (119879) = 119860 sdot exp( 119861

119862 + (119879 minus 27315K)) (11)

The pressure dependency of the viscosity 120578(119879 119901) is modelledby the Roelands [17] equation

120578 (119879 119901) = 120578 (119879) sdot exp(ln (120578 (119879)) + 967)

sdot [minus1 + (1 +119901

1199011205780

)

119911(119879)

]

(12)

with a temperature dependent pressure exponent 119911(119879)

according to

119911 (119879) =120572

119901 (119879) sdot 1199011205780

ln (120578 (119879)) + 967 (13)

The pressure-viscosity coefficient 120572119901(119879) is temperature

dependent

120572119901 (119879) = 119864120572

1199011sdot exp (119864

1205721199012sdot 119879) (14)


The rheological behaviour of the lubricant is described by anEyring model [18]

120578 (119879 119901 120574119909) =

120591119888

120574119909

asinh(120578 (119879 119901) sdot 120574

119909

120591119888

)

with 120591 (119879 119901 120574119909) = 120578 (119879 119901 120574) sdot 120574

119909

(15)

The temperature and pressure dependency of lubricant den-sity 120588(119879 119901) is modelled by the Bode model [19]

120588 (119879 119901)

=120588119904sdot (1 minus 120572

119904sdot 119879)

1 minus 1198631205880sdot ln ((119863

1205881+ 119863

1205882sdot 119879 + 119901) (119863

1205881+ 119863

1205882sdot 119879))

(16)

Based on measurements Larsson and Andersson [20]derived a model for the temperature and pressure depen-dency of the thermal conductivity 120582(119901) and of the specificheat capacity per volume (119888

119901sdot 120588)(119879 119901) Temperature depen-

dency of the thermal conductivity is not considered as in[20] the influence is assessed as negligible The equations for120582(119901) and (119888

119901sdot 120588)(119879 119901) are

120582 (119901) = 1205820sdot (1 +

1198891205821sdot 119901

1 + 1198891205822sdot 119901) (17)

(119888119901sdot 120588) (119879 119901) = (119888

119901sdot 120588) (295K) sdot (1 +

1198601198881sdot 119901

1 + 1198601198882sdot 119901)

sdot [1 + 1198601198883sdot (1 + 119860

1198884sdot 119901 + 119860

1198885sdot 119901

2) sdot (119879 minus 295K)]

(18)

Among others parameters for paraffinic mineral oil areavailable for (17) and (18) in [20]

3 Model Description and Implementation

A guideline for the implementation of a TEHL simulationmodel in COMSOL Multiphysics (abbr by COMSOL) andMathWorks MATLAB [21] (abbr by MATLAB) is given inthe following sections All COMSOL-specific notations arewritten in italics

31 Dimensionless Parameters For a convenient numericalsolution procedure of the nonlinear system of equationswith good conditioning the variables are transferred to a

dimensionless form Following Habchi [1] Tan et al [8]Venner [4] and Lubrecht [3] the definitions are

119883 =119909

119887119867

119879 =119879

119879119872

119863 =119889

119887119867

119875 =119901

119901119867

119867 =ℎ sdot 119877

119909

1198872119867

120575 =120575 sdot 119877

119909

1198872119867

120588 =120588

120588 (119879119872)

120578 =120578

120578 (119879119872)

119905 =119905 sdot V

Σ

2119887119867

119862V =VΣ (119905)

VΣ (119905 = 0)

119862119877=

119877119909 (119905)

119877119909 (119905 = 0)

119862119908=

(119865119873119897eff) (119905)

(119865119873119897eff) (119905 = 0)

119885 =

119911

119887119867

for solids

119911

ℎfor lubricant

(19)

32 Numerical Solution Procedure The numerical solutionscheme shown in Figure 2 is similar to other schemes of forexample Habchi [1] and Wang et al [22] In this work itis based on two self-developed decoupled COMSOL FEM-models denoted as FEM-model (119875119867) and FEM-model (119879)and a MATLAB sequence controller which also performsadditional calculations

After reading all the required input parameters an initialsolution based on a simple steady-state isothermal Newto-nian approach is calculated in the FEM-model (119875119867) forthe initial operating conditions The procedure thereby is thesame as described in the following paragraph Based on theinitial solution the time loop is launched and repeated untilthe last time-step has been calculated The coupling of twoconsecutive time-steps is realised by an implicit backwarddifferentiation formula (BDE) scheme of the first order asused in [23] It is implemented by passing the required


Output

Calculate a non-Newtonian elastohydrodynamic

Update lubricant properties integral

Calculate an elastohydrodynamic temperature

Update lubricant properties integral terms

All time-steps calculated

Save TEHL solution for time-step

Update operating conditions

Calculate an initial solution based on an isothermal Newtonian approach for initial operating conditions

Input

No

No

No

No

Yes

Yes

Yes

Yes

Glo

bal l

oop

Tim

e loo

p

FEM-model (PH)

FEM-model (PH)

FEM-model (T)

pressure P and film thickness H for a giventemperature distribution T

terms and velocity distribution x

Converged P

distribution T corresponding to given pressure Pand film thickness H

Converged P and T

Converged T

velocity distribution x and heat sources

Figure 2 Numerical solution scheme

solution variables of the previous time-step for calculatingthe time derivatives of the generalised Reynolds equation andenergy equations from the sequence controller to the FEM-model (119875119867) and FEM-model (119879)

Within the time loop a global loop is launched for eachtime-step Firstly the pressure and film thickness distributionunder consideration of non-Newtonian fluid behaviour iscalculated for a given temperature distribution 119879 in theFEM-model (119875119867) described in Section 331 Then thelubricant properties acc to Section 25 the integral terms in(5) and the velocity distribution V

119909(4) are updated in the

sequence controller Thereby the shear stress 120591 is analyticallyderived (15) After calling up the FEM-model (119875119867) againthe new calculated pressure distribution is compared withthe previous calculated pressure distribution The iterativeprocedure is repeated until the maximum absolute differenceof two consecutive pressure distributions is smaller than 10minus3

After the convergence of 119875 the pressure and film thick-ness distributions 119875 and119867 are kept constant and an iterativecalculation of the temperature distribution 119879 in lubricantand solid bodies based on the FEM-model (119879) describedin Section 332 is launched For each new temperaturedistribution 119879 calculated the lubricant properties acc toSection 25 the integral terms in (5) the heat sources in(9) and the velocity distribution V


sequence controller Thereby the shear stress 120591 is analyticallyderived (15) The iterative calculation is repeated until themaximum absolute difference of two consecutive tempera-ture distributions in the lubricant is smaller than 10minus3

Finally the converged pressure distribution and temper-ature distribution in the lubricant of two consecutive globalloops are compared Again convergence is assumed whenthe maximum absolute difference between two consecutivesolutions of pressure and temperature distributions is smallerthan 10minus3 After saving the solution of the considered time-step the next global loop is calculatedwith updated operatingconditions and time derivatives of the generalised Reynoldsequation and energy equations

33 Implementation in COMSOL In the following the char-acteristics of the FEM-model (119875119867) and FEM-model (119879)as well as of the solver settings and sequence controllerare described in detail To a great extent the structure ofthe FEM-models follows the outstanding work of Habchi[1] who introduces the FEM based full-system approachto TEHL achieving very high convergence rates withoutunderrelaxation required Thereby the FEM approach offersthe possibility of using a mesh with nonregular unstructuredelements and high order approximation functions As shownin the following it is possible to implement the approach ofHabchi in commercial multiphysics software

331 FEM-Model (119875119867) Figure 3 (top left) shows the com-putational domains Ω

120575and Ω

119875on which the pressure

and film thickness distribution are calculated fully coupledFigure 3 (bottom left) illustrates in parts the correspondingmodel tree For each new call-up of FEM-model (119875119867) thesolution of the previous call is used as an initial value

The elastic deformation of the equivalent body in 119885-direction120575(119883 119905) is calculated on a quadratic two-dimensionalcomputational domainΩ

120575with the extensions119883 isin [minus30 30]

and 119885 isin [minus60 0] Hence an influence of the boundaryconditions of Ω

120575on the contact area can be excluded The

physics Solid Mechanics is used with the displacement vector119880(119883 119905) as the dependent variable A linear elastic isotropicmaterial specified by Youngrsquos Modulus 119864eq and Poissonrsquosratio ]eq is assumed The boundary conditions described inSection 23 are implemented as a fixed constraint conditionon the bottom boundary of Ω

120575 a boundary load condition

with the load type pressure 119875 on Ω119875 and a free boundary

condition on the remaining boundaries of Ω120575 Lagrange

quadratic elements are used for the linear elasticity equationsThe pressure and film thickness distributions 119875 and119867 as

well as the constant parameter1198670based on the load balance

in (8) are calculated on the one-dimensional computationaldomain Ω

119875with an extension from 119883

119894119899= minus45 to 119883

119890119909= 3

Thereby119883119890119909ofΩ

119875has already been enlarged for simulations

with varying load over time for example [5] The dependentvariables are the pressure 119875 the constant parameter 119867

0

and the displacement vector 119880 The film thickness in (7) isimplemented as a variable and the deviation from the smoothprofile R is read as the input parameter by an interpolationfunction The load balance in (8) is implemented as a Global


Z = 1 + D

Z = 1

Z = 0

Z = minusD60

60

0

Lubricant

Solid 1

Solid 2

Z

X

Z

X

Ω120575

ΩP

Xin Xex

Xin Xex

ΩT1

ΩT

ΩT2

Figure 3 FEM-model (119875119867) for pressure and film thickness calculation with corresponding COMSOL model tree (left) and FEM-model(119879) for temperature calculation with corresponding COMSOL model tree (right) element numbers are adjusted for presentation purposes


Equation with a component coupling integration on Ω119875for

the left-hand side of the load balance equation Its result isthe constant parameter 119867

0 As no specific module for the

generalisedReynolds equation in (5) is available inCOMSOLit is defined in the physics Weak Form Boundary PDE byusing Lagrange quintic elements Based on the dimensionlessparameters in (19) the weak formof the generalised Reynoldsequation in dimensionless form is written as

minus intΩ119875

((120576 + 119868119863)120597119875

120597119883)120597119882

119875

120597119883119889119883

minus intΩ119875

(120588119867)119867

120597119882119875

120597119883119889119883 + int

Ω119875

120597 (120588119867)119905

120597119905119882

119875119889Ω

119875

+ intΩ119875

119876120585119882

119875119889Ω

119875+

119878119880119875119866

119866119871119878 = 0

with 119878119880119875119866

119866119871119878 = 119878119880119875119866 or 119866119871119878

(20)

119882119875is the test function for the pressure 119875 and is denoted as

119905119890119904119905(119875) in COMSOL 119868119863 and 119878119880119875119866119866119871119878 represent numericalstabilisation terms which are implemented as variables andgiven here for completeness Note that for moderate casesthe TEHL simulation model can work without stabilisationtermsThe numerical stabilisation terms have been applied tothe generalised Reynolds equation by Habchi [1 24] in orderto avoid the oscillatory behaviour of the Reynolds equationat high loads 119878119880119875119866 (Streamline-Upwind-Petrov-Galerkin)andGLS (Galerkin-Least-Squares) are consistent stabilisationterms whereas ID (Isotropic Diffusion) is a nonconsistentstabilisation term For implementation of 119868119863 and 119878119880119875119866119866119871119878the reader is referred to [1 24] The penalty term 119876

120585=

119875119891sdot 119875

minus as an additional source term with 119875119891in the order of

106 [1] ensures the Reynolds cavitation boundary conditions[13] and represents the cavitation model according to Wu[12] Zero pressure on the boundaries of Ω

119875is ensured by a

Dirichlet boundary condition The coefficients 120576 (120588119867)119867 and

(120588119867)119905represent the first second and third term of (5) in

dimensionless form

120576 =119867

3

120582int

1

0

120588(int

119885

0

119885

120578119889119885 minus int

119885

0

1

120578119889119885

int1

0(119885120578) 119889119885

int1

0(1120578) 119889119885

)119889119885

with 120582 =119877

2

119909sdot 120578 (119879

119872) sdot V

Σ

2 sdot 119901119867sdot 1198873

119867

(120588119867)119867=2119867

VΣ

(V2minus V

1

int1

0(1120578) 119889119885

int

1

0

120588int

119885

0

1

120578119889119885119889119885

+ int

1

0

120588V1119889119885)

(120588119867)119905= 119867int

1

0

120588 119889119885

(21)

Note thereby that all the parts of the coefficients in (21) and ofthe stabilisation terms in (20) can despite the film thickness119867 be prepared in the sequence controller and transferred tothe FEM-model (119875119867) by interpolation functions

Themesh in the FEM-model (119875119867) is free triangular witha normal element size and a refinement atΩ

119875by a distribution

with typically 1000 elements

332 FEM-Model (119879) Figure 3 (top right) shows the compu-tational domains Ω

119879 Ω

1198791 and Ω

1198792on which the tempera-

ture distribution 119879 in lubricant and solid bodies is calculatedfully coupled Figure 3 (bottom right) illustrates in parts thecorresponding model tree The lubricant domain Ω

119879has

the dimensionless height of 1 with 119885 isin [0 1] whereasthe solid domains Ω

1198791and Ω

1198792have the dimensionless

heights of 119863 = 315 with 119885 isin [minus119863 0] and 119885 isin [1119863 +

1] respectively The length of the computational domainsextends from119883

119894119899to119883

119890119909(see Figure 3 top left) For each new

call-up of the FEM-model (119879) the solution of the previouscall-up is used as initial valueThe transient energy equationsin dimensionless form are implemented and calculated inthe physics Heat Transfer in Fluids for the lubricant domainΩ

