Transient Elastohydrodynamic Lubrication Analysis with time varying Entrainment Velocity using Magne by Allied Journals

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

Transient Elastohydrodynamic Lubrication Analysis with time varying Entrainment Velocity using Magnetorheological Fluids Rajbeer Singh Anand, Punit Kumar  Abstract— Transient Elastohydrodynamic Lubrication (EHL) analysis using Magneto rheological (MR) fluids is carried out in the present paper. A time varying entrainment velocity is taken into account and its effect on pressure distribution and film thickness are considered. MR fluid is defined by the Bingham Model and the Reynolds Equation is also modified to consider the effect of MR fluids. Effect of variation in oscillating frequency, amplitude and yield stress have been considered and it is found that film thickness is affected by these variations. Increase in yield stress increases the film thickness thus increasing the load carrying capacity of the fluid film. Also, the effect of speed variation is incorporated and it is seen that on increasing 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 also called field-responsive colloids. These fluids are basically formed by dispersion of magnetisable carbonyl iron particles (micron-sized) and on the application of external magnetic field the viscosity of these fluids increases rapidly. This is due to the formation of clusters of particles aligned in the direction of the field [1-4]. In the past, most of the efforts were focused in understanding bulk rheological behaviour of materials due to viscometric flows in the presence of external magnetic field. Wong et al. [5] in 2001, published the tribological results pertaining to an MR fluid (MRF132) for different concentrations of iron particles in a block-on-ring tester. Leung et al. [6] later said that “the block has relatively less damage in tests with very high (iron) particle concentrations” using two different viscosity base fluids in a block-on-ring tester. In EHL, elastic deformation of surfaces is considered under hydrodynamic lubrication and in rolling or sliding contact. Here, the surfaces are non-conforming and load is higher. A high pressure area is created due to elastic deformation of either one or both of the surfaces. With pressure, there is an increase in lubricant viscosity. Spur Gears, Cylindrical Roller Manuscript received February 20, 2015 Bearings, Cams and Tappets etc. are few examples of EHL contacts. Dowson [7] described the experimental

determination and film thickness development in Elastohydrodynamic Lubrication line contacts in his paper. Wang and Cheng [8, 9] considered transient problem in elastohydrodynamic lubrication and developed a numerical arrangement using Grubin-type Inlet Zone Analysis. They predicted the minimum film thickness at various points on the line of action and also bulk surface temperatures. In present study, the effect of magnetorheological (MR) fluids is taken into consideration and its effect on pressure distribution and film thickness is depicted. To consider the effect of MR fluids, Bingham Model is employed. II. MATHEMATICAL MODEL The equations used for the modelling of EHL problem are given in dimensionless form below. A. Modified Bingham Model Magnetorheological Fluids follow Bingham Plastic Model which is difficult to incorporate in Reynolds Equation in its actual form. Therefore, the Modified Bingham Model is given as,  a  /   om 1  e  om      e  



TABLE I.

   

(1)

NOMENCLATURE

DIMENSIONAL PARAMETERS b

Half width of Hertzian contact zone (m)

Effective elastic modulus (Pa)

Film thickness (m)

hmin

Minimum film thickness (m)

Offset film thickness (m)

Pressure (Pa)

Maximum Hertzian Pressure (Pa)

Equivalent radius of contact (m).

Local fluid velocity (m/s)

Average rolling speed (m/s)

ua, ub

Velocities of lower and upper surfaces.

Surface displacement (m).

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Transient Elastohydrodynamic Lubrication Analysis with time varying Entrainment Velocity using Magnetorheological Fluids DIMENSIONAL PARAMETERS w

Applied load per unit length (N/m)

Abscissa along rolling direction (m)

Ordinate across the fluid film (m)



Piezo-viscous coefficient (P a - 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)

Fluid viscosity (Pa–s)

om

Yield Stress for MR fluid.

Where, Dij = Influence Coefficients for a uniform mesh size ∆X   1   1  Dij   i  j  X ln  i  j  X   1 2    2      1   1    i  j  X ln  i  j  X   1 2    2   

D. Density Pressure Relationship  0 

   1 

NON- DIMENSIONAL PARAMETERS H

Non-dimensional film thickness

Hmin

Non-dimensional minimum film thickness

Non-dimensional abscissa

Non-dimensional speed parameters

Non-dimensional pressure



Non-dimensional fluid density

Non-dimensional load parameter

Slide to roll ratio

Non-dimensional displacement

Non-dimensional offset film thickness



Non-dimensional shear stress



Non-dimensional viscosity of Newtonian fluid

Xin

Inlet boundary co-ordinate.

