Post on 29-Apr-2023
International Journal of Engineering, Management & Sciences (IJEMS)ISSN-2348 –3733, Volume-2, Issue-5, May 2015
18 www.alliedjournals.com
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
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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).
International Journal of Engineering, Management & Sciences (IJEMS)ISSN-2348 –3733, Volume-2, Issue-5, May 2015
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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.
Transient Elastohydrodynamic Lubrication Analysis with time varying Entrainment Velocity usingMagnetorheological Fluids
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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.
International Journal of Engineering, Management & Sciences (IJEMS)ISSN-2348 –3733, Volume-2, Issue-5, May 2015
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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).
Transient Elastohydrodynamic Lubrication Analysis with time varying Entrainment Velocity usingMagnetorheological Fluids
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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)
International Journal of Engineering, Management & Sciences (IJEMS)ISSN-2348 –3733, Volume-2, Issue-5, May 2015
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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
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[6] Leung, W.C., Bullough, W.A., Wong, P.L., Feng, C. (2004). “Theeffect of particle concentration in a magneto rheological suspension on
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[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.