119879and in Heat Transfer in Solids for the solid domains

Ω1198791

and Ω1198792 The dependent variable 119879 is solved based

on Lagrange quadratic elements The numerical stabilisationterms 119868119863 119878119880119875119866 and 119866119871119878 are already implemented in theconsidered heat transfer physics in COMSOL and appliedwhen required The velocity distribution (4) viscosity distri-bution (15) density distribution (16) thermal conductivitydistribution (17) and heat capacity distribution (18) in Ω

119879

are calculated in the sequence controller and transferred tothe FEM-model (119879) by interpolation functions Furthermorethe heat sources in the physics Heat Transfer in Fluids due tothe shearing and compression of the lubricant are calculatedin the sequence controller and transferred to FEM-model(119879) Note thereby that similar to the coefficients in (21) 119879as a dependent variable needs to be applied in the FEM-model (119879) The implementation of the energy equations onΩ

1198791andΩ

1198792in theHeat Transfer in Solidsmodules with the

translation motions of solids V1and V

2as input parameters is

straightforwardThe boundary conditions in Section 24 are implemented

by defining bulk temperature 119879119872= 1 at the left and upper

boundary of Ω1198792

and at the left and lower boundary of Ω1198791

anOutflow boundary condition at the right-hand boundariesofΩ

1198791andΩ

1198792and an Open boundary condition at the left-

and right-hand boundary of the lubricant domain Ω119879with

lubricant entering Ω119879having a bulk temperature 119879

119872= 1 As

a common coordinate system in the FEM-model (119879) is usedmanual adjustments are necessary to respect the differentdimensionless definitions of 119885 in Ω

119879and Ω

1198791and Ω

1198792

respectively (19) To be precise the dimensionless scale of the119885 axis of the solid domains Ω

1198791and Ω

1198792has to be adjusted

according to the dimensionless scale of the 119885 axis of thelubricant domain Ω

119879 so that the continuity condition of the

conductive heat flux and temperature across the lubricant-solid boundaries is respectedThis is handled in COMSOL by


multiplying the dimensionless energy equations of the solidsby the conversion factor 119885cf This factor is the ratio of thedimensionless scales of the 119885 axes of the lubricant domainand the solid domains

119885cf =119885 (lubricant)119885 (solid)

=119911ℎ

119911119887119867

=119877

119909

119867 sdot 119887119867

(22)

Themesh in the FEM-model (119879) is divided into the lubricantdomain Ω

119879and the solid domains Ω

1198791and Ω

1198792 Whereas

Ω119879is meshed by a mapped equidistant mesh Ω

1198791and Ω

1198792

are meshed with free triangular elements with a refinementon the solid-lubricant boundaries and an extra-fine elementsize The lubricant domain Ω

119879may consist of 40 elements in

height and 1000 elements in the length direction

333 Sequence Controller The communication between theMATLAB sequence controller and the FEM-model (119875119867)and FEM-model (119879) is achieved by COMSOLrsquos LiveLink forMATLAB function The MATLAB sequence controller basi-cally represents the implementation of the global numericalscheme shown in Figure 2 There it is very convenient forexample to implement the models for the lubricant proper-ties calculate the integrals by adaptive Simpson quadraturerule with typically 40 elements in 119885-direction and performother calculations with the current values for pressurefilm thickness and temperature As no other possibility isavailable the data exchange of vectors and matrices fromMATLAB to COMSOL is handled in a somewhat cumber-some manner The quantities calculated in the sequencecontroller for example velocity distribution are written intoa text file and read by the COMSOL interpolation functionsFor data exchange from COMSOL to MATLAB themphevalcommand is used

334 Solver For solving the nonlinear system of equationstwo separateCOMSOL solvers for the FEM-model (119875119867) andfor the FEM-model (119879) are requiredThe solver for the FEM-model (119875119867) includes the dependent solution variables119875119867

0

and 119880 whereas the solver for the FEM-model (119879) includesthe temperature 119879 as the dependent solution variable Forboth FEM-models a direct COMSOLMUMPS (MultifrontalMassively Parallel Direct Solver) [25] is used with a dampedNewton-Raphson approach [26] and 10

minus3 as the tolerancefactor as defined in [7]

4 Model Validation

In order to show the functionality and the plausibility of theTEHL simulation model described in Section 3 its solutionsare compared with steady-state solutions from Hartinger etal [2] and with transient solutions presented by Venner [4]

41 Steady-State Validation Hartinger et al [2] investigatedthe TEHL contact by CFD simulations for various conditionsThey show that the assumptions of the generalised Reynoldsequation are reasonable and provide representative resultsfor thermal and isothermal contacts For comparison of their

results in [2] with the presented TEHL simulation model thesame lubricant property models and properties for lubricantand solids are used Note that cavitation effects considered bytheCFDapproach cannot be depicted by the cavitationmodelof Wu [12] Figure 4 shows the calculated pressure and filmthickness distributions for a thermal (TEHL) and isothermal(EHL) contact with high sliding proportion (V

Σ= 5ms V

119892=

25ms) in comparison to the presented TEHL simulationmodel (a) and CFD model of Hartinger et al [2] (b) Theresults show very good agreement Furthermore the localdistributions of shear rate temperature and viscosity (notshown) correlate very well with the results of Hartinger et al[2] The calculation time for the TEHL solution in Figure 4 isapproximately 6 to 8 minutes on a computer with a 35 GHzprocessor for the presented TEHL simulation model and inthe order of 24 hours for the CFD model [2]

42 Transient Validation Venner [4] investigated an EHLcontact with a moving surface indentation under high slid-ing conditions by using a transient isothermal Newtonianapproach For the sake of comparison the authors alsoassume isothermal and Newtonian simplifications in theirpresented TEHL simulationmodel Input parameters and theinvestigated surface indentation are thereby adopted from[4] Figure 5 shows the comparison of pressure and filmthickness distribution for the example time when the surfaceindentation is located at the dimensionless position 119883

119889=

minus025 It is thereby shown that themoving surface indentationinitiates a significant film thickness change which hurriesahead the actual position of the indentation This can occurunder high sliding conditions The comparison in Figure 5shows very good agreement Small differences in the resultsmay be caused by different elastic deformation calculations(FEM in the TEHL simulationmodel in this study and elastichalf-space assumption in [4])

The validation of the TEHL simulation model shows itsfunctionality and plausibility

5 Example Results

In this section examples of possible application opportu-nities such as simulating coated surfaces coefficients offriction and nonsmooth surfaces of the described TEHLsimulation model are demonstrated

51 Thin Surface Coatings Firstly the possibility of inves-tigating the effect of thin surface coatings on the TEHLcontact behaviour is shown Results based on the TEHLmodel described in this study with slightly different lubricantproperties have already been published by Lohner et al [27]There the influence of thin surface coatings on the steady-state TEHL contact temperature was analysed by addingcoating layers with separate thermophysical properties to theFEM-model (119879)The FEM-model (119875119867) is unchanged In theFEM-model (119879) additional geometries with the physics HeatTransfer in Solids including adopted boundary and continuityconditions as described in Section 332 additional interpo-lation functions in COMSOL and additional meshes for


minus015 minus005minus01minus02 005 01501 020

x (mm)

0

01

02

03

04

05

06p

(GPa

)

p (EHL)h (EHL)

p (TEHL)h (TEHL)

h(120583

m)

0

01

02

03

04

05pH = 570Nmm2

Rx = 001mΣ = 5msg = 25ms1205780 = 001Pas

(a)

Hartinger et al [2]

x (mm)

h(120583

m)


01

02

03

04

05SRR = 11205780 = 001

Pressure thermalFilm thermal

Pressure isothermalFilm isothermal

0

01

02

03

04

05

06

p(G

Pa)

(b)

Figure 4 Comparison of simulated pressure and film thickness distributions for a thermal (TEHL) and an isothermal (EHL) contact betweenthe TEHL simulation model of this study (a) and the CFD model of Hartinger et al [2] (b)

0

03

06

09

12

15

18

21

P

0

003

006

009

012

015

018

021

H

Xd = minus025

minus35 minus05minus25 minus15 05 15minus2minus3 minus1minus4 0 1

X

pH = 2000Nmm2

Rx = 00139mΣ = 194msg = 097ms1205780 = 004Pas

(a)

Venner [4]

0

03

06

09

12

15

18

21

P

0

003

006

009

012

015

018

021

H


X

(b)

Figure 5 Comparison of simulated pressure and film thickness distribution for a transient isothermal Newtonian EHL contact between theTEHL simulation model of this study (a) and the solution of Venner [4] (b)

the thin surface coatings are integrated The correspondinghandling in the sequence controller is the same as for solidsΩ

1198791and Ω

1198792 This is very convenient to implement and

shows the flexibility of the presented TEHL model approachFor simplification it has been assumed that the deformationof the solid bodies is not affected by the very thin coatings(15 120583m) [6]

Figure 6 shows an example of the pressure and film thick-ness (a) and the corresponding temperature distribution ofthe lubricant and solid bodies (b) for a TEHL contact includ-ing a coating with a thickness of 15 120583m on the upper solidbody The bulk material is case-carburised steel 16MnCr5and the coating material SiO

2with a significant different

thermal conductivity (120582(16MnCr5) = 44WmK 120582(SiO2)

= 14WmK) The SiO2coating shows an insulating effect

and leads to significantly higher temperatures compared to

an uncoated TEHL contactThis also affects the coefficient offriction as shown by for example Habchi [6] for TEHL pointcontacts

52 Coefficient of Friction For comparing measurement andsimulation results the coefficient of friction is a suitablequantity with integral character Coefficient of friction mea-surements in fluid film lubrication regime can be performedat the FZG twin disk test rig with for example cylindricaldiscs (119903

1= 119903

2= 40mm 119897eff = 5mm) made from case-

carburised steel (16MnCr5) with a polished surface structureFurther information about the test rig is documented in[28] for example Table 1 shows the main properties of theconsidered lubricant MIN100 and the physical propertiesof the bulk material Model constants of Section 25 of the


h(120583

m)

0

02

04

06

08

1

12

14

16

18

2

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0420 085minus085 minus042

x (mm)

p(N

mm

2)

pH = 1200Nmm2

Rx = 002mΣ = 8msg = 089ms120599M = 895∘C

(a)

16MnCr5

16MnCr5

16MnCr5

Surface

Depth of

Gap

hei

ght d

irect

ionz

(120583m

)

ΩΩ2 (2)

Ω1 (1)

126∘C

SiO2

15 120583m

Depth of 15 120583m

90

95

100

105

110

115

120

125

130

135

Tem

pera

ture

(∘C)

minus3

minus2

minus1

0

1

2

3

minus085 0 042 085minus042

x (mm)

(b)

Figure 6 Pressure and film thickness (a) and temperature distribu-tion (b) of a TEHL contact with coated (SiO

2) upper solid acc to

Lohner et al [27]

considered lubricant have been obtained from regression ofmeasurement data The Eyring shear stress of 120591

119888= 6Nmm2

has been adopted from [29] as there a mineral oil of the sameviscosity has been considered Note that based on the Eyringshear model the increase of the shear stress with shear rate isnot limited To achieve a more realistic representation of therheological behaviour of the lubricant a shear stress modelbased on a limiting shear stress is currently being developedby the authors However this is not the focus of this study asit can be easy implemented once it is ready

A comparison of measured and simulated coefficients ofthe friction of lubricant MIN100 with very good correlationis shown in Figure 7 The considered sliding velocities V

119892=

0 008 021 042 064 089 141 200 267ms are thereby

Table 1 Lubricant and bulk material properties (119894 = 1 2)

Bulk material properties16MnCr5

119864119894in Nmm2 206000

]119894

030120588119894in kgm3 7760

120582119894in WmK 44

119888119901119894

in JkgK 431Lubricant properties

MIN100120592 (40∘C) in mm2s 95120592 (100∘C) in mm2s 105120588 (15∘C) in kgm3 885120582

0in WmK 0137

119888119901(15∘C) in JkgK 1921

120591119888in Nmm2 6

MeasuredSimulated

Coe

ffici

ent o

f fric

tion120583

pH = 1200Nmm2

Σ = 8ms120599oil = 40∘C

0

001

002

003

004

005

006

Rx = 002m

0 089 141 200 267042Sliding velocity g (ms)

Figure 7 Simulated andmeasured coefficients of friction 120583 over thesliding velocity V

119892

accompanied with corresponding bulk temperatures of theslower disc of 120599

119872= 42 47 56 69 80 90 109 123 140

∘C

53 Nonsmooth Surfaces Surface features have a significantinfluence onTEHL contact behaviour [30 31] As an examplethe contact of a deterministically structured surface (upperbody) with repetitive protruding bars with a height of 02 120583mand a length of 50120583m every 30 120583m shown in Figure 8with a smooth surface (lower body) is investigated Thedeterministically structured surface is represented by thetermR in (7)

The considered lubricant and material parameters areshown in Table 1 and the operation conditions are 119901

119867=

1200Nmm2 V119892= 089ms V

Σ= 8ms and 120599

119872= 895

∘CThe model constants of the considered lubricant MIN100as well as the macrogeometry of the discs are the same asin Section 52 Figure 9 shows the simulated pressure film


02 120583m50120583m 30120583m

Figure 8 Schematic representation of the deterministically struc-tured surface

h(120583

m)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

p(N

mm

2)

0

02

04

06

08

1

12

14

16

18

2

0 042 085minus042minus085

x (mm)

pH = 1200Nmm2

Rx = 002mΣ = 8msg = 089ms120599M = 895∘C

(a)

Surface

Gap

hei

ght d

irect

ionz

(120583m

)