Outlet boundary co-ordinate.

Non-dimensional central film thickness.

   h p / x       u o h   ( h) x  12  m  x t

 

Outlet boundary condition P

P  0 at X  X 0 X

C. Film Thickness Equation H X   H  0

X2 1 N   Dij Pj 2  j 1





z 

    exp  In  9.67   1  1  5.1 10  9 P. p   0 H 

z 





  

(5)

 5 . 1  10

9

(ln  0  9 . 67 )

F. Load Equilibrium Equation

x 0  PdX   2 xin

(6)

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



j 2

W   C j Pj 

Where, X 3  C j  4 X 3 (2) 2 X 3 

Inlet boundary condition P  0 at X  X in

(4)

E. Viscosity Pressure Relationship We use the Roelands Equation as it can be used for many types of lubricants.

B. Reynolds Equation The following Reynolds equation is obtained for MR fluids for which perturbation scheme employed is given in the Appendix. 3

9

p   9  1  1.7  10 p  0.6  10

0

(7)

j 1 j  2,4,6... j  3,5,7...

III. SOLUTION PROCEDURE In the present simulation, the solution domain (X) has its range from -4 to 1.5 and a grid size (∆X) equal to 0.02. Finite Difference Method is used to discretize the Reynolds equation and then Newton-Raphson technique is used to solve the system of equations. The simulations are carried out keeping the load, rolling speed and pressure-viscosity coefficient at a constant value. An initial guess for pressure distribution (P) and offset film thickness (H0) is made at the beginning of solution, and these are used to calculate film (3) thickness and the fluid properties (density and viscosity).

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International Journal of Engineering, Management & Sciences (IJEMS) ISSN-2348 –3733, Volume-2, Issue-5, May 2015 Magnetorheological fluids are best described by Bingham model which is used in the present analysis. Due to the 1.0 transient conditions, pressure distribution and film shape are f=50Hz,  =80kPa calculated for several cycles of sinusoidally varying f=50Hz,  =400kPa 0.8 f=100Hz,  =80kPa entrainment velocity till the cyclic pattern of film thickness f=100Hz,  =400kPa variation repeats itself. om om

TABLE II.

Pressure, P

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

0.6

0.4

0.2

0.0 -4

-3

-2

-1

Fig. 2. Pressure Distribution (P v/s X) for the given combinations of Frequency (f) and Yield Stress (τom) at umax. 1.0 f=50Hz, om=80kPa f=50Hz, om=400kPa

0.8

f=100Hz, om=80kPa

From the figures given below it is seen that as the yield stress of the fluid is increased, the film thickness also increases in all the cases. However, there is minimal effect of yield stress on pressure distribution. A. Effect of Oscillating Frequency Variation of frequency has minimal effect on the pressure distribution for different fluid velocities as shown in Figs. 1 to 3. For film thickness at fluid velocities umean and umax, as shown in Fig. 4 and Fig. 5 respectively, as we increase the frequency of oscillation the film thickness decreases slightly. But at umin, as shown in Fig. 6, film thickness increases with an increase in frequency.

Pressure, P

f=100Hz, om=400kPa

IV. RESULTS AND DISCUSSIONS

0.6

0.4

0.2

0.0 -4

-3

-2

-1

Fig. 3. Pressure Distribution (P v/s X) for the given combinations of Frequency (f) and Yield Stress (τom) at umin. 0.10 f=50Hz, om=80kPa f=50Hz, om=400kPa

0.08

f=100Hz, om=80kPa f=100Hz, om=400kPa

1.0 Film Thickness, H

f=50Hz, om=80kPa f=50Hz, om=400kPa

0.8

f=100Hz, om=80kPa f=100Hz, om=400kPa

Pressure, P

0.06

0.04

0.6

0.02 0.4

0.00 -1.5

0.2

-1.0

-0.5

0.0

0.5

1.0

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

0.0 -4

-3

-2

-1

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

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Transient Elastohydrodynamic Lubrication Analysis with time varying Entrainment Velocity using Magnetorheological Fluids

