I Taylor Modern Approaches Tribological Modelling

Shell Global Solutions

Modern Approaches to Tribological Modelling

Ian Taylor

IET Meeting on Green Tribology

9th June 2009 © Shell Petroleum Company Ltd. All rights reserved. Do not distribute without consent of

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Outline

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Reynolds’ equation & analytical solutions

Numerical solution of Reynolds’ equation

Using Excel for lubrication problems

Using MATLAB for lubrication problems

Using general purpose finite element software for lubrication problems

Conclusions


Reynolds’ Equation & Analytical Solutions

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Original paper can be downloaded from:http://www.archive.org/details/papersonmechanic02reynrich

th

xhU

yPh

yxPh

x ∂∂

+∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

21212

33

ηη

• h(x,y) = oil film thickness• η= viscosity• P(x,y) = pressure• U = U1+U2 where U1, U2 are speeds of surfaces in x-direction

http://www.archive.org/details/papersonmechanic02reynrich


Reynolds’ Equation & Analytical Solutions Following the derivation of the Reynolds’ equation, many analytical

studieswere carried out:

Sommerfeld (1904) – solution of journal bearing problemMichell (1905) – solution of thrust bearing problemMartin’s analysis (1916) of gear lubrication showed that oil film

thicknesses were too small for effective lubricationWe had to wait until 1953 (!) for the “short bearing” approximation*.

As may be expected if the bearing width is small compared to it’s diameter, it is a “short bearing” – this is typical of many journal bearings. Therefore the pressure variation across the width of the bearing is much greater than that along the bearing. Simplification of the Reynolds’ equation is possible and a simple equation relates the eccentricity ratio to the load

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http://naca.central.cranfield.ac.uk/reports/1953/naca-report-1157.pdf

11164)1(

22222

3+⎟

⎠⎞

⎜⎝⎛ −

−= ε

ππ

εηε

cLUW

http://naca.central.cranfield.ac.uk/reports/1953/naca-report-1157.pdf


Reynolds’ Equation & Analytical Solutions Martin’s observations about gear lubrication led Grubin (1949) to

propose that in certain high pressure lubricated contacts, surfaces may deform under pressure and lubricant properties may change. He then found an analytical expression for the oil film thickness in the contact under such conditions

5

( )11/1

11/811/8

95.1−

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

ERwE

ERU

Rhcen αη

Even recently (2002), analytical work can lead to new results e.g. friction loss of short bearing in high load limit*

* R.I. Taylor, SAE 2002-01-3355


Numerical Solution of Reynolds’Equation

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Most complicated numerical solutions are for ElastohydrodynamicLubrication – graph below shows approx computation time versus year

Simple static EHD line contact, no thermal effects1959 – Dowson quotes a time of 18 months for EHD line contact direct

solver*Lubrecht, in his PhD thesis quotes a computation time of 120 minutes for

an EHD line contact, with 240 nodes: http://www.tr.ctw.utwente.nl/Research/Publications/PhDTheses/thesis_lubrecht.pdf

Shell EHD line contact direct solver with adaptive mesh) – 10 secs for 322 points

* R.W Snidle & H.P. Evans (Eds.), “IUTAM Symposium on Elastohydrodynamics....”, Springer 2006

http://www.tr.ctw.utwente.nl/Research/Publications/PhDTheses/thesis_lubrecht.pdf


Numerical Solution of Reynolds’EquationDirect solvers – generally computation time takes O(N3) for N mesh

points (and memory requirement is O(N2)) – although these can be smaller for modern iterative direct solvers

Trend over last 20 years has been to use Multigrid solvers, whose computation time is O(NlogN)

Shell have Windows™ based EHD line contact solvers using both (1) a direct solver with an adaptive mesh, and (2) a multigrid solver

Shell EHD Line Contact Direct Solver: Written in C, number of files = 58, total number of lines of code = 31,040 (includes solver code)

EHD Line Contact Multigrid Solver (from Leeds University): Written in C, number of files = 146, total number of lines of code = 20,599

A lot of effort & expertise is needed to maintain these codes

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Numerical Solution of Reynolds’Equation

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* G.W. Roper & R.I. Taylor, STLE Meeting, Calgary 2006

Shell direct solver with adaptive mesh: comparison with multigridDesktop PC, Intel E6750 Duo Core, 2 x 2.66 GHz, 4 GB RAM, Vista

PremiumNote times are approx 3-4x faster than reported in 2006*

M=14.89L = 10.83

Shell direct solver, adaptive mesh Multigrid methodN Time

(secs)hmin (μm) hcen (μm)

162 1.31 0.397 0.457322 10.17 0.399 0.460642 140.3 0.400 0.461

N Time (secs)

hmin (μm) hcen (μm)

512 9.99 0.417 0.4812048 17.21 0.410 0.4708192 33.16 0.410 0.468


Numerical Solution of Reynolds’EquationShell direct solver with adaptive mesh: comparison with multigrid

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Numerical Solution of Reynolds’EquationFor the prediction of friction, we need to include both thermal effects

AND the effects of lubricant shear thinning: therefore speed androbustness are important

Simulations below are for a line contact*

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* G.W. Roper & R.I. Taylor, STLE Meeting, Calgary 2006

0.00

0.02

0.04

0.06

0.08

0.01 0.1 1 10 100

Entrainment speed (m.s-1)

Fric

tion

coef

ficie

nt

High Vk mineral oil, 0.1 MN/m, 70°C


Low Vk synthetic oil, 0.1 MN/m, 70°C


Effect of temperature Effect of load

0.00

0.02

0.04

0.06

0.08

0.01 0.1 1 10 100

Entrainment speed (m.s-1)Fr

ictio

n co

effic

ient






Numerical Solution of Reynolds’EquationShell Multigrid Point Contact SolverWritten in C and

