On the Role of Smart Lubricants in EHD Lubrication of Rollers under Heavy Loads by Allied Journals

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

On the Role of Smart Lubricants in EHD Lubrication of Rollers under Heavy Loads Manish Kaushik, Punit Kumar  Abstract— Elastohydrodynamic lubrication (EHL) analysis of line contact lubricated with smart fluids has been carried out in present work. The aim is to observe the effect of smart fluids on EHL performance. The simulation is based upon modified Bingham model which is used to describe the smart fluids. Perturbation scheme is used to modify classical Reynolds equation to incorporate the effect of smart lubricants. It is observed that film thickness increases with increase in yield stress of smart fluid due to which load carrying capacity of lubricant film is improved. Index Terms— Bingham model, EHL, Film thickness, Line contact, Perturbation scheme, Smart fluids, Yield stress

I. INTRODUCTION Checking the properties of material has attracted an appreciable attention over the last few decades. Electrorheological fluids are smart materials whose rheological properties can be changed by applying an electric field. An electrorheological (ER) fluids are commonly a suspension of solid dielectric particles diffused in non-conducting liquid, however, ER fluids can also be prepared by blending phosphorated starch particles and silicone oil. By applying electric field their resistance to flow can be altered very quickly. The