0.10

1.0 A=0.2, om=80kPa

f=50Hz, om=80kPa f=50Hz, om=400kPa

0.08

A=0.2, om=400kPa

0.8

f=100Hz, om=80kPa

A=0.4, om=80kPa A=0.4, om=400kPa

0.06 Pressure, P

Film Thickness, H

f=100Hz, om=400kPa 0.6

0.04

0.4

0.02

0.2

0.00 -1.5

0.0

-1.0

-0.5

0.0

0.5

-4

1.0

-3

-2

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

1.0 A=0.2, om=80kPa

f=50Hz, om=80kPa f=50Hz, om=400kPa

0.08

A=0.2, om=400kPa

0.8

f=100Hz, om=80kPa

A=0.4, om=80kPa A=0.4, om=400kPa

f=100Hz, om=400kPa 0.06 Pressure, P

Film Thickness, H

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

0.10

0.04

0.6

0.4

0.02

0.00 -1.5

-1 X

0.2

-1.0

-0.5

0.0

0.5

0.0

1.0

-4

-3

-2

B. Effect of Oscillation Amplitude There is minimal effect of amplitude on the pressure distribution as observed in Figs. 7 to 9. But there is effect of amplitude on film thickness which is given in Figs. 10 to 12. At umean, the film thickness increases with decreasing amplitude. But at umax, the film thickness increases as the oscillation amplitude increases. Whereas at umin, the film thickness first increases with decreasing amplitude and then becomes almost equal in the latter part of the domain at a particular yield stress. C. Central Film Thickness At lower frequency, the film remains thick for a longer period of time as compared to the film at higher frequency (for a particular value of yield stress) as shown in Fig. 13. But as illustrated by Fig. 14, there is no such effect of amplitude on central film thickness.

Fig. 8. Pressure Distribution (P v/s X) for the given combinations of Amplitude (A) and Yield Stress (τom) at umax. 1.0 A=0.2, om=80kPa A=0.2, om=400kPa

0.8

A=0.4, om=80kPa A=0.4, om=400kPa

Pressure, P

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

-1 X

0.6

0.4

0.2

0.0 -4

-3

-2

-1

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

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International Journal of Engineering, Management & Sciences (IJEMS) ISSN-2348 –3733, Volume-2, Issue-5, May 2015 0.10

700 A=0.2, om=80kPa

0.08

f=100Hz, om=80kPa Central Film Thickness, Hc(nm)

Film Thickness, H

f=50Hz, om=400kPa

A=0.4, om=80kPa A=0.4, om=400kPa

0.06

0.04

0.02

0.00 -1.5

f=50Hz, om=80kPa

600

A=0.2, om=400kPa

f=100Hz, om=400kPa

500

400

300

200

100 -1.0

-0.5

0.0

0.5

1.0

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

700 A=0.2, om=80kPa A=0.2, om=400kPa

0.08

A=0.2, om=400kPa A=0.4, om=80kPa

Central Film Thickness, Hc(nm)

Film Thickness, H

A=0.2, om=80kPa

600

A=0.4, om=80kPa A=0.4, om=400kPa

0.06

0.04

0.02

A=0.4, om=400kPa

500

400

300

200

-1.0

-0.5

0.0

0.5

100

1.0

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

A=0.2, om=80kPa A=0.2, om=400kPa

0.08

Time, t

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 Thickness The observations are same as those made for central film thickness and are observed from Fig. 15 and Fig. 16. Here also, at lower frequency the film remains thick for a longer period of time but there is no effect of amplitude on minimum film thickness.

0.10

A=0.4, om=80kPa A=0.4, om=400kPa

Film Thickness, H

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

0.10

0.00 -1.5

Time, t

0.06

0.04

500 f=50Hz, om=80kPa f=50Hz, om=400kPa

0.00 -1.5

-1.0

-0.5

0.0

0.5

1.0

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

Minimum Film Thickness, Hmin(nm)

0.02

f=100Hz, om=80kPa

400

f=100Hz, om=400kPa

300

200

100 0

Time, t

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

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Transient Elastohydrodynamic Lubrication Analysis with time varying Entrainment Velocity using Magnetorheological Fluids Now, expansion of velocity in terms of  is given by, 500 A=0.2, om=80kPa

Minimum Film Thickness, Hmin(nm)

450

A=0.2, om=400kPa A=0.4, om=80kPa

400

A=0.4, om=400kPa

u u0 u1

(A2)

I  I 0   I1

(A3)

350

u u Where, I  0 , I  1 0 y 1 y

300 250 200

Taylor Series expansion of Equivalent Viscosity e in the vicinity of I 0 is given below,

150 100 0

Time, t

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

Also, for umax the fluid layer formed remains thicker throughout the domain as compared to umean and umin. This can be seen in Figs. 4 to 6 and Figs. 10 to 12. Fluid velocity also plays a vital role in film thickness.

