Transient Elastohydrodynamic Lubrication Analysis with time varying Entrainment Velocity using...

International Journal of Engineering, Management & Sciences (IJEMS)ISSN-2348 –3733, Volume-2, Issue-5, May 2015

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Abstract— Transient Elastohydrodynamic Lubrication

(EHL) analysis using Magneto rheological (MR) fluids is carriedout in the present paper. A time varying entrainment velocity istaken into account and its effect on pressure distribution andfilm thickness are considered. MR fluid is defined by theBingham Model and the Reynolds Equation is also modified toconsider the effect of MR fluids. Effect of variation in oscillatingfrequency, amplitude and yield stress have been considered andit is found that film thickness is affected by these variations.Increase in yield stress increases the film thickness thusincreasing the load carrying capacity of the fluid film. Also, theeffect of speed variation is incorporated and it is seen that onincreasing the velocity the film thickness also increases.

Index Terms— Magneto rheological Fluids,Elastohydrodynamic Lubrication, Transient, Bingham Model,Film Thickness.

I. INTRODUCTION

Magnetorheological fluids are the types of fluids alsocalled field-responsive colloids. These fluids are basicallyformed by dispersion of magnetisable carbonyl iron particles(micron-sized) and on the application of external magneticfield the viscosity of these fluids increases rapidly. This is dueto the formation of clusters of particles aligned in the directionof the field [1-4].

In the past, most of the efforts were focused inunderstanding bulk rheological behaviour of materials due toviscometric flows in the presence of external magnetic field.Wong et al. [5] in 2001, published the tribological resultspertaining to an MR fluid (MRF132) for differentconcentrations of iron particles in a block-on-ring tester.Leung et al. [6] later said that “the block has relatively lessdamage in tests with very high (iron) particle concentrations”using two different viscosity base fluids in a block-on-ringtester.

In EHL, elastic deformation of surfaces is considered underhydrodynamic lubrication and in rolling or sliding contact.Here, the surfaces are non-conforming and load is higher. Ahigh pressure area is created due to elastic deformation ofeither one or both of the surfaces. With pressure, there is anincrease in lubricant viscosity. Spur Gears, Cylindrical Roller

Manuscript received February 20, 2015Bearings, Cams and Tappets etc. are few examples of EHL

contacts. Dowson [7] described the experimental

determination and film thickness development inElastohydrodynamic Lubrication line contacts in his paper.Wang and Cheng [8, 9] considered transient problem inelastohydrodynamic lubrication and developed a numericalarrangement using Grubin-type Inlet Zone Analysis. Theypredicted the minimum film thickness at various points on theline of action and also bulk surface temperatures.

In present study, the effect of magnetorheological (MR)fluids is taken into consideration and its effect on pressuredistribution and film thickness is depicted. To consider theeffect of MR fluids, Bingham Model is employed.

II. MATHEMATICAL MODEL

The equations used for the modelling of EHL problem aregiven in dimensionless form below.

A. Modified Bingham ModelMagnetorheological Fluids follow Bingham Plastic Model

which is difficult to incorporate in Reynolds Equation in itsactual form. Therefore, the Modified Bingham Model is givenas,

omaeom

e

/1

(1)

TABLE I. NOMENCLATURE

DIMENSIONAL PARAMETERS

b Half width of Hertzian contact zone (m)

E Effective elastic modulus (Pa)

h Film thickness (m)

h m i n Minimum film thickness (m)

h0 Offset film thickness (m)

p Pressure (Pa)

pH Maximum Hertzian Pressure (Pa)

R Equivalent radius of contact (m).

u Local fluid velocity (m/s)

u0 Average rolling speed (m/s)

ua, ub Velocities of lower and upper surfaces.

v Surface displacement (m).

Transient Elastohydrodynamic LubricationAnalysis with time varying Entrainment Velocity

using Magnetorheological FluidsRajbeer Singh Anand, Punit Kumar

Transient Elastohydrodynamic Lubrication Analysis with time varying Entrainment Velocity usingMagnetorheological Fluids


DIMENSIONAL PARAMETERS

w Applied load per unit length (N/m)

x Abscissa along rolling direction (m)

y Ordinate across the fluid film (m)

Piezo-viscous coefficient (Pa - l )

Shear strain rate across the fluid film

0 Inlet density of the lubricant (k g /m 3 )

Lubricant density at the local pressure (kg/m3)

Shear stress in fluid (Pa)

µ0 Inlet viscosity of the Newtonian fluid (Pa–s)

µ Fluid viscosity (Pa–s)

om Yield Stress for MR fluid.