ΩΩ2 (2)

Ω1 (1)

120∘C

90

95

100

105

110

115

120

125

130

135

Tem

pera

ture

(∘C)


x (mm)

minus2

minus1

0

1

2

(b)

Figure 9 Temperature (a) and pressure and film thickness dis-tribution (b) of a transient TEHL contact with a deterministicallystructured surface shown in Figure 8

thickness and temperature distribution which are signifi-cantly influenced by the deterministically structured surfaceVery local changes of the pressure film thickness and tem-perature distributions are observed thereby The maximumtemperature rise in the lubricant is simulated to be 4K higherfor the deterministically structured surface than for a TEHLcontact with smooth surfaces The transient calculation time

Lubricant

Ω2 (2)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

0

01

02

03

04

05

Mat

eria

l dep

th (m

m)

minus085 minus042 0 042 085x (mm)

(a)

Lubricant

Ω1 (1)minus042 0 042 085minus085

x (mm)

05

04

03

02

01

0

Mat

eria

l dep

th (m

m)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

(b)

Figure 10 von Mises stress in the lower solid body (a) and in theupper solid body (b) correlating to Figure 9

is approximately 5 to 10 hours on a computer with a 35 GHzprocessor Postprocessing the simulated pressure and shearstress distribution by applying it to the upper and lowersolid body with bulk material properties acc to Table 1easily delivers the stress distributions in the solid bodiesbased on linear elastic isotropic material behaviour Figure 10shows an example of the von Mises stress distribution inthe lower (a) and upper (b) solid body corresponding to thepressure and shear stress distribution in Figure 9 Stressesdue to temperature are omitted Due to the deterministicallystructured surface stress maxima close to the surface occurin addition to the well-known stress maxima in the deepermaterial depth

6 Conclusion

In this paper guidelines for the implementation of a TEHLmodel in commercial multiphysics software are presentedThe introduced approach can be implemented withmoderateeffort and the resulting TEHL model is very easy to extendfor various applications The TEHL model divided in asequence controller inMathWorksMATLAB and in two self-developed decoupled COMSOLMultiphysics FEM-models isalso capable of calculating the transient TEHL contact along


the path of contact of gears Even more challenging taskstoo such as the extension of the described model to mixedlubrication regimes are possible By providing implementa-tion guidelines for a TEHL simulation model in commercialmultiphysics software the authors are confident that theresearch in computational tribology has been stimulated andaccelerated

Nomenclature

11986011988815

Coefficients of lubricant heatcapacity model

119860 119861 and 119862 Coefficients of lubricant Vogeltemperature model

119887119867 Hertzian contact half-width in m

119887119867= 4 sdot 119877

119909sdot 119901

119867119864

1015840

Compliance matrix in Pa119888119901 Lubricant specific heat capacity in

J(kgK)1198881199011 119888

1199012 Specific heat capacity of solids 1

and 2 in J(kg K)119889 Height of solids in the FEM-model

(119879) in m119889

1205821 119889

1205822 Pressure coefficients of lubricant

thermal conductivity model in 1Pa1199031 119903

2 Radius of solids 1 and 2 in m

1198631205880 119863

1205881 and 119863

1205882 Coefficients of lubricant Bodedensity model

1198641 119864

2 Youngrsquos Modulus of solids 1 and 2

in Pa119864

1205721199011 119864

1205721199012 Coefficients of lubricant

pressure-viscosity coefficientmodel in 1Pa and K

119864eq Equivalent Youngrsquos Modulus in Pa119864eq = (119864

2

1119864

2(1 + 120592

2)2+ 119864

2

2119864

1(1 +

1205921)2)[119864

1(1 + 120592

2) + 119864

2(1 + 120592

1)]

2sdot

119887119867(119877

119909sdot 119901

119867)

1198641015840 Reduced Youngrsquos Modulus in Pa

1198641015840= 2 sdot 119864

2sdot 119864

1(119864

1sdot (1 minus 120592

2

2) +

1198642sdot (1 minus 120592

2

1))

119865119873 Normal force in N

119865119877 Friction force in N

ℎ Film thickness in mℎ0 Constant parameter of film

thickness in m119897eff Effective contact length in width

direction in m119901 Pressure in Pa119901

119867 Hertzian pressure in Pa

1199011205780 Coefficient Roelands equation

(1199010Roe = 196 sdot 10

8 Pa)119877

119909 Reduced radius in m

119877119909= 119903

1sdot 119903

2(119903

1+ 119903

2)

R Deviation from the smooth profilein m

119905 Time in s119879 Temperature in K119879

119872 Bulk temperature in K

= (119880119906

119880V) Displacement vector in m

VΣ Sum velocity in ms

V1 V

2 Velocity of solids 1 and 2 in ms

V119892 Sliding velocity in ms

V119909 Lubricant velocity distribution in

ms119909 Space coordinate in gap length

direction in m119909

119894119899 119909

119890119909 Left and right boundary ofΩ

119875in m

119911 Space coordinate in gap heightdirection in m

119911(119879) Roelands pressure-viscosityparameter

Greek Symbols

120572119901 Pressure-viscosity coefficient in 1Pa

120572119904 Coefficient of lubricant Bode density

model in 1K120574119909 Shear rate in 1s

120575 Deformation of the equivalent body in m120576 Strain tensor120578 Lubricant viscosity in Pas120599119872 Bulk temperature in ∘C

120599oil Oil inlet temperature in ∘C120582 Lubricant thermal conductivity in

W(mK)120582

1 120582

2 Thermal conductivity of solids 1 and 2 in

W(mK)120583 Coefficient of friction]1 ]

2 Poissonrsquos ratios of solids 1 and 2

]eq Equivalent Poissonrsquos ratio ]eq = (1198641]2(1 +

]2) +119864

2]1(1 + ]

1))(119864

1(1 + ]

2) +119864

2(1 + ]

1))

120588 Lubricant density in kgm3

1205881 120588

2 Density of solids 1 and 2 in kgm3

120588119904 Coefficient of the lubricant Bode density

model in kgm3

Stress tensor of equivalent body in Pa120591 Shear stress in Pa 120591 = 120578 sdot 120574

120591119888 Eyring shear stress in Pa

120592 Kinematic viscosity in mm2sΩ

119879 Lubricant domain in the FEM-model (119879)

Ω1198791 Ω

1198792 Solid domains 1 and 2 in the FEM-model(119879)

Ω119875 Domain of Reynolds equation in the

FEM-model (119875119867)Ω

120575 Equivalent body domain in the

FEM-model (119875119867)

Disclosure

Upon request the authors are pleased to offer to provide allthe input parameters for recalculation of the result examplesin Section 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


References

[1] W Habchi A full-system finite element approach to elastohydro-dynamic lubrication problems application to ultra-low-viscosityfluids [Dissertation] Institut National des Sciences Appliqueesde Lyon Lyon France 2008

[2] M Hartinger M-L Dumont S Ioannides D Gosman andH Spikes ldquoCFD modeling of a thermal and shear-thinningelastohydrodynamic line contactrdquo Journal of Tribology vol 130no 4 Article ID 041503 2008

[3] A A LubrechtThenumerical solution of the elastohydrodynam-ically lubricated line- and point contact problem using multigridtechniques [PhD dissertation] University of Twente EnschedeThe Netherlands 1987

[4] C H VennerMultilevel solution of the EHL line and point con-tact problems [Dissertation] University of Twente EnschedeThe Netherlands 1991

[5] J Raisin N Fillot D Dureisseix P Vergne and V LacourldquoCharacteristic times in transient thermal elastohydrodynamicline contactsrdquoTribology International vol 82 pp 472ndash483 2015

[6] W Habchi ldquoA numerical model for the solution of thermalelastohydrodynamic lubrication in coated circular contactsrdquoTribology International vol 73 pp 57ndash68 2014

[7] Comsol COMSOLMultiphysics Reference Manual version 512015

[8] X Tan C E Goodyer P K Jimack R I Taylor and M AWalkley ldquoComputational approaches for modelling elastohy-drodynamic lubrication using multiphysics softwarerdquo Proceed-ings of the Institution of Mechanical Engineers Part J Journal ofEngineering Tribology vol 226 no 6 pp 463ndash480 2012

[9] H Hertz ldquoContact theory of rigid and elastic bodiesrdquo JournalFur die Reine undAngewandteMathematik vol 92 pp 156ndash1711881

[10] D Bartel Simulation von Tribosystemen Grundlagen undAnwendungen Habilitation Treatise Vieweg+Teubner Wies-baden Germany 1st edition 2009

[11] P Yang and S Wen ldquoA generalized Reynolds equation for non-Newtonian thermal elastohydrodynamic lubricationrdquo Journal ofTribology vol 112 no 4 pp 631ndash636 1990

[12] S R Wu ldquoA penalty formulation and numerical approximationof the Reynolds-Hertz problem of elastohydrodynamic lubrica-tionrdquo International Journal of Engineering Science vol 24 no 6pp 1001ndash1013 1986

[13] O Reynolds ldquoOn the theory of the lubrication and its appli-cation to Mr Beauchamps Towerrsquos experiments including anexperimental determination of the viscosity of olive oilrdquo Philo-sophical Transactions of the Royal Society of London vol 177 pp157ndash234 1886

[14] J Wang and P Yang ldquoA numerical analysis for TEHL ofeccentric-tappet pair subjected to transient loadrdquo Journal ofTribology vol 125 no 4 pp 770ndash779 2003

[15] P Hepermann and R Beilicke Ortliche FresstragfahigkeitmdashBestimmung der Ortlichen Fresstragfahigkeit Einfluss vonSchrag- und Hochverzahnungen Vorhaben-Nr 598 I FVA-Heft 1024 Forschungsvereinigung Antriebstechnik FrankfurtGermany 2012

[16] H Vogel ldquoPrinciple of temperature dependency of viscosity offluidsrdquo Zeitschrift fur Physik vol 22 pp 645ndash647 1921

[17] C J A RoelandsCorrelation aspects of the viscosity-temperaturerelationship of lubricating oil [PhD thesis] TechnischeHogeschool Delft 1966

[18] H Eyring ldquoViscosity plasticity and diffusion as examples ofabsolute reaction ratesrdquoThe Journal of Chemical Physics vol 4no 4 pp 283ndash291 1936

[19] B Bode ldquoModell zur Beschreibung des Flieszligverhaltens vonFlussigkeiten unter hohem Druckrdquo Tribologie und Schmierung-stechnik vol 36 pp 182ndash189 1989

[20] R Larsson and O Andersson ldquoLubricant thermal conductivityand heat capacity under high pressurerdquo Journal of EngineeringTribology vol 214 no 4 pp 337ndash342 2000

[21] MathWorks Matlab and Statistic Toolbox Release 2014 Math-Works Natick Mass USA 2014

[22] Y Wang H Li J Tong and P Yang ldquoTransient thermoelas-tohydrodynamic lubrication analysis of an involute spur gearrdquoTribology International vol 37 no 10 pp 773ndash782 2004

[23] P Yang and S Wen ldquoBehavior of non-newtonian thermal EHLfilm in line contacts at dynamic loadsrdquo Journal of Tribology vol114 no 1 pp 81ndash85 1992

[24] WHabchi D Eyheramendy P Vergne andGMorales-EspejelldquoStabilized fully-coupled finite elements for elastohydrody-namic lubrication problemsrdquoAdvances in Engineering Softwarevol 46 no 1 pp 4ndash18 2012

[25] MUMPS Documentation MUMPSmdashaMUltifrontal MassivelyParallel sparse direct Solver httpmumps-solverorg

[26] P Deuflhard ldquoA modified Newton method for the solution ofill-conditioned systems of nonlinear equations with applicationtomultiple shootingrdquoNumerischeMathematik vol 22 pp 289ndash315 1974

[27] T Lohner J Mayer and K Stahl ldquoEHL contact temperaturemdashcomparison of theoretical and experimental determinationrdquo inProceedings of the STLE 70th Annual Meeting amp ExhibitionDallas Tex USA May 2015

[28] T Lohner R Merz J Mayer K Michaelis M Kopnarski andK Stahl ldquoOn the effect of plastic deformation (PD) additives inlubricantsrdquoTribologie und Schmierungstechnik vol 62 no 2 pp13ndash24 2015

[29] L Bobach D Bartel R Beilicke et al ldquoReduction in EHLfriction by a DLC coatingrdquo Tribology Letters vol 60 article 172015

[30] T Almqvist and R Larsson ldquoThermal transient rough EHL linecontact simulations by aid of computational fluid dynamicsrdquoTribology International vol 41 no 8 pp 683ndash693 2008

[31] Q J Wang D Zhu H S Cheng T Yu X Jiang and S LiuldquoMixed lubrication analyses by a macro-micro approach and afull-scale mixed EHL modelrdquo Journal of Tribology vol 126 no1 pp 81ndash91 2004

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Control Scienceand Engineering

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Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



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Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



v

Gear contact

(2)

(1)

EA

Wheel (2)

Pinion (1)

A

Model contact Equivalent contact

1 12 2

ET2

T2

T1T1

FNleff

Rx

1 2Eeq eq

Figure 1 Line contact in a spur gear and its simplification to a model (disc) contact and to an equivalent contact

software COMSOL Multiphysics to stimulate and acceler-ate research in computational tribology The possibility ofdeveloping a complex TEHL model with moderate effort isdemonstrated in this paper

2 Theory

In this section the theory and governing equations for sim-ulation of TEHL contacts are addressed For simplification atwo-dimensional line contact is considered