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FORTRANNumber of lines of code approx 14,000 , number of files = 148

Grid is N x N/2: (symmetry is assumed in program)

w = 100 N, uA = 0, uB = 2 m/s, R = 0.0125 m, ηo = 85.6 mPa.s , α = 1.79e-8

M ≈ 134.4, L ≈ 10.3

N Time (secs)

hmin (μm) hcen (μm)

64 ≈ 5 0.0205 0.0453128 ≈ 9 0.0209 0.0442256 ≈ 28 0.0206 0.0429


Numerical Solution of Reynolds’EquationShell Multigrid Point Contact Solver

Typical results shown below (N = 128) – graphics done in gnuplot

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Oil Film Thickness Pressure


Using Excel for Lubrication Problems

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Excel is useful for:

Solving the finite difference form of the Reynolds equation, eg for journal bearings

It can be used as a “front-end” for legacy programs which run from a Windows command line



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Finite difference form of Reynolds equation:

(j, i) (j, i+1)(j, i-1)

(j-1, i)

(j+1, i)

Δx

Δy

( )

( ) ( )⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

Δ

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

Δ

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

Δ

−−

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

Δ

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

Δ

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

=

−+−+

−+−

−+

+−

−+

+

2,2/1

3

,2/1

3

22/1,

3

2/1,

3

1,1,2

,,2/1

3

,1,2/1

3

2

1,2/1,

3

1,2/1,

3

,

)2(6

)()(

y

hh

x

hh

xhh

Uy

PhPh

x

PhPh

P

ijijijij

ijijij

ijij

ijij

ijij

ij

ij

ηηηη

ηηηη

Typical VBA code (in Excel) for implementing the above is:fact1 = 0.5 * ((Sheet2.Cells(j + 1, i + 1) ^ 3 / vis) + (Sheet2.Cells(j + 1, i + 2) ^ 3) / vis)fact2 = 0.5 * ((Sheet2.Cells(j + 1, i + 1) ^ 3 / vis) + (Sheet2.Cells(j + 1, i) ^ 3 / vis))fact3 = 0.5 * ((Sheet2.Cells(j + 1, i + 1) ^ 3 / vis) + (Sheet2.Cells(j + 2, i + 1) ^ 3 / vis))fact4 = 0.5 * ((Sheet2.Cells(j + 1, i + 1) ^ 3 / vis) + (Sheet2.Cells(j, i + 1) ^ 3 / vis))old_value = a(j, i)new_value = ((fact1 * a(j, i + 1) + fact2 * a(j, i - 1)) / (dx ^ 2)) + _

((fact3 * a(j + 1, i) + fact4 * a(j - 1, i)) / (dy ^ 2)) - _(6# * speed) * (Sheet2.Cells(j + 1, i + 2) - Sheet2.Cells(j + 1, i)) / (2# * dx)

new_value = new_value / (((fact1 + fact2) / (dx ^ 2)) + ((fact3 + fact4) / (dy ^ 2)))



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Using Excel as “Front-End” for legacy programs

Example below shows Excel being used as “Front-End” for EHD Line Solver


Using MATLAB for Lubrication ProblemsSimple MATLAB code for EHD line lubrication

Number of MATLAB m-files = 17

Total number of lines of code = 506 Approx 40 times less code than for C code

Uses MATLAB in-built direct solver - gmres

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Using MATLAB for Lubrication ProblemsScreenshot of running code: 257 grid points, 18 secondsRun times would be longer for more grid points

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Using General Purpose Finite Element Solvers for Lubrication Problems

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General purpose finite element solvers are now available commercially

Habchi has reported the use of COMSOL™ for elastohydrodynamic line and point contact lubrication problems: http://www.civil-comp.com/conf/habchi.pdf

He has reported that an EHD point contact, with lubricant shearthinning takes 10-20 minutes to run on a modern PC with a 2 GHz processor. Inclusion of thermal effects would result in a run-time of 30-60 minutes

http://www.civil-comp.com/conf/habchi.pdf

http://www.civil-comp.com/conf/habchi.pdf



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Example below shows the use of COMSOL™ for a simple slider bearingAdvantage of this approach is that very little “code” needs writingComplicated geometries can also be handled easily



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Equations can be inserted into COMSOL without any explicit programming

Different options for the numerical solutions can be specified easilyGood graphics capabilities for plotting out results

Reynolds’ Equation


Discussion

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Tools are becoming available which will enable the more efficient generation of lubrication models – less code, more efficient solvers etc

Direct solvers which are currently available, particular those which can use adaptive meshes, can be used effectively for EHD lubricationproblems

As computer speeds and memory increase, it may be expected thatthere will be a move back towards direct solvers for EHD, from multigrid

Tools such as MATLAB and COMSOL may be increasingly used for lubrication problems, where the numerical side of the code is handled by MATLAB or COMSOL, and the user concentrates on incorporating thecorrect physical properties (of the lubricant and the materials)


Conclusions

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For modelling lubricated contacts, there is no longer a need to develop C or FORTRAN code from scratch

Tools such as Excel, MATLAB, and more recently COMSOL, can greatly improve the efficiency of developing such models

Using these tools will enable the scientist to concentrate on the physics going into the models, rather than the detailed numerical algorithms which are needed to solve the equations

Our experience with EHD software is that a direct solver with an adaptive mesh is much more robust than a multigrid solver, and it is a possibility that there will be a move back to direct solvers for such problems in the coming years


Thank you for your attention

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Any Questions ?

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I Taylor Modern Approaches Tribological Modelling

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