NON- DIMENSIONAL PARAMETERS

H Non-dimensional film thickness

Hmin Non-dimensional minimum film thickness

X Non-dimensional abscissa

U Non-dimensional speed parameters

P Non-dimensional pressure

Non-dimensional fluid density

W Non-dimensional load parameter

S Slide to roll ratio

v Non-dimensional displacement

H0 Non-dimensional offset film thickness

Non-dimensional shear stress

Non-dimensional viscosity of Newtonian fluid

Xin Inlet boundary co-ordinate.

X0 Outlet boundary co-ordinate.

Hc Non-dimensional central film thickness.

B. Reynolds EquationThe following Reynolds equation is obtained for MR fluids

for which perturbation scheme employed is given in theAppendix.

)(12

/3

ht

hx

uxphx o

m

(2)

Inlet boundary condition

inXXatP 0

Outlet boundary condition

00 XXatXPP

C. Film Thickness Equation

N

jjij PDXHXH

1

12

2

0 (3)

Where, Dij = Influence Coefficients for a uniform mesh size∆X

121ln

21

121ln

21

XjiXji

XjiXjiijD

D. Density Pressure Relationship

p

p9107.11

9106.01

0 (4)

E. Viscosity Pressure RelationshipWe use the Roelands Equation as it can be used for many

types of lubricants.

zpPIn

H.9101.51167.90exp (5)

)67.9(ln101.5 09

z

F. Load Equilibrium Equation

2

0

x

inxPdX (6)

We use Simpson’s Rule for the calculation of aboveintegral which is written in the form,

022

N

jjjPCW (7)

Where,

...7,5,332...6,4,234

13

jXjXjX

jC

III. SOLUTION PROCEDURE

In the present simulation, the solution domain (X) has itsrange from -4 to 1.5 and a grid size (∆X) equal to 0.02. FiniteDifference Method is used to discretize the Reynoldsequation and then Newton-Raphson technique is used to solvethe system of equations. The simulations are carried outkeeping the load, rolling speed and pressure-viscositycoefficient at a constant value. An initial guess for pressuredistribution (P) and offset film thickness (H0) is made at thebeginning of solution, and these are used to calculate filmthickness and the fluid properties (density and viscosity).



Magnetorheological fluids are best described by Binghammodel which is used in the present analysis. Due to thetransient conditions, pressure distribution and film shape arecalculated for several cycles of sinusoidally varyingentrainment velocity till the cyclic pattern of film thicknessvariation repeats itself.

TABLE II. VALUES OF INPUT PARAMETERS

Maximum Hertzian Pressure (pH) 1.3 GPa

Pressure-Viscosity Coefficient () 17*10-9 Pa-1

Equivalent Elastic Modulus (E) 2.2*1011 Pa

Equivalent Radius (R) 0.02 m

Yield Stress (om) 80-400kPa

Inlet Density of fluid (ρ0) 846 kg/m3

Slide to Roll Ratio (S) 0.5

Rolling Speed (u0) 0.5 m/s

Domain, X -4≤X≤1.5

Grid Size, ∆X 0.02

IV. RESULTS AND DISCUSSIONS

From the figures given below it is seen that as the yieldstress of the fluid is increased, the film thickness alsoincreases in all the cases. However, there is minimal effect ofyield stress on pressure distribution.

A. Effect of Oscillating FrequencyVariation of frequency has minimal effect on the pressure

distribution for different fluid velocities as shown in Figs. 1 to3. For film thickness at fluid velocities umean and umax, asshown in Fig. 4 and Fig. 5 respectively, as we increase thefrequency of oscillation the film thickness decreases slightly.But at umin, as shown in Fig. 6, film thickness increases withan increase in frequency.