21 General Contact conditions of for example gears arecharacterised by varying load motion and geometry alongthe path of contact This transient contact can be interpretedas a transient model contact of two rolling elements withvarying load motion and geometry for each meshing pointIn TEHL simulations the transient contact is usually furthersimplified to an equivalent contact between a single (inelas-tic) roller and an elastic flat bodyThis is adopted in this studyand exemplarily shown for a line contact of a spur gear inFigure 1

Characteristic quantities of the equivalent contact aredescribed by the Hertzian contact parameters for example119901

119867 119887

119867 and 119877

119909[9] Poissonrsquos ratio ]eq and Youngrsquos Modulus

119864eq of the equivalent elastic flat body are calculated accordingto Habchi [1] The kinematics are characterised by the sumvelocity V

Σand the siding velocity V

119892 whereby for all analyses

in this study the velocity of the lower body is higher than thevelocity of the upper body V

1gt V

2

V119892= V

1minus V

2

VΣ= V

1+ V

2

(1)

The coefficient of friction 120583 of the TEHL contact is evaluatedby integrating the shear stress in the middle of the lubricantfilm (see also Habchi [1])

120583 =

int119909119890119909

119909119894119899

120591119911119909

100381610038161003816100381610038161003816119911=ℎ2

119889119909

119865119873119897eff

(2)

22 Generalised Reynolds Equation Derived from the two-dimensional compressible Navier-Stokes equations by con-sidering reasonable assumptions for TEHL contacts [1 10]and no-slip boundary conditions the shear rate 120574

119909and

velocity distribution V119909in the lubricant film can be expressed

by

120574119909 (119909 119911 119905) =

120597V119909

120597119911

=1

120578

120597119901

120597119909(119911 minus

intℎ

0(119911120578) 119889119911

intℎ

0(1120578) 119889119911

)

+V2minus V

1

120578 intℎ

0(1120578) 119889119911

(3)

V119909 (119909 119911 119905) =

120597119901

120597119909(int

119911

0

120578119889 minus int

119911

0

1

120578119889int

ℎ

0(119911120578) 119889119911

intℎ

0(1120578) 119889119911

)

+V2minus V

1

intℎ

0(1120578) 119889119911

int

119911

0

1

120578119889 + V

1

(4)

Inserting the velocity equation in the integrated transientcompressible continuity equation results in the generalisedReynolds equation according to Yang and Wen [11]

120597

120597119909[

[

120597119901

120597119909int

ℎ

0

120588(int

119911

0

120578119889 minus int

119911

0

1

120578119889int

ℎ

0(119911120578) 119889119911

intℎ

0(1120578) 119889119911

)119889119911]

]

+120597

120597119909[

[

V2minus V

1

intℎ

0(1120578) 119889119911

int

ℎ

0

120588(int

119911

0

1

120578119889) 119889119911

+ int

ℎ

0

120588 119889119911 sdot V1]

]

+120597

120597119905int

ℎ

0

120588 119889119911 = 0

(5)

In TEHL line contacts the generalised Reynolds equationdetermines the hydrodynamic pressure Due to the inte-grations when deriving the generalised Reynolds equation



119894119899) = 119901(119909



119875and119901(119909cav) =

120597119901120597119909(119909cav) = 0



120575 (119909 119905) =1003816100381610038161003816119880V (119909 119905)

1003816100381610038161003816


119906

119880V)

(6)


120575


119890119909on



120575




ℎ (119909 119905) = ℎ0+119909

2

2119877119909

+ 120575 minusR (7)


0in (7)

intΩ119875

119901 (119909 119905) 119889Ω119875= (

119865119873

119897eff) (119905) (8)


120588119888119901sdot (120597119879

120597119905+ V

119909

120597119879

120597119909) minus (

120597

120597119909(120582

120597119879

120597119909) +

120597

120597119911(120582

120597119879

120597119911))

= minus119879

120588

120597120588

120597119879(120597119901

120597119905+ V

119909

120597119901

120597119909) + 120578(

120597V119909

120597119911)

2

(9)



120588119894119888119901119894sdot (120597119879

120597119905+ V

119894

120597119879

120597119909)

minus (120597

120597119909(120582

119894

120597119879

120597119909) +

120597

120597119911(120582

119894

120597119879

120597119911)) = 0

with 119894 = 1 2

(10)


119879(V

119909(119909

119894119899 119911 119905) ge 0 and V

119909(119909

119890119909 119911 119905) le



(V119909(119909

119894119899 119911 119905) le 0 and V

119909(119909



1198791and Ω

1198792(V

119894(119909

119894119899 119905) ge 0 and

V119894(119909



leaving Ω1198791

and Ω1198792

(V119894(119909

119894119899 119905) le 0 and V

119894(119909

119890119909 119905) ge 0) the


of solids Ω1198791

and Ω1198792





120578 (119879) = 119860 sdot exp( 119861

119862 + (119879 minus 27315K)) (11)


120578 (119879 119901) = 120578 (119879) sdot exp(ln (120578 (119879)) + 967)

sdot [minus1 + (1 +119901

1199011205780

)

119911(119879)

]

(12)


according to

119911 (119879) =120572

119901 (119879) sdot 1199011205780

ln (120578 (119879)) + 967 (13)


dependent

120572119901 (119879) = 119864120572

1199011sdot exp (119864

1205721199012sdot 119879) (14)



120578 (119879 119901 120574119909) =

120591119888

120574119909

asinh(120578 (119879 119901) sdot 120574

119909

120591119888

)

with 120591 (119879 119901 120574119909) = 120578 (119879 119901 120574) sdot 120574

119909

(15)


120588 (119879 119901)

=120588119904sdot (1 minus 120572

119904sdot 119879)

1 minus 1198631205880sdot ln ((119863

1205881+ 119863

1205882sdot 119879 + 119901) (119863

1205881+ 119863

1205882sdot 119879))

(16)




119901sdot 120588)(119879 119901) are

120582 (119901) = 1205820sdot (1 +

1198891205821sdot 119901

1 + 1198891205822sdot 119901) (17)

(119888119901sdot 120588) (119879 119901) = (119888

119901sdot 120588) (295K) sdot (1 +

1198601198881sdot 119901

1 + 1198601198882sdot 119901)

sdot [1 + 1198601198883sdot (1 + 119860

1198884sdot 119901 + 119860

1198885sdot 119901

2) sdot (119879 minus 295K)]

(18)






119883 =119909

119887119867

119879 =119879

119879119872

119863 =119889

119887119867

119875 =119901

119901119867

119867 =ℎ sdot 119877

119909

1198872119867

120575 =120575 sdot 119877

119909

1198872119867

120588 =120588

120588 (119879119872)

120578 =120578

120578 (119879119872)

119905 =119905 sdot V

Σ

2119887119867

119862V =VΣ (119905)

VΣ (119905 = 0)

119862119877=

119877119909 (119905)

119877119909 (119905 = 0)

119862119908=

(119865119873119897eff) (119905)

(119865119873119897eff) (119905 = 0)

119885 =

119911

119887119867

for solids

119911

ℎfor lubricant

(19)




Output









Input

No

No

No

No

Yes

Yes

Yes

Yes

Glo

bal l

oop

Tim

e loo

p

FEM-model (PH)

FEM-model (PH)

FEM-model (T)



Converged P


Converged P and T

Converged T













120575and Ω















119894119899= minus45 to 119883

119890119909= 3

Thereby119883119890119909ofΩ



0



Z = 1 + D

Z = 1

Z = 0

Z = minusD60

60

0

Lubricant

Solid 1

Solid 2

Z

X

Z

X

Ω120575

ΩP

Xin Xex

Xin Xex

ΩT1

ΩT

ΩT2







minus intΩ119875

((120576 + 119868119863)120597119875

120597119883)120597119882

119875

120597119883119889119883

minus intΩ119875

(120588119867)119867

120597119882119875

120597119883119889119883 + int

Ω119875

120597 (120588119867)119905

120597119905119882

119875119889Ω

119875

+ intΩ119875

119876120585119882

119875119889Ω

119875+

119878119880119875119866

119866119871119878 = 0

with 119878119880119875119866

119866119871119878 = 119878119880119875119866 or 119866119871119878

(20)



120585=

119875119891sdot 119875






dimensionless form

120576 =119867

3

120582int

1

0

120588(int

119885

0

119885

120578119889119885 minus int

119885

0

1

120578119889119885

int1

0(119885120578) 119889119885

int1

0(1120578) 119889119885

)119889119885

with 120582 =119877

2

119909sdot 120578 (119879

119872) sdot V

Σ

2 sdot 119901119867sdot 1198873

119867

(120588119867)119867=2119867

VΣ

(V2minus V

1

int1

0(1120578) 119889119885

int

1

0

120588int

119885

0

1

120578119889119885119889119885

+ int

1

0

120588V1119889119885)

(120588119867)119905= 119867int

1

0

120588 119889119885

(21)






119879 Ω

1198791 and Ω



119879has


1198791and Ω




119894119899to119883




Ω1198791



119879


1198791andΩ









1198791andΩ




119872= 1 As


119879and Ω

1198791and Ω

1198792


1198791and Ω








=119911ℎ

119911119887119867

=119877

119909

119867 sdot 119887119867

(22)



1198791and Ω

1198792 Whereas


1198791and Ω

1198792






0



4 Model Validation




Σ= 5ms V

119892=



119889=



5 Example Results





x (mm)

0

01

02

03

04

05

06p

(GPa

)

p (EHL)h (EHL)

p (TEHL)h (TEHL)

h(120583

m)

0

01

02

03

04

05pH = 570Nmm2

Rx = 001mΣ = 5msg = 25ms1205780 = 001Pas

(a)

Hartinger et al [2]

x (mm)

h(120583

m)


01

02

03

04

05SRR = 11205780 = 001



0

01

02

03

04

05

06

p(G

Pa)

(b)


0

03

06

09

12

15

18

21

P

0

003

006

009

012

015

018

021

H

Xd = minus025


X

pH = 2000Nmm2

Rx = 00139mΣ = 194msg = 097ms1205780 = 004Pas

(a)

Venner [4]

0

03

06

09

12

15

18

21

P

0

003

006

009

012

015

018

021

H


X

(b)



1198791and Ω










1= 119903




h(120583

m)

0

02

04

06

08

1

12

14

16

18

2

0

200

400

600

800

1000

1200

1400

1600

1800

2000


x (mm)

p(N

mm

2)

pH = 1200Nmm2

Rx = 002mΣ = 8msg = 089ms120599M = 895∘C

(a)

16MnCr5

16MnCr5

16MnCr5

Surface

Depth of

Gap

hei

ght d

irect

ionz

(120583m

)

ΩΩ2 (2)

Ω1 (1)

126∘C

SiO2

15 120583m

Depth of 15 120583m

90

95

100

105

110

115

120

125

130

135

Tem

pera

ture

(∘C)

minus3

minus2

minus1

0

1

2

3

minus085 0 042 085minus042

x (mm)

(b)



Lohner et al [27]


119888= 6Nmm2



119892=

0 008 021 042 064 089 141 200 267ms are thereby



119864119894in Nmm2 206000

]119894

030120588119894in kgm3 7760

120582119894in WmK 44

119888119901119894



0in WmK 0137

119888119901(15∘C) in JkgK 1921

120591119888in Nmm2 6

MeasuredSimulated

Coe

ffici

ent o

f fric

tion120583

pH = 1200Nmm2

Σ = 8ms120599oil = 40∘C

0

001

002

003

004

005

006

Rx = 002m



119892


119872= 42 47 56 69 80 90 109 123 140

∘C



119867=

1200Nmm2 V119892= 089ms V

Σ= 8ms and 120599

119872= 895



02 120583m50120583m 30120583m


h(120583

m)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

p(N

mm

2)

0

02

04

06

08

1

12

14

16

18

2


x (mm)

pH = 1200Nmm2

Rx = 002mΣ = 8msg = 089ms120599M = 895∘C

(a)

Surface

Gap

hei

ght d

irect

ionz

(120583m

)

ΩΩ2 (2)

Ω1 (1)

120∘C

90

95

100

105

110

115

120

125

130

135

Tem

pera

ture

(∘C)


x (mm)

minus2

minus1

0

1

2

(b)



Lubricant

Ω2 (2)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

0

01

02

03

04

05

Mat

eria

l dep

th (m

m)

minus085 minus042 0 042 085x (mm)

(a)

Lubricant

Ω1 (1)minus042 0 042 085minus085

x (mm)

05

04

03

02

01

0

Mat

eria

l dep

th (m

m)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

(b)



6 Conclusion




Nomenclature

11986011988815




119887119867= 4 sdot 119877

119909sdot 119901

119867119864

1015840


J(kgK)1198881199011 119888



(119879) in m119889

1205821 119889




1198631205880 119863

1205881 and 119863


1198641 119864


in Pa119864

1205721199011 119864




2

1119864

2(1 + 120592

2)2+ 119864

2

2119864

1(1 +

1205921)2)[119864

1(1 + 120592

2) + 119864

2(1 + 120592

1)]

2sdot

119887119867(119877

119909sdot 119901

119867)


1198641015840= 2 sdot 119864

2sdot 119864

1(119864


2

2) +

1198642sdot (1 minus 120592

2

1))








(1199010Roe = 196 sdot 10

8 Pa)119877


119877119909= 119903

1sdot 119903

2(119903

1+ 119903

2)




= (119880119906



V1 V






119894119899 119909


119875in m



Greek Symbols






W(mK)120582

1 120582





]2) +119864

2]1(1 + ]

1))(119864

1(1 + ]

2) +119864

2(1 + ]

1))


1205881 120588



model in kgm3





Ω1198791 Ω



FEM-model (119875119867)Ω


FEM-model (119875119867)

Disclosure





References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119894119899) = 119901(119909