In the present study, transient EHL analysis using MR fluids is carried out. Bingham model is used to consider the effect of MR fluids. 1) As observed, there is minimal effect of variation of frequency, amplitude and yield stress on pressure distribution. 2) Film thickness increases with an increase in yield stress of the fluid which can be depicted from the above results. 3) The film thickness increases with a decrease in the oscillating frequency for the fluid velocities umean and umax. But for umin, film thickness increases with an increase in frequency. 4) For umean, the film thickness increases with decreasing amplitude. But for umax, the film thickness increases with an increase in the oscillation amplitude. And for umin, the film thickness first increases with a decrease in amplitude and then becomes almost equal for the same yield stress. 5) Central and Minimum Film Thickness remain thick for a longer time at lower frequency for same yield stress. But this is not the case with varying the amplitude. 6) Fluid velocity also has a significant effect on film thickness, as at umax the film formed is thicker as compared to those forming at umean and umin.

(A4)

Where 0   eI 0  and  1  I 1   e 

 I  I 0

(A5)

The Momentum Equation is given by, 

V. CONCLUSIONS

y



p x

(A6)

Using (A1), (A3) and (A5) and neglecting  2 ,    0    1  I 0   I 1`    0 I 0    1 I 0   0 I 1` 

(A7)

Expanding p , p  0   ˆ

(A8)

Substituting (A7) and (A8) in (A6), 0

0

I 0   1 I 0   0 I 1`   ˆ   y y x  2u 0 y 2

(A9)

(A10)

0

ˆ And   1 I 0   0 I 1`     y

(A11)

x

Integrating (A10) for the boundary conditions, u0  ua at y  0 and u0  ub at y  h , we have

APPENDIX A. Perturbation Scheme For 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 of lubricant mixture.

e   / I

e  o   1

(A1)

u0  ua 

(A12)

Here ua , ub are the velocities of lower and upper surfaces respectively and h is the film thickness. Substituting (A5) in (A11), we get 

Here, e = Equivalent Viscosity, and I  u / y

u b  u a  y

 2 u1 y



 ˆ x

(A13)

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International Journal of Engineering, Management & Sciences (IJEMS) ISSN-2348 –3733, Volume-2, Issue-5, May 2015  Where,     I 0   e   

 I  I0

  0   

(A14) [7]

Integrating (A13) for the boundary conditions, u1  0 at y  0 and ub  0 at y  h , gives u1 

y

[9]



 hy  ˆ 2  x

[8]

(A15)

the performance of a boundary lubricated contact”, Proc. Inst. Mech. Eng. J., 218, 251–263. D. Dowson (1995). “Elastohydrodynamic and Microelastohydrodynamic Lubrication”, Wear, 190, 125-138. Wang KL, Cheng HS (1981). “A numerical solution to the dynamic load, film thickness and surface temperatures in spur gears”, ASME Journal of Mechanical Design, 103, 177–187. Wang KL, Cheng HS (1981). “A numerical solution to the dynamic load, film thickness and surface temperatures in spur gears”, ASME Journal of Mechanical Design, 103, 188–94.

From (A2), (A8), (A12) and (A14) u  ua 

u b  u a  y  y 2  hy  p 2 

(A16)

x

 u u b  u a  2 y  y   p   y h 2   x

(A17)

For Bingham Fluid,

e   





 o 1  e  a I /  0 I



aI 0 e   e      I  I

 aI 0 / 0

(A18)

 1  e I 02



  

 o  aI 0 e

aI0 / o

I 02

 aI 0 / 0

 o

   e   a I    0    1  ae   I  I 



  I0 



/  om



(A19)



Where,  m  1  ae  aI 0 /  om



(A20)

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

   h 3p / x       u o h   ( h) x  12 m  x t



REFERENCES [1] [2] [3]

[4] [5] [6]

Ginder, J.M. (1998). “Behavior of magnetorheological fluids”, MRS Bulletin, 26–29. Rankin, P.J., Ginder, J.M. and Klingenberg, D.J. (1998). “Electro- and magnetorheology. Curr. Opin”, Colloid Interface, 3, 373–381. Bossis, G., Volkova, O., Lacis, S. and Meunier, A. (2002). “Magnetorheology: fluids, structures and rheology. In: Odenbach, S. (ed.) Ferrofluids”, Magnetically Controllable Fluids and Their Applications Lecture Notes in Physics, 594, 202–230. de Vicente, J., Klingenberg and D.J., Hidalgo-A´ lvarez, R.(2011). “Magnetorheological fluids”, a review. Soft Matter, 7, 3701–3710. Wong, P.L., Bullough, W.A., Feng, C., Lingard, S. (2001). “Tribological performance of a magneto-rheological suspension”, Wear, 247, 33–40. Leung, W.C., Bullough, W.A., Wong, P.L., Feng, C. (2004). “The effect of particle concentration in a magneto rheological suspension on