X

-4 -3 -2 -1 0 1

Pres

sure

,P

0.0

0.2

0.4

0.6

0.8

1.0

f=50Hz, om=80kPa

f=50Hz, om=400kPa

f=100Hz, om=80kPa

f=100Hz, om=400kPa

Fig. 1. Pressure Distribution (P v/s X) for the given combinations ofFrequency (f) and Yield Stress (τom) at umean.

X

-4 -3 -2 -1 0 1

Pres

sure

,P

0.0

0.2

0.4

0.6

0.8

1.0

f=50Hz, om=80kPa

f=50Hz, om=400kPa

f=100Hz, om=80kPa

f=100Hz, om=400kPa

Fig. 2. Pressure Distribution (P v/s X) for the given combinations ofFrequency (f) and Yield Stress (τom) at umax.

X

-4 -3 -2 -1 0 1

Pres

sure

,P

0.0

0.2

0.4

0.6

0.8

1.0

f=50Hz, om=80kPa

f=50Hz, om=400kPa

f=100Hz, om=80kPa

f=100Hz, om=400kPa

Fig. 3. Pressure Distribution (P v/s X) for the given combinations ofFrequency (f) and Yield Stress (τom) at umin.

X

-1.5 -1.0 -0.5 0.0 0.5 1.0

Film

Thi

ckne

ss,H

0.00

0.02

0.04

0.06

0.08

0.10

f=50Hz, om=80kPa

f=50Hz, om=400kPa

f=100Hz, om=80kPa

f=100Hz, om=400kPa

Fig. 4. Film Shape (H v/s X) for the given combinations of Frequency (f)and Yield Stress (τom) at umean.



X

-1.5 -1.0 -0.5 0.0 0.5 1.0

Film

Thi

ckne

ss,H

0.00

0.02

0.04

0.06

0.08

0.10

f=50Hz, om=80kPa

f=50Hz, om=400kPa

f=100Hz, om=80kPa

f=100Hz, om=400kPa

Fig. 5. Film Shape (H v/s X) for the given combinations of Frequency (f)and Yield Stress (τom) at umax.

X

-1.5 -1.0 -0.5 0.0 0.5 1.0

Film

Thi

ckne

ss,H

0.00

0.02

0.04

0.06

0.08

0.10

f=50Hz, om=80kPa

f=50Hz, om=400kPa

f=100Hz, om=80kPa

f=100Hz, om=400kPa

Fig. 6. Film Shape (H v/s X) for the given combinations of Frequency (f)and Yield Stress (τom) at umin.

B. Effect of Oscillation AmplitudeThere is minimal effect of amplitude on the pressure

distribution as observed in Figs. 7 to 9. But there is effect ofamplitude on film thickness which is given in Figs. 10 to 12.At umean, the film thickness increases with decreasingamplitude. But at umax, the film thickness increases as theoscillation amplitude increases. Whereas at umin, the filmthickness first increases with decreasing amplitude and thenbecomes almost equal in the latter part of the domain at aparticular yield stress.

C. Central Film ThicknessAt lower frequency, the film remains thick for a longer

period of time as compared to the film at higher frequency (fora particular value of yield stress) as shown in Fig. 13. But asillustrated by Fig. 14, there is no such effect of amplitude oncentral film thickness.

X

-4 -3 -2 -1 0 1

Pres

sure

,P

0.0

0.2

0.4

0.6

0.8

1.0

A=0.2, om=80kPa

A=0.2, om=400kPa

A=0.4, om=80kPa

A=0.4, om=400kPa

Fig. 7. Pressure Distribution (P v/s X) for the given combinations ofAmplitude (A) and Yield Stress (τom) at umean.

X

-4 -3 -2 -1 0 1

Pres

sure

, P

0.0

0.2

0.4

0.6

0.8

1.0

A=0.2, om=80kPa

A=0.2, om=400kPa

A=0.4, om=80kPa

A=0.4, om=400kPa

Fig. 8. Pressure Distribution (P v/s X) for the given combinations ofAmplitude (A) and Yield Stress (τom) at umax.

X

-4 -3 -2 -1 0 1

Pres

sure

,P

0.0

0.2

0.4

0.6

0.8

1.0

A=0.2, om=80kPa

A=0.2, om=400kPa

A=0.4, om=80kPa

A=0.4, om=400kPa

Fig. 9. Pressure Distribution (P v/s X) for the given combinations ofAmplitude (A) and Yield Stress (τom) at umin.