119875and119901(119909cav) =

120597119901120597119909(119909cav) = 0



120575 (119909 119905) =1003816100381610038161003816119880V (119909 119905)

1003816100381610038161003816


119906

119880V)

(6)


120575


119890119909on



120575




ℎ (119909 119905) = ℎ0+119909

2

2119877119909

+ 120575 minusR (7)


0in (7)

intΩ119875

119901 (119909 119905) 119889Ω119875= (

119865119873

119897eff) (119905) (8)


120588119888119901sdot (120597119879

120597119905+ V

119909

120597119879

120597119909) minus (

120597

120597119909(120582

120597119879

120597119909) +

120597

120597119911(120582

120597119879

120597119911))

= minus119879

120588

120597120588

120597119879(120597119901

120597119905+ V

119909

120597119901

120597119909) + 120578(

120597V119909

120597119911)

2

(9)



120588119894119888119901119894sdot (120597119879

120597119905+ V

119894

120597119879

120597119909)

minus (120597

120597119909(120582

119894

120597119879

120597119909) +

120597

120597119911(120582

119894

120597119879

120597119911)) = 0

with 119894 = 1 2

(10)


119879(V

119909(119909

119894119899 119911 119905) ge 0 and V

119909(119909

119890119909 119911 119905) le



(V119909(119909

119894119899 119911 119905) le 0 and V

119909(119909



1198791and Ω

1198792(V

119894(119909

119894119899 119905) ge 0 and

V119894(119909



leaving Ω1198791

and Ω1198792

(V119894(119909

119894119899 119905) le 0 and V

119894(119909

119890119909 119905) ge 0) the


of solids Ω1198791

and Ω1198792





120578 (119879) = 119860 sdot exp( 119861

119862 + (119879 minus 27315K)) (11)


120578 (119879 119901) = 120578 (119879) sdot exp(ln (120578 (119879)) + 967)

sdot [minus1 + (1 +119901

1199011205780

)

119911(119879)

]

(12)


according to

119911 (119879) =120572

119901 (119879) sdot 1199011205780

ln (120578 (119879)) + 967 (13)


dependent

120572119901 (119879) = 119864120572

1199011sdot exp (119864

1205721199012sdot 119879) (14)



120578 (119879 119901 120574119909) =

120591119888

120574119909

asinh(120578 (119879 119901) sdot 120574

119909

120591119888

)

with 120591 (119879 119901 120574119909) = 120578 (119879 119901 120574) sdot 120574

119909

(15)


120588 (119879 119901)

=120588119904sdot (1 minus 120572

119904sdot 119879)

1 minus 1198631205880sdot ln ((119863

1205881+ 119863

1205882sdot 119879 + 119901) (119863

1205881+ 119863

1205882sdot 119879))

(16)




119901sdot 120588)(119879 119901) are

120582 (119901) = 1205820sdot (1 +

1198891205821sdot 119901

1 + 1198891205822sdot 119901) (17)

(119888119901sdot 120588) (119879 119901) = (119888

119901sdot 120588) (295K) sdot (1 +

1198601198881sdot 119901

1 + 1198601198882sdot 119901)

sdot [1 + 1198601198883sdot (1 + 119860

1198884sdot 119901 + 119860

1198885sdot 119901

2) sdot (119879 minus 295K)]

(18)






119883 =119909

119887119867

119879 =119879

119879119872

119863 =119889

119887119867

119875 =119901

119901119867

119867 =ℎ sdot 119877

119909

1198872119867

120575 =120575 sdot 119877

119909

1198872119867

120588 =120588

120588 (119879119872)

120578 =120578

120578 (119879119872)

119905 =119905 sdot V

Σ

2119887119867

119862V =VΣ (119905)

VΣ (119905 = 0)

119862119877=

119877119909 (119905)

119877119909 (119905 = 0)

119862119908=

(119865119873119897eff) (119905)

(119865119873119897eff) (119905 = 0)

119885 =

119911

119887119867

for solids

119911

ℎfor lubricant

(19)




Output









Input

No

No

No

No

Yes

Yes

Yes

Yes

Glo

bal l

oop

Tim

e loo

p

FEM-model (PH)

FEM-model (PH)

FEM-model (T)



Converged P


Converged P and T

Converged T













120575and Ω















119894119899= minus45 to 119883

119890119909= 3

Thereby119883119890119909ofΩ



0



Z = 1 + D

Z = 1

Z = 0

Z = minusD60

60

0

Lubricant

Solid 1

Solid 2

Z

X

Z

X

Ω120575

ΩP

Xin Xex

Xin Xex

ΩT1

ΩT

ΩT2







minus intΩ119875

((120576 + 119868119863)120597119875

120597119883)120597119882

119875

120597119883119889119883

minus intΩ119875

(120588119867)119867

120597119882119875

120597119883119889119883 + int

Ω119875

120597 (120588119867)119905

120597119905119882

119875119889Ω

119875

+ intΩ119875

119876120585119882

119875119889Ω

119875+

119878119880119875119866

119866119871119878 = 0

with 119878119880119875119866

119866119871119878 = 119878119880119875119866 or 119866119871119878

(20)



120585=

119875119891sdot 119875






dimensionless form

120576 =119867

3

120582int

1

0

120588(int

119885

0

119885

120578119889119885 minus int

119885

0

1

120578119889119885

int1

0(119885120578) 119889119885

int1

0(1120578) 119889119885

)119889119885

with 120582 =119877

2

119909sdot 120578 (119879

119872) sdot V

Σ

2 sdot 119901119867sdot 1198873

119867

(120588119867)119867=2119867

VΣ

(V2minus V

1

int1

0(1120578) 119889119885

int

1

0

120588int

119885

0

1

120578119889119885119889119885

+ int

1

0

120588V1119889119885)

(120588119867)119905= 119867int

1

0

120588 119889119885

(21)






119879 Ω

1198791 and Ω



119879has


1198791and Ω




119894119899to119883




Ω1198791



119879


1198791andΩ









1198791andΩ




119872= 1 As


119879and Ω

1198791and Ω

1198792


1198791and Ω








=119911ℎ

119911119887119867

=119877

119909

119867 sdot 119887119867

(22)



1198791and Ω

1198792 Whereas


1198791and Ω

1198792






0



4 Model Validation




Σ= 5ms V

119892=



119889=



5 Example Results





x (mm)

0

01

02

03

04

05

06p

(GPa

)

p (EHL)h (EHL)

p (TEHL)h (TEHL)

h(120583

m)

0

01

02

03

04

05pH = 570Nmm2

Rx = 001mΣ = 5msg = 25ms1205780 = 001Pas

(a)

Hartinger et al [2]

x (mm)

h(120583

m)


01

02

03

04

05SRR = 11205780 = 001



0

01

02

03

04

05

06

p(G

Pa)

(b)


0

03

06

09

12

15

18

21

P

0

003

006

009

012

015

018

021

H

Xd = minus025


X

pH = 2000Nmm2

Rx = 00139mΣ = 194msg = 097ms1205780 = 004Pas

(a)

Venner [4]

0

03

06

09

12

15

18

21

P

0

003

006

009

012

015

018

021

H


X

(b)



1198791and Ω










1= 119903




h(120583

m)

0

02

04

06

08

1

12

14

16

18

2

0

200

400

600

800

1000

1200

1400

1600

1800

2000


x (mm)

p(N

mm

2)

pH = 1200Nmm2

Rx = 002mΣ = 8msg = 089ms120599M = 895∘C

(a)

16MnCr5

16MnCr5

16MnCr5

Surface

Depth of

Gap

hei

ght d

irect

ionz

(120583m

)

ΩΩ2 (2)

Ω1 (1)

126∘C

SiO2

15 120583m

Depth of 15 120583m

90

95

100

105

110

115

120

125

130

135

Tem

pera

ture

(∘C)

minus3

minus2

minus1

0

1

2

3

minus085 0 042 085minus042

x (mm)

(b)



Lohner et al [27]


119888= 6Nmm2



119892=

0 008 021 042 064 089 141 200 267ms are thereby



119864119894in Nmm2 206000

]119894

030120588119894in kgm3 7760

120582119894in WmK 44

119888119901119894



0in WmK 0137

119888119901(15∘C) in JkgK 1921

120591119888in Nmm2 6

MeasuredSimulated

Coe

ffici

ent o

f fric

tion120583

pH = 1200Nmm2

Σ = 8ms120599oil = 40∘C

0

001

002

003

004

005

006

Rx = 002m



119892


119872= 42 47 56 69 80 90 109 123 140

∘C



119867=

1200Nmm2 V119892= 089ms V

Σ= 8ms and 120599

119872= 895



02 120583m50120583m 30120583m


h(120583

m)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

p(N

mm

2)

0

02

04

06

08

1

12

14

16

18

2


x (mm)

pH = 1200Nmm2

Rx = 002mΣ = 8msg = 089ms120599M = 895∘C

(a)

Surface

Gap

hei

ght d

irect

ionz

(120583m

)

ΩΩ2 (2)

Ω1 (1)

120∘C

90

95

100

105

110

115

120

125

130

135

Tem

pera

ture

(∘C)


x (mm)

minus2

minus1

0

1

2

(b)



Lubricant

Ω2 (2)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

0

01

02

03

04

05

Mat

eria

l dep

th (m

m)

minus085 minus042 0 042 085x (mm)

(a)

Lubricant

Ω1 (1)minus042 0 042 085minus085

x (mm)

05

04

03

02

01

0

Mat

eria

l dep

th (m

m)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

(b)



6 Conclusion




Nomenclature

11986011988815




119887119867= 4 sdot 119877

119909sdot 119901

119867119864

1015840


J(kgK)1198881199011 119888



(119879) in m119889

1205821 119889




1198631205880 119863

1205881 and 119863


1198641 119864


in Pa119864

1205721199011 119864




2

1119864

2(1 + 120592

2)2+ 119864

2

2119864

1(1 +

1205921)2)[119864

1(1 + 120592

2) + 119864

2(1 + 120592

1)]

2sdot

119887119867(119877

119909sdot 119901

119867)


1198641015840= 2 sdot 119864

2sdot 119864

1(119864


2

2) +

1198642sdot (1 minus 120592

2

1))








(1199010Roe = 196 sdot 10

8 Pa)119877


119877119909= 119903

1sdot 119903

2(119903

1+ 119903

2)




= (119880119906



V1 V






119894119899 119909


119875in m



Greek Symbols






W(mK)120582

1 120582





]2) +119864

2]1(1 + ]

1))(119864

1(1 + ]

2) +119864

2(1 + ]

1))


1205881 120588



model in kgm3





Ω1198791 Ω



FEM-model (119875119867)Ω


FEM-model (119875119867)

Disclosure





References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120578 (119879 119901 120574119909) =

120591119888

120574119909

asinh(120578 (119879 119901) sdot 120574

119909

120591119888

)

with 120591 (119879 119901 120574119909) = 120578 (119879 119901 120574) sdot 120574

119909

(15)


120588 (119879 119901)

=120588119904sdot (1 minus 120572

119904sdot 119879)

1 minus 1198631205880sdot ln ((119863

1205881+ 119863

1205882sdot 119879 + 119901) (119863

1205881+ 119863

1205882sdot 119879))

(16)




119901sdot 120588)(119879 119901) are

120582 (119901) = 1205820sdot (1 +

1198891205821sdot 119901

1 + 1198891205822sdot 119901) (17)

(119888119901sdot 120588) (119879 119901) = (119888

119901sdot 120588) (295K) sdot (1 +

1198601198881sdot 119901

1 + 1198601198882sdot 119901)

sdot [1 + 1198601198883sdot (1 + 119860

1198884sdot 119901 + 119860

1198885sdot 119901

2) sdot (119879 minus 295K)]

(18)






119883 =119909

119887119867

119879 =119879

119879119872

119863 =119889

119887119867

119875 =119901

119901119867

119867 =ℎ sdot 119877

119909

1198872119867

120575 =120575 sdot 119877

119909

1198872119867

120588 =120588

120588 (119879119872)

120578 =120578

120578 (119879119872)

119905 =119905 sdot V

Σ

2119887119867

119862V =VΣ (119905)

VΣ (119905 = 0)

119862119877=

119877119909 (119905)

119877119909 (119905 = 0)

119862119908=

(119865119873119897eff) (119905)

(119865119873119897eff) (119905 = 0)

119885 =

119911

119887119867

for solids

119911

ℎfor lubricant

(19)




Output









Input

No

No

No

No

Yes

Yes

Yes

Yes

Glo

bal l

oop

Tim

e loo

p

FEM-model (PH)

FEM-model (PH)

FEM-model (T)



Converged P


Converged P and T

Converged T













120575and Ω















119894119899= minus45 to 119883

119890119909= 3

Thereby119883119890119909ofΩ



0



Z = 1 + D

Z = 1

Z = 0

Z = minusD60

60

0

Lubricant

Solid 1

Solid 2

Z

X

Z

X

Ω120575

ΩP

Xin Xex

Xin Xex

ΩT1

ΩT

ΩT2







minus intΩ119875

((120576 + 119868119863)120597119875

120597119883)120597119882

119875

120597119883119889119883

minus intΩ119875

(120588119867)119867

120597119882119875

120597119883119889119883 + int

Ω119875

120597 (120588119867)119905

120597119905119882

119875119889Ω

119875

+ intΩ119875

119876120585119882

119875119889Ω

119875+

119878119880119875119866

119866119871119878 = 0

with 119878119880119875119866

119866119871119878 = 119878119880119875119866 or 119866119871119878

(20)



120585=

119875119891sdot 119875






dimensionless form

120576 =119867

3

120582int

1

0

120588(int

119885

0

119885

120578119889119885 minus int

119885

0

1

120578119889119885

int1

0(119885120578) 119889119885

int1

0(1120578) 119889119885

)119889119885

with 120582 =119877

2

119909sdot 120578 (119879

119872) sdot V

Σ

2 sdot 119901119867sdot 1198873

119867

(120588119867)119867=2119867

VΣ

(V2minus V

1

int1

0(1120578) 119889119885

int

1

0

120588int

119885

0

1

120578119889119885119889119885

+ int

1

0

120588V1119889119885)

(120588119867)119905= 119867int

1

0

120588 119889119885

(21)






119879 Ω

1198791 and Ω



119879has


1198791and Ω




119894119899to119883




Ω1198791



119879


1198791andΩ









1198791andΩ




119872= 1 As


119879and Ω

1198791and Ω

1198792


1198791and Ω








=119911ℎ

119911119887119867

=119877

119909

119867 sdot 119887119867

(22)



1198791and Ω

1198792 Whereas


1198791and Ω

1198792






0



4 Model Validation




Σ= 5ms V

119892=



119889=



5 Example Results





x (mm)

0

01

02

03

04

05

06p

(GPa

)

p (EHL)h (EHL)

p (TEHL)h (TEHL)

h(120583

m)

0

01

02

03

04

05pH = 570Nmm2

Rx = 001mΣ = 5msg = 25ms1205780 = 001Pas

(a)

Hartinger et al [2]

x (mm)

h(120583

m)


01

02

03

04

05SRR = 11205780 = 001



0

01

02

03

04

05

06

p(G

Pa)

(b)


0

03

06

09

12

15

18

21

P

0

003

006

009

012

015

018

021

H

Xd = minus025


X

pH = 2000Nmm2

Rx = 00139mΣ = 194msg = 097ms1205780 = 004Pas

(a)

Venner [4]

0

03

06

09

12

15

18

21

P

0

003

006

009

012

015

018

021

H


X

(b)



1198791and Ω










1= 119903




h(120583

m)

0

02

04

06

08

1

12

14

16

18

2

0

200

400

600

800

1000

1200

1400

1600

1800

2000


x (mm)

p(N

mm

2)

pH = 1200Nmm2

Rx = 002mΣ = 8msg = 089ms120599M = 895∘C

(a)

16MnCr5

16MnCr5

16MnCr5

Surface

Depth of

Gap

hei

ght d

irect

ionz

(120583m

)

ΩΩ2 (2)

Ω1 (1)

126∘C

SiO2

15 120583m

Depth of 15 120583m

90

95

100

105

110

115

120

125

130

135

Tem

pera

ture

(∘C)

minus3

minus2

minus1

0

1

2

3

minus085 0 042 085minus042

x (mm)

(b)



Lohner et al [27]


119888= 6Nmm2



119892=

0 008 021 042 064 089 141 200 267ms are thereby



119864119894in Nmm2 206000

]119894

030120588119894in kgm3 7760

120582119894in WmK 44

119888119901119894



0in WmK 0137

119888119901(15∘C) in JkgK 1921

120591119888in Nmm2 6

MeasuredSimulated

Coe

ffici

ent o

f fric

tion120583

pH = 1200Nmm2

Σ = 8ms120599oil = 40∘C

0

001

002

003

004

005

006

Rx = 002m



119892


119872= 42 47 56 69 80 90 109 123 140

∘C



119867=

1200Nmm2 V119892= 089ms V

Σ= 8ms and 120599

119872= 895



02 120583m50120583m 30120583m


h(120583

m)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

p(N

mm

2)

0

02

04

06

08

1

12

14

16

18

2


x (mm)

pH = 1200Nmm2

Rx = 002mΣ = 8msg = 089ms120599M = 895∘C

(a)

Surface

Gap

hei

ght d

irect

ionz

(120583m

)

ΩΩ2 (2)

Ω1 (1)

120∘C

90

95

100

105

110

115

120

125

130

135

Tem

pera

ture

(∘C)


x (mm)

minus2

minus1

0

1

2

(b)



Lubricant

Ω2 (2)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

0

01

02

03

04

05

Mat

eria

l dep

th (m

m)

minus085 minus042 0 042 085x (mm)

(a)

Lubricant

Ω1 (1)minus042 0 042 085minus085

x (mm)

05

04

03

02

01

0

Mat

eria

l dep

th (m

m)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

(b)



6 Conclusion




Nomenclature

11986011988815




119887119867= 4 sdot 119877

119909sdot 119901

119867119864

1015840


J(kgK)1198881199011 119888



(119879) in m119889

1205821 119889




1198631205880 119863

1205881 and 119863


1198641 119864


in Pa119864

1205721199011 119864




2

1119864

2(1 + 120592

2)2+ 119864

2

2119864

1(1 +

1205921)2)[119864

1(1 + 120592

2) + 119864

2(1 + 120592

1)]

2sdot

119887119867(119877

119909sdot 119901

119867)


1198641015840= 2 sdot 119864

2sdot 119864

1(119864


2

2) +

1198642sdot (1 minus 120592

2

1))








(1199010Roe = 196 sdot 10

8 Pa)119877


119877119909= 119903

1sdot 119903

2(119903

1+ 119903

2)




= (119880119906



V1 V






119894119899 119909


119875in m



Greek Symbols






W(mK)120582

1 120582





]2) +119864

2]1(1 + ]

1))(119864

1(1 + ]

2) +119864

2(1 + ]

1))


1205881 120588



model in kgm3





Ω1198791 Ω



FEM-model (119875119867)Ω


FEM-model (119875119867)

Disclosure





References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Output









Input

No

No

No

No

Yes

Yes

Yes

Yes

Glo

bal l

oop

Tim

e loo

p

FEM-model (PH)

FEM-model (PH)

FEM-model (T)



Converged P


Converged P and T

Converged T













120575and Ω















119894119899= minus45 to 119883

119890119909= 3

Thereby119883119890119909ofΩ



0



Z = 1 + D

Z = 1

Z = 0

Z = minusD60

60

0

Lubricant

Solid 1

Solid 2

Z

X

Z

X

Ω120575

ΩP

Xin Xex

Xin Xex

ΩT1

ΩT

ΩT2







minus intΩ119875

((120576 + 119868119863)120597119875

120597119883)120597119882

119875

120597119883119889119883

minus intΩ119875

(120588119867)119867

120597119882119875

120597119883119889119883 + int

Ω119875

120597 (120588119867)119905

120597119905119882

119875119889Ω

119875

+ intΩ119875

119876120585119882

119875119889Ω

119875+

119878119880119875119866

119866119871119878 = 0

with 119878119880119875119866

119866119871119878 = 119878119880119875119866 or 119866119871119878

(20)



120585=

119875119891sdot 119875






dimensionless form

120576 =119867

3

120582int

1

0

120588(int

119885

0

119885

120578119889119885 minus int

119885

0

1

120578119889119885

int1

0(119885120578) 119889119885

int1

0(1120578) 119889119885

)119889119885

with 120582 =119877

2

119909sdot 120578 (119879

119872) sdot V

Σ

2 sdot 119901119867sdot 1198873

119867

(120588119867)119867=2119867

VΣ

(V2minus V

1

int1

0(1120578) 119889119885

int

1

0

120588int

119885

0

1

120578119889119885119889119885

+ int

1

0

120588V1119889119885)

(120588119867)119905= 119867int

1

0

120588 119889119885

(21)






119879 Ω

1198791 and Ω



119879has


1198791and Ω




119894119899to119883




Ω1198791



119879


1198791andΩ









1198791andΩ




119872= 1 As


119879and Ω

1198791and Ω

1198792


1198791and Ω








=119911ℎ

119911119887119867

=119877

119909

119867 sdot 119887119867

(22)



1198791and Ω

1198792 Whereas


1198791and Ω

1198792






0



4 Model Validation




Σ= 5ms V

119892=



119889=



5 Example Results





x (mm)

0

01

02

03

04

05

06p

(GPa

)

p (EHL)h (EHL)

p (TEHL)h (TEHL)

h(120583

m)

0

01

02

03

04

05pH = 570Nmm2

Rx = 001mΣ = 5msg = 25ms1205780 = 001Pas

(a)

Hartinger et al [2]

x (mm)

h(120583

m)


01

02

03

04

05SRR = 11205780 = 001



0

01

02

03

04

05

06

p(G

Pa)

(b)


0

03

06

09

12

15

18

21

P

0

003

006

009

012

015

018

021

H

Xd = minus025


X

pH = 2000Nmm2

Rx = 00139mΣ = 194msg = 097ms1205780 = 004Pas

(a)

Venner [4]

0

03

06

09

12

15

18

21

P

0

003

006

009

012

015

018

021

H


X

(b)



1198791and Ω










1= 119903




h(120583

m)

0

02

04

06

08

1

12

14

16

18

2

0

200

400

600

800

1000

1200

1400

1600

1800

2000


x (mm)

p(N

mm

2)

pH = 1200Nmm2

Rx = 002mΣ = 8msg = 089ms120599M = 895∘C

(a)

16MnCr5

16MnCr5

16MnCr5

Surface

Depth of

Gap

hei

ght d

irect

ionz

(120583m

)

ΩΩ2 (2)

Ω1 (1)

126∘C

SiO2

15 120583m

Depth of 15 120583m

90

95

100

105

110

115

120

125

130

135

Tem

pera

ture

(∘C)

minus3

minus2

minus1

0

1

2

3

minus085 0 042 085minus042

x (mm)

(b)



Lohner et al [27]


119888= 6Nmm2



119892=

0 008 021 042 064 089 141 200 267ms are thereby



119864119894in Nmm2 206000

]119894

030120588119894in kgm3 7760

120582119894in WmK 44

119888119901119894



0in WmK 0137

119888119901(15∘C) in JkgK 1921

120591119888in Nmm2 6

MeasuredSimulated

Coe

ffici

ent o

f fric

tion120583

pH = 1200Nmm2

Σ = 8ms120599oil = 40∘C

0

001

002

003

004

005

006

Rx = 002m



119892


119872= 42 47 56 69 80 90 109 123 140

∘C



119867=

1200Nmm2 V119892= 089ms V

Σ= 8ms and 120599

119872= 895



02 120583m50120583m 30120583m


h(120583

m)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

p(N

mm

2)

0

02

04

06

08

1

12

14

16

18

2


x (mm)

pH = 1200Nmm2

Rx = 002mΣ = 8msg = 089ms120599M = 895∘C

(a)

Surface

Gap

hei

ght d

irect

ionz

(120583m

)

ΩΩ2 (2)

Ω1 (1)

120∘C

90

95

100

105

110

115

120

125

130

135

Tem

pera

ture

(∘C)


x (mm)

minus2

minus1

0

1

2

(b)



Lubricant

Ω2 (2)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

0

01

02

03

04

05

Mat

eria

l dep

th (m

m)

minus085 minus042 0 042 085x (mm)

(a)

Lubricant

Ω1 (1)minus042 0 042 085minus085

x (mm)

05

04

03

02

01

0

Mat

eria

l dep

th (m

m)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

(b)



6 Conclusion




Nomenclature

11986011988815




119887119867= 4 sdot 119877

119909sdot 119901

119867119864

1015840


J(kgK)1198881199011 119888



(119879) in m119889

1205821 119889




1198631205880 119863

1205881 and 119863


1198641 119864


in Pa119864

1205721199011 119864




2

1119864

2(1 + 120592

2)2+ 119864

2

2119864

1(1 +

1205921)2)[119864

1(1 + 120592

2) + 119864

2(1 + 120592

1)]

2sdot

119887119867(119877

119909sdot 119901

119867)


1198641015840= 2 sdot 119864

2sdot 119864

1(119864


2

2) +

1198642sdot (1 minus 120592

2

1))








(1199010Roe = 196 sdot 10

8 Pa)119877


119877119909= 119903

1sdot 119903

2(119903

1+ 119903

2)




= (119880119906



V1 V






119894119899 119909


119875in m



Greek Symbols






W(mK)120582

1 120582





]2) +119864

2]1(1 + ]

1))(119864

1(1 + ]

2) +119864

2(1 + ]

1))


1205881 120588



model in kgm3





Ω1198791 Ω



FEM-model (119875119867)Ω


FEM-model (119875119867)

Disclosure





References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Z = 1 + D

Z = 1

Z = 0

Z = minusD60

60

0

Lubricant

Solid 1

Solid 2

Z

X

Z

X

Ω120575

ΩP

Xin Xex

Xin Xex

ΩT1

ΩT

ΩT2







minus intΩ119875

((120576 + 119868119863)120597119875

120597119883)120597119882

119875

120597119883119889119883

minus intΩ119875

(120588119867)119867

120597119882119875

120597119883119889119883 + int

Ω119875

120597 (120588119867)119905

120597119905119882

119875119889Ω

119875

+ intΩ119875

119876120585119882

119875119889Ω

119875+

119878119880119875119866

119866119871119878 = 0

with 119878119880119875119866

119866119871119878 = 119878119880119875119866 or 119866119871119878

(20)



120585=

119875119891sdot 119875






dimensionless form

120576 =119867

3

120582int

1

0

120588(int

119885

0

119885

120578119889119885 minus int

119885

0

1

120578119889119885

int1

0(119885120578) 119889119885

int1

0(1120578) 119889119885

)119889119885

with 120582 =119877

2

119909sdot 120578 (119879

119872) sdot V

Σ

2 sdot 119901119867sdot 1198873

119867

(120588119867)119867=2119867

VΣ

(V2minus V

1

int1

0(1120578) 119889119885

int

1

0

120588int

119885

0

1

120578119889119885119889119885

+ int

1

0

120588V1119889119885)