X

-1.5 -1.0 -0.5 0.0 0.5 1.0

Film

Thi

ckne

ss,H

0.00

0.02

0.04

0.06

0.08

0.10

A=0.2, om=80kPa

A=0.2, om=400kPa

A=0.4, om=80kPa

A=0.4, om=400kPa

Fig. 10. Film Shape (H v/s X) for the given combinations of Amplitude (A)and Yield Stress (τom) at umean.

X

-1.5 -1.0 -0.5 0.0 0.5 1.0

Film

Thi

ckne

ss,H

0.00

0.02

0.04

0.06

0.08

0.10

A=0.2, om=80kPa

A=0.2, om=400kPa

A=0.4, om=80kPa

A=0.4, om=400kPa

Fig. 11. Film Shape (H v/s X) for the given combinations of Amplitude (A)and Yield Stress (τom) at umax.

X

-1.5 -1.0 -0.5 0.0 0.5 1.0

Film

Thi

ckne

ss,H

0.00

0.02

0.04

0.06

0.08

0.10

A=0.2, om=80kPa

A=0.2, om=400kPa

A=0.4, om=80kPa

A=0.4, om=400kPa

Fig. 12. Film Shape (H v/s X) for the given combinations of Amplitude (A)and Yield Stress (τom) at umin.

Time, t

0 5 10 15 20 25

Cen

tral F

ilm T

hick

ness

,Hc(n

m)

100

200

300

400

500

600

700

f=50Hz, om=80kPa

f=50Hz, om=400kPa

f=100Hz, om=80kPa

f=100Hz, om=400kPa

Fig. 13. Comparison of Central Film Thickness (Hc) with respect to Time (t)for the given combinations of Yield Stress (τom) and Frequency(f).

Time, t

0 5 10 15 20 25

Cent

ral F

ilm T

hick

ness

,Hc(n

m)

100

200

300

400

500

600

700

A=0.2, om=80kPa

A=0.2, om=400kPa

A=0.4, om=80kPa

A=0.4, om=400kPa

Fig. 14. Comparison of Central Film Thickness (Hc) with respect to Time (t)for the given combinations of Yield Stress (τom) and Amplitude(A).

D. Minimum Film ThicknessThe observations are same as those made for central film

thickness and are observed from Fig. 15 and Fig. 16. Herealso, at lower frequency the film remains thick for a longerperiod of time but there is no effect of amplitude on minimumfilm thickness.

Time, t

0 5 10 15 20 25

Min

imum

Film

Thi

ckne

ss,H

min

(nm

)

100

200

300

400

500

f=50Hz, om=80kPa

f=50Hz, om=400kPa

f=100Hz, om=80kPa

f=100Hz, om=400kPa

Fig. 15. Comparison of Minimum Film Thickness (Hmin) with respect toTime (t) for the given combinations of Yield Stress (τom) and Frequency(f).



Time, t

0 5 10 15 20 25

Min

imum

Film

Thi

ckne

ss,H

min

(nm

)

100

150

200

250

300

350

400

450

500

A=0.2, om=80kPa

A=0.2, om=400kPa

A=0.4, om=80kPa

A=0.4, om=400kPa

Fig. 16. Comparison of Minimum Film Thickness (Hmin) with respect toTime (t) for the given combinations of Yield Stress (τom) and Amplitude(A).

Also, for umax the fluid layer formed remains thickerthroughout the domain as compared to umean and umin. This canbe seen in Figs. 4 to 6 and Figs. 10 to 12. Fluid velocity alsoplays a vital role in film thickness.

V. CONCLUSIONS

In the present study, transient EHL analysis using MRfluids is carried out. Bingham model is used to consider theeffect of MR fluids.1) As observed, there is minimal effect of variation of

frequency, amplitude and yield stress on pressuredistribution.

2) Film thickness increases with an increase in yield stressof the fluid which can be depicted from the above results.

3) The film thickness increases with a decrease in theoscillating frequency for the fluid velocities umean andumax. But for umin, film thickness increases with anincrease in frequency.

4) For umean, the film thickness increases with decreasingamplitude. But for umax, the film thickness increases withan increase in the oscillation amplitude. And for umin, thefilm thickness first increases with a decrease in amplitudeand then becomes almost equal for the same yield stress.