(120588119867)119905= 119867int

1

0

120588 119889119885

(21)






119879 Ω

1198791 and Ω



119879has


1198791and Ω




119894119899to119883




Ω1198791



119879


1198791andΩ









1198791andΩ




119872= 1 As


119879and Ω

1198791and Ω

1198792


1198791and Ω








=119911ℎ

119911119887119867

=119877

119909

119867 sdot 119887119867

(22)



1198791and Ω

1198792 Whereas


1198791and Ω

1198792






0



4 Model Validation




Σ= 5ms V

119892=



119889=



5 Example Results





x (mm)

0

01

02

03

04

05

06p

(GPa

)

p (EHL)h (EHL)

p (TEHL)h (TEHL)

h(120583

m)

0

01

02

03

04

05pH = 570Nmm2

Rx = 001mΣ = 5msg = 25ms1205780 = 001Pas

(a)

Hartinger et al [2]

x (mm)

h(120583

m)


01

02

03

04

05SRR = 11205780 = 001



0

01

02

03

04

05

06

p(G

Pa)

(b)


0

03

06

09

12

15

18

21

P

0

003

006

009

012

015

018

021

H

Xd = minus025


X

pH = 2000Nmm2

Rx = 00139mΣ = 194msg = 097ms1205780 = 004Pas

(a)

Venner [4]

0

03

06

09

12

15

18

21

P

0

003

006

009

012

015

018

021

H


X

(b)



1198791and Ω










1= 119903




h(120583

m)

0

02

04

06

08

1

12

14

16

18

2

0

200

400

600

800

1000

1200

1400

1600

1800

2000


x (mm)

p(N

mm

2)

pH = 1200Nmm2

Rx = 002mΣ = 8msg = 089ms120599M = 895∘C

(a)

16MnCr5

16MnCr5

16MnCr5

Surface

Depth of

Gap

hei

ght d

irect

ionz

(120583m

)

ΩΩ2 (2)

Ω1 (1)

126∘C

SiO2

15 120583m

Depth of 15 120583m

90

95

100

105

110

115

120

125

130

135

Tem

pera

ture

(∘C)

minus3

minus2

minus1

0

1

2

3

minus085 0 042 085minus042

x (mm)

(b)



Lohner et al [27]


119888= 6Nmm2



119892=

0 008 021 042 064 089 141 200 267ms are thereby



119864119894in Nmm2 206000

]119894

030120588119894in kgm3 7760

120582119894in WmK 44

119888119901119894



0in WmK 0137

119888119901(15∘C) in JkgK 1921

120591119888in Nmm2 6

MeasuredSimulated

Coe

ffici

ent o

f fric

tion120583

pH = 1200Nmm2

Σ = 8ms120599oil = 40∘C

0

001

002

003

004

005

006

Rx = 002m



119892


119872= 42 47 56 69 80 90 109 123 140

∘C



119867=

1200Nmm2 V119892= 089ms V

Σ= 8ms and 120599

119872= 895



02 120583m50120583m 30120583m


h(120583

m)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

p(N

mm

2)

0

02

04

06

08

1

12

14

16

18

2


x (mm)

pH = 1200Nmm2

Rx = 002mΣ = 8msg = 089ms120599M = 895∘C

(a)

Surface

Gap

hei

ght d

irect

ionz

(120583m

)

ΩΩ2 (2)

Ω1 (1)

120∘C

90

95

100

105

110

115

120

125

130

135

Tem

pera

ture

(∘C)


x (mm)

minus2

minus1

0

1

2

(b)



Lubricant

Ω2 (2)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

0

01

02

03

04

05

Mat

eria

l dep

th (m

m)

minus085 minus042 0 042 085x (mm)

(a)

Lubricant

Ω1 (1)minus042 0 042 085minus085

x (mm)

05

04

03

02

01

0

Mat

eria

l dep

th (m

m)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

(b)



6 Conclusion




Nomenclature

11986011988815




119887119867= 4 sdot 119877

119909sdot 119901

119867119864

1015840


J(kgK)1198881199011 119888



(119879) in m119889

1205821 119889




1198631205880 119863

1205881 and 119863


1198641 119864


in Pa119864

1205721199011 119864




2

1119864

2(1 + 120592

2)2+ 119864

2

2119864

1(1 +

1205921)2)[119864

1(1 + 120592

2) + 119864

2(1 + 120592

1)]

2sdot

119887119867(119877

119909sdot 119901

119867)


1198641015840= 2 sdot 119864

2sdot 119864

1(119864


2

2) +

1198642sdot (1 minus 120592

2

1))








(1199010Roe = 196 sdot 10

8 Pa)119877


119877119909= 119903

1sdot 119903

2(119903

1+ 119903

2)




= (119880119906



V1 V






119894119899 119909


119875in m



Greek Symbols






W(mK)120582

1 120582





]2) +119864

2]1(1 + ]

1))(119864

1(1 + ]

2) +119864

2(1 + ]

1))


1205881 120588



model in kgm3





Ω1198791 Ω



FEM-model (119875119867)Ω


FEM-model (119875119867)

Disclosure





References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














minus intΩ119875

((120576 + 119868119863)120597119875

120597119883)120597119882

119875

120597119883119889119883

minus intΩ119875

(120588119867)119867

120597119882119875

120597119883119889119883 + int

Ω119875

120597 (120588119867)119905

120597119905119882

119875119889Ω

119875

+ intΩ119875

119876120585119882

119875119889Ω

119875+

119878119880119875119866

119866119871119878 = 0

with 119878119880119875119866

119866119871119878 = 119878119880119875119866 or 119866119871119878

(20)



120585=

119875119891sdot 119875






dimensionless form

120576 =119867

3

120582int

1

0

120588(int

119885

0

119885

120578119889119885 minus int

119885

0

1

120578119889119885

int1

0(119885120578) 119889119885

int1

0(1120578) 119889119885

)119889119885

with 120582 =119877

2

119909sdot 120578 (119879

119872) sdot V

Σ

2 sdot 119901119867sdot 1198873

119867

(120588119867)119867=2119867

VΣ

(V2minus V

1

int1

0(1120578) 119889119885

int

1

0

120588int

119885

0

1

120578119889119885119889119885

+ int

1

0

120588V1119889119885)

(120588119867)119905= 119867int

1

0

120588 119889119885

(21)






119879 Ω

1198791 and Ω



119879has


1198791and Ω




119894119899to119883




Ω1198791



119879


1198791andΩ









1198791andΩ




119872= 1 As


119879and Ω

1198791and Ω

1198792


1198791and Ω








=119911ℎ

119911119887119867

=119877

119909

119867 sdot 119887119867

(22)



1198791and Ω

1198792 Whereas


1198791and Ω

1198792






0



4 Model Validation




Σ= 5ms V

119892=



119889=



5 Example Results





x (mm)

0

01

02

03

04

05

06p

(GPa

)

p (EHL)h (EHL)

p (TEHL)h (TEHL)

h(120583

m)

0

01

02

03

04

05pH = 570Nmm2

Rx = 001mΣ = 5msg = 25ms1205780 = 001Pas

(a)

Hartinger et al [2]

x (mm)

h(120583

m)


01

02

03

04

05SRR = 11205780 = 001



0

01

02

03

04

05

06

p(G

Pa)

(b)


0

03

06

09

12

15

18

21

P

0

003

006

009

012

015

018

021

H

Xd = minus025


X

pH = 2000Nmm2

Rx = 00139mΣ = 194msg = 097ms1205780 = 004Pas

(a)

Venner [4]

0

03

06

09

12

15

18

21

P

0

003

006

009

012

015

018

021

H


X

(b)



1198791and Ω










1= 119903




h(120583

m)

0

02

04

06

08

1

12

14

16

18

2

0

200

400

600

800

1000

1200

1400

1600

1800

2000


x (mm)

p(N

mm

2)

pH = 1200Nmm2

Rx = 002mΣ = 8msg = 089ms120599M = 895∘C

(a)

16MnCr5

16MnCr5

16MnCr5

Surface

Depth of

Gap

hei

ght d

irect

ionz

(120583m

)

ΩΩ2 (2)

Ω1 (1)

126∘C

SiO2

15 120583m

Depth of 15 120583m

90

95

100

105

110

115

120

125

130

135

Tem

pera

ture

(∘C)

minus3

minus2

minus1

0

1

2

3

minus085 0 042 085minus042

x (mm)

(b)



Lohner et al [27]


119888= 6Nmm2



119892=

0 008 021 042 064 089 141 200 267ms are thereby



119864119894in Nmm2 206000

]119894

030120588119894in kgm3 7760

120582119894in WmK 44

119888119901119894



0in WmK 0137

119888119901(15∘C) in JkgK 1921

120591119888in Nmm2 6

MeasuredSimulated

Coe

ffici

ent o

f fric

tion120583

pH = 1200Nmm2

Σ = 8ms120599oil = 40∘C

0

001

002

003

004

005

006

Rx = 002m



119892


119872= 42 47 56 69 80 90 109 123 140

∘C



119867=

1200Nmm2 V119892= 089ms V

Σ= 8ms and 120599

119872= 895



02 120583m50120583m 30120583m


h(120583

m)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

p(N

mm

2)

0

02

04

06

08

1

12

14

16

18

2


x (mm)

pH = 1200Nmm2

Rx = 002mΣ = 8msg = 089ms120599M = 895∘C

(a)

Surface

Gap

hei

ght d

irect

ionz

(120583m

)

ΩΩ2 (2)

Ω1 (1)

120∘C

90

95

100

105

110

115

120

125

130

135

Tem

pera

ture

(∘C)


x (mm)

minus2

minus1

0

1

2

(b)



Lubricant

Ω2 (2)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

0

01

02

03

04

05

Mat

eria

l dep

th (m

m)

minus085 minus042 0 042 085x (mm)

(a)

Lubricant

Ω1 (1)minus042 0 042 085minus085

x (mm)

05

04

03

02

01

0

Mat

eria

l dep

th (m

m)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

(b)



6 Conclusion




Nomenclature

11986011988815




119887119867= 4 sdot 119877

119909sdot 119901

119867119864

1015840


J(kgK)1198881199011 119888



(119879) in m119889

1205821 119889




1198631205880 119863

1205881 and 119863


1198641 119864


in Pa119864

1205721199011 119864




2

1119864

2(1 + 120592

2)2+ 119864

2

2119864

1(1 +

1205921)2)[119864

1(1 + 120592

2) + 119864

2(1 + 120592

1)]

2sdot

119887119867(119877

119909sdot 119901

119867)


1198641015840= 2 sdot 119864

2sdot 119864

1(119864


2

2) +

1198642sdot (1 minus 120592

2

1))








(1199010Roe = 196 sdot 10

8 Pa)119877


119877119909= 119903

1sdot 119903

2(119903

1+ 119903

2)




= (119880119906



V1 V






119894119899 119909


119875in m



Greek Symbols






W(mK)120582

1 120582





]2) +119864

2]1(1 + ]

1))(119864

1(1 + ]

2) +119864

2(1 + ]

1))


1205881 120588



model in kgm3





Ω1198791 Ω



FEM-model (119875119867)Ω


FEM-model (119875119867)

Disclosure





References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












=119911ℎ

119911119887119867

=119877

119909

119867 sdot 119887119867

(22)



1198791and Ω

1198792 Whereas


1198791and Ω

1198792






0



4 Model Validation




Σ= 5ms V

119892=



119889=



5 Example Results





x (mm)

0

01

02

03

04

05

06p

(GPa

)

p (EHL)h (EHL)

p (TEHL)h (TEHL)

h(120583

m)

0

01

02

03

04

05pH = 570Nmm2

Rx = 001mΣ = 5msg = 25ms1205780 = 001Pas

(a)

Hartinger et al [2]

x (mm)

h(120583

m)


01

02

03

04

05SRR = 11205780 = 001



0

01

02

03

04

05

06

p(G

Pa)

(b)


0

03

06

09

12

15

18

21

P

0

003

006

009

012

015

018

021

H

Xd = minus025


X

pH = 2000Nmm2

Rx = 00139mΣ = 194msg = 097ms1205780 = 004Pas

(a)

Venner [4]

0

03

06

09

12

15

18

21

P

0

003

006

009

012

015

018

021

H


X

(b)



1198791and Ω










1= 119903




h(120583

m)

0

02

04

06

08

1

12

14

16

18

2

0

200

400

600

800

1000

1200

1400

1600

1800

2000


x (mm)

p(N

mm

2)

pH = 1200Nmm2

Rx = 002mΣ = 8msg = 089ms120599M = 895∘C

(a)

16MnCr5

16MnCr5

16MnCr5

Surface

Depth of

Gap

hei

ght d

irect

ionz

(120583m

)

ΩΩ2 (2)

Ω1 (1)

126∘C

SiO2

15 120583m

Depth of 15 120583m

90

95

100

105

110

115

120

125

130

135

Tem

pera

ture

(∘C)

minus3

minus2

minus1

0

1

2

3

minus085 0 042 085minus042

x (mm)

(b)



Lohner et al [27]


119888= 6Nmm2



119892=

0 008 021 042 064 089 141 200 267ms are thereby



119864119894in Nmm2 206000

]119894

030120588119894in kgm3 7760

120582119894in WmK 44

119888119901119894



0in WmK 0137

119888119901(15∘C) in JkgK 1921

120591119888in Nmm2 6

MeasuredSimulated

Coe

ffici

ent o

f fric

tion120583

pH = 1200Nmm2

Σ = 8ms120599oil = 40∘C

0

001

002

003

004

005

006

Rx = 002m



119892


119872= 42 47 56 69 80 90 109 123 140

∘C



119867=

1200Nmm2 V119892= 089ms V

Σ= 8ms and 120599

119872= 895



02 120583m50120583m 30120583m


h(120583

m)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

p(N

mm

2)

0

02

04

06

08

1

12

14

16

18

2


x (mm)

pH = 1200Nmm2

Rx = 002mΣ = 8msg = 089ms120599M = 895∘C

(a)

Surface

Gap

hei

ght d

irect

ionz

(120583m

)

ΩΩ2 (2)

Ω1 (1)

120∘C

90

95

100

105

110

115

120

125

130

135

Tem

pera

ture

(∘C)


x (mm)

minus2

minus1

0

1

2

(b)



Lubricant

Ω2 (2)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

0

01

02

03

04

05

Mat

eria

l dep

th (m

m)

minus085 minus042 0 042 085x (mm)

(a)

Lubricant

Ω1 (1)minus042 0 042 085minus085

x (mm)

05

04

03

02

01

0

Mat

eria

l dep

th (m

m)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

(b)



6 Conclusion




Nomenclature

11986011988815




119887119867= 4 sdot 119877

119909sdot 119901

119867119864

1015840


J(kgK)1198881199011 119888



(119879) in m119889

1205821 119889




1198631205880 119863

1205881 and 119863


1198641 119864


in Pa119864

1205721199011 119864




2

1119864

2(1 + 120592

2)2+ 119864

2

2119864

1(1 +

1205921)2)[119864

1(1 + 120592

2) + 119864

2(1 + 120592

1)]

2sdot

119887119867(119877

119909sdot 119901

119867)


1198641015840= 2 sdot 119864

2sdot 119864

1(119864


2

2) +

1198642sdot (1 minus 120592

2

1))








(1199010Roe = 196 sdot 10

8 Pa)119877


119877119909= 119903

1sdot 119903

2(119903

1+ 119903

2)




= (119880119906



V1 V






119894119899 119909


119875in m



Greek Symbols






W(mK)120582

1 120582





]2) +119864

2]1(1 + ]

1))(119864

1(1 + ]

2) +119864

2(1 + ]

1))


1205881 120588



model in kgm3





Ω1198791 Ω



FEM-model (119875119867)Ω


FEM-model (119875119867)

Disclosure





References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











x (mm)

0

01

02

03

04

05

06p

(GPa

)

p (EHL)h (EHL)

p (TEHL)h (TEHL)

h(120583

m)

0

01

02

03

04

05pH = 570Nmm2

Rx = 001mΣ = 5msg = 25ms1205780 = 001Pas

(a)

Hartinger et al [2]

x (mm)

h(120583

m)


01

02

03

04

05SRR = 11205780 = 001



0

01

02

03

04

05

06

p(G

Pa)

(b)


0

03

06

09

12

15

18

21

P

0

003

006

009

012

015

018

021

H

Xd = minus025


X

pH = 2000Nmm2

Rx = 00139mΣ = 194msg = 097ms1205780 = 004Pas

(a)

Venner [4]

0

03

06

09

12

15

18

21

P

0

003

006

009

012

015

018

021

H


X

(b)



1198791and Ω










1= 119903




h(120583

m)

0

02

04

06

08

1

12

14

16

18

2

0

200

400

600

800

1000

1200

1400

1600

1800

2000


x (mm)

p(N

mm

2)

pH = 1200Nmm2

Rx = 002mΣ = 8msg = 089ms120599M = 895∘C

(a)

16MnCr5

16MnCr5

16MnCr5

Surface

Depth of

Gap

hei

ght d

irect

ionz

(120583m

)

ΩΩ2 (2)

Ω1 (1)

126∘C

SiO2

15 120583m

Depth of 15 120583m

90

95

100

105

110

115

120

125

130

135

Tem

pera

ture

(∘C)

minus3

minus2

minus1

0

1

2

3

minus085 0 042 085minus042

x (mm)

(b)



Lohner et al [27]


119888= 6Nmm2



119892=

0 008 021 042 064 089 141 200 267ms are thereby



119864119894in Nmm2 206000

]119894

030120588119894in kgm3 7760

120582119894in WmK 44

119888119901119894



0in WmK 0137

119888119901(15∘C) in JkgK 1921

120591119888in Nmm2 6

MeasuredSimulated

Coe

ffici

ent o

f fric

tion120583

pH = 1200Nmm2

Σ = 8ms120599oil = 40∘C

0

001

002

003

004

005

006

Rx = 002m



119892


119872= 42 47 56 69 80 90 109 123 140

∘C



119867=

1200Nmm2 V119892= 089ms V

Σ= 8ms and 120599

119872= 895



02 120583m50120583m 30120583m


h(120583

m)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

p(N

mm

2)

0

02

04

06

08

1

12

14

16

18

2


x (mm)

pH = 1200Nmm2

Rx = 002mΣ = 8msg = 089ms120599M = 895∘C

(a)

Surface

Gap

hei

ght d

irect

ionz

(120583m

)

ΩΩ2 (2)

Ω1 (1)

120∘C

90

95

100

105

110

115

120

125

130

135

Tem

pera

ture

(∘C)


x (mm)

minus2

minus1

0

1

2

(b)



Lubricant

Ω2 (2)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

0

01

02

03

04

05

Mat

eria

l dep

th (m

m)

minus085 minus042 0 042 085x (mm)

(a)

Lubricant

Ω1 (1)minus042 0 042 085minus085

x (mm)

05

04

03

02

01

0

Mat

eria

l dep

th (m

m)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

(b)



6 Conclusion




Nomenclature

11986011988815




119887119867= 4 sdot 119877

119909sdot 119901

119867119864

1015840


J(kgK)1198881199011 119888



(119879) in m119889

1205821 119889




1198631205880 119863

1205881 and 119863


1198641 119864


in Pa119864

1205721199011 119864




2

1119864

2(1 + 120592

2)2+ 119864

2

2119864

1(1 +

1205921)2)[119864

1(1 + 120592

2) + 119864

2(1 + 120592

1)]

2sdot

119887119867(119877

119909sdot 119901

119867)


1198641015840= 2 sdot 119864

2sdot 119864

1(119864


2

2) +

1198642sdot (1 minus 120592

2

1))








(1199010Roe = 196 sdot 10

8 Pa)119877


119877119909= 119903

1sdot 119903

2(119903

1+ 119903

2)




= (119880119906



V1 V






119894119899 119909


119875in m



Greek Symbols






W(mK)120582

1 120582





]2) +119864

2]1(1 + ]

1))(119864

1(1 + ]

2) +119864

2(1 + ]

1))


1205881 120588



model in kgm3





Ω1198791 Ω



FEM-model (119875119867)Ω


FEM-model (119875119867)

Disclosure





References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










h(120583

m)

0

02

04

06

08

1

12

14

16

18

2

0

200

400

600

800

1000

1200

1400

1600

1800

2000


x (mm)

p(N

mm

2)

pH = 1200Nmm2

Rx = 002mΣ = 8msg = 089ms120599M = 895∘C

(a)

16MnCr5

16MnCr5

16MnCr5

Surface

Depth of

Gap

hei

ght d

irect

ionz

(120583m

)

ΩΩ2 (2)

Ω1 (1)

126∘C

SiO2

15 120583m

Depth of 15 120583m

90

95

100

105

110

115

120

125

130

135

Tem

pera

ture

(∘C)

minus3

minus2

minus1

0

1

2

3

minus085 0 042 085minus042

x (mm)

(b)



Lohner et al [27]


119888= 6Nmm2



119892=

0 008 021 042 064 089 141 200 267ms are thereby



119864119894in Nmm2 206000

]119894

030120588119894in kgm3 7760

120582119894in WmK 44

119888119901119894



0in WmK 0137

119888119901(15∘C) in JkgK 1921

120591119888in Nmm2 6

MeasuredSimulated

Coe

ffici

ent o

f fric

tion120583

pH = 1200Nmm2

Σ = 8ms120599oil = 40∘C

0

001

002

003

004

005

006

Rx = 002m



119892


119872= 42 47 56 69 80 90 109 123 140

∘C



119867=

1200Nmm2 V119892= 089ms V

Σ= 8ms and 120599

119872= 895



02 120583m50120583m 30120583m


h(120583

m)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

p(N

mm

2)

0

02

04

06

08

1

12

14

16

18

2


x (mm)

pH = 1200Nmm2

Rx = 002mΣ = 8msg = 089ms120599M = 895∘C

(a)

Surface

Gap

hei

ght d

irect

ionz

(120583m

)

ΩΩ2 (2)

Ω1 (1)

120∘C

90

95

100

105

110

115

120

125

130

135

Tem

pera

ture

(∘C)


x (mm)

minus2

minus1

0

1

2

(b)



Lubricant

Ω2 (2)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

0

01

02

03

04

05

Mat

eria

l dep

th (m

m)

minus085 minus042 0 042 085x (mm)

(a)

Lubricant

Ω1 (1)minus042 0 042 085minus085

x (mm)

05

04

03

02

01

0

Mat

eria

l dep

th (m

m)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

(b)



6 Conclusion




Nomenclature

11986011988815




119887119867= 4 sdot 119877

119909sdot 119901

119867119864

1015840


J(kgK)1198881199011 119888



(119879) in m119889

1205821 119889




1198631205880 119863

1205881 and 119863


1198641 119864


in Pa119864

1205721199011 119864




2

1119864

2(1 + 120592

2)2+ 119864

2

2119864

1(1 +

1205921)2)[119864

1(1 + 120592

2) + 119864

2(1 + 120592

1)]

2sdot

119887119867(119877

119909sdot 119901

119867)


1198641015840= 2 sdot 119864

2sdot 119864

1(119864


2

2) +

1198642sdot (1 minus 120592

2

1))








(1199010Roe = 196 sdot 10

8 Pa)119877


119877119909= 119903

1sdot 119903

2(119903

1+ 119903

2)




= (119880119906



V1 V






119894119899 119909


119875in m



Greek Symbols






W(mK)120582

1 120582





]2) +119864

2]1(1 + ]

1))(119864

1(1 + ]

2) +119864

2(1 + ]

1))


1205881 120588



model in kgm3





Ω1198791 Ω



FEM-model (119875119867)Ω


FEM-model (119875119867)

Disclosure





References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










02 120583m50120583m 30120583m


h(120583

m)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

p(N

mm

2)

0

02

04

06

08

1

12

14

16

18

2


x (mm)

pH = 1200Nmm2

Rx = 002mΣ = 8msg = 089ms120599M = 895∘C

(a)

Surface

Gap

hei

ght d

irect

ionz

(120583m

)

ΩΩ2 (2)

Ω1 (1)

120∘C

90

95

100

105

110

115

120

125

130

135

Tem

pera

ture

(∘C)


x (mm)

minus2

minus1

0

1

2

(b)



Lubricant

Ω2 (2)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

0

01

02

03

04

05

Mat

eria

l dep

th (m

m)

minus085 minus042 0 042 085x (mm)

(a)

Lubricant

Ω1 (1)minus042 0 042 085minus085

x (mm)

05

04

03

02

01

0

Mat

eria

l dep

th (m

m)

0

100

200

300

400

500

600

700

von

Mise

s stre

ss (M

Pa)

(b)



6 Conclusion




Nomenclature

11986011988815




119887119867= 4 sdot 119877

119909sdot 119901

119867119864

1015840


J(kgK)1198881199011 119888



(119879) in m119889

1205821 119889




1198631205880 119863

1205881 and 119863


1198641 119864


in Pa119864

1205721199011 119864




2

1119864

2(1 + 120592

2)2+ 119864

2

2119864

1(1 +

1205921)2)[119864

1(1 + 120592

2) + 119864

2(1 + 120592

1)]

2sdot

119887119867(119877

119909sdot 119901

119867)


1198641015840= 2 sdot 119864

2sdot 119864

1(119864


2

2) +

1198642sdot (1 minus 120592

2

1))








(1199010Roe = 196 sdot 10

8 Pa)119877


119877119909= 119903

1sdot 119903

2(119903

1+ 119903

2)




= (119880119906



V1 V






119894119899 119909


119875in m



Greek Symbols






W(mK)120582

1 120582





]2) +119864

2]1(1 + ]

1))(119864

1(1 + ]

2) +119864

2(1 + ]

1))


1205881 120588



model in kgm3





Ω1198791 Ω



FEM-model (119875119867)Ω


FEM-model (119875119867)

Disclosure





References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Nomenclature

11986011988815




119887119867= 4 sdot 119877

119909sdot 119901

119867119864

1015840


J(kgK)1198881199011 119888



(119879) in m119889

1205821 119889




1198631205880 119863

1205881 and 119863


1198641 119864


in Pa119864

1205721199011 119864




2

1119864

2(1 + 120592

2)2+ 119864

2

2119864

1(1 +

1205921)2)[119864

1(1 + 120592

2) + 119864

2(1 + 120592

1)]

2sdot

119887119867(119877

119909sdot 119901

119867)


1198641015840= 2 sdot 119864

2sdot 119864

1(119864


2

2) +

1198642sdot (1 minus 120592

2

1))








(1199010Roe = 196 sdot 10

8 Pa)119877


119877119909= 119903

1sdot 119903

2(119903

1+ 119903

2)




= (119880119906



V1 V






119894119899 119909


119875in m



Greek Symbols






W(mK)120582

1 120582





]2) +119864

2]1(1 + ]

1))(119864

1(1 + ]

2) +119864

2(1 + ]

1))


1205881 120588



model in kgm3





Ω1198791 Ω



FEM-model (119875119867)Ω


FEM-model (119875119867)

Disclosure





References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article engineering software solution for thermal...

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