5) Central and Minimum Film Thickness remain thick for alonger time at lower frequency for same yield stress. Butthis is not the case with varying the amplitude.

6) Fluid velocity also has a significant effect on filmthickness, as at umax the film formed is thicker ascompared to those forming at umean and umin.

APPENDIX

A. Perturbation SchemeFor MR fluids, Bingham Model has to be used for which

perturbation scheme is necessary to include its effect.Perturbation method is used to derive the velocity profile oflubricant mixture.

Ie / (A1)

Here, e = Equivalent Viscosity, and yuI /

Now, expansion of velocity in terms of is given by,

10uuu (A2)

10 III (A3)

Where,y

uI

y

uI

1

1,00

Taylor Series expansion of Equivalent Viscosity e in thevicinity of 0I is given below,

1 oe (A4)

Where 00 Ie and0

11I

eI

I

(A5)

The Momentum Equation is given by,

xp

y

(A6)

Using (A1), (A3) and (A5) and neglecting 2 ,

`100100`1010 IIIII (A7)

Expanding p ,

ˆ0 p (A8)

Substituting (A7) and (A8) in (A6),

xy

IIyI

ˆ`10010

0(A9)

020

2

0

yu (A10)

And xy

II

ˆ

`1001 (A11)

Integrating (A10) for the boundary conditions, auu 0

at 0y and buu 0 at hy , we have

yh

uuuu aba

0 (A12)

Here au , bu are the velocities of lower and upper surfacesrespectively and h is the film thickness.Substituting (A5) in (A11), we get

xyu

ˆ21

2(A13)



Where,

000

I

e

II (A14)

Integrating (A13) for the boundary conditions, 01 u at0y and 0bu at hy , gives

x

hyyu

ˆ

2

2

1 (A15)

From (A2), (A8), (A12) and (A14)

xphyyy

huuuu ab

a

2

2(A16)

xpyy

huu

yu ab

22 (A17)

For Bingham Fluid,

I

e Iao

e

0/1

(A18)

20

0

20

000

000

/0

// 1

0

IeIa

IeeIa

I

oo

I

e

oIa

IaIa

omIa

I

e aeI

I /

000

0

1

(A19)

Where, omIam ae /01 (A20)

Using the velocity distribution in the mass continuityequation, the following Reynolds equation is obtained

)(12

/3

ht

hx

uxphx o

m

REFERENCES

[1] Ginder, J.M. (1998). “Behavior of magnetorheological fluids”, MRSBulletin, 26–29.

[2] Rankin, P.J., Ginder, J.M. and Klingenberg, D.J. (1998). “Electro- andmagnetorheology. Curr. Opin”, Colloid Interface, 3, 373–381.

[3] Bossis, G., Volkova, O., Lacis, S. and Meunier, A. (2002).“Magnetorheology: fluids, structures and rheology. In: Odenbach, S.(ed.) Ferrofluids”, Magnetically Controllable Fluids and TheirApplications Lecture Notes in Physics, 594, 202–230.

[4] de Vicente, J., Klingenberg and D.J., Hidalgo-A´ lvarez, R.(2011).“Magnetorheological fluids”, a review. Soft Matter, 7, 3701–3710.

[5] Wong, P.L., Bullough, W.A., Feng, C., Lingard, S. (2001).“Tribological performance of a magneto-rheological suspension”,Wear, 247, 33–40.

[6] Leung, W.C., Bullough, W.A., Wong, P.L., Feng, C. (2004). “Theeffect of particle concentration in a magneto rheological suspension on

the performance of a boundary lubricated contact”, Proc. Inst. Mech.Eng. J., 218, 251–263.

[7] D. Dowson (1995). “Elastohydrodynamic andMicroelastohydrodynamic Lubrication”, Wear, 190, 125-138.

[8] Wang KL, Cheng HS (1981). “A numerical solution to the dynamicload, film thickness and surface temperatures in spur gears”, ASMEJournal of Mechanical Design, 103, 177–187.

[9] Wang KL, Cheng HS (1981). “A numerical solution to the dynamicload, film thickness and surface temperatures in spur gears”, ASMEJournal of Mechanical Design, 103, 188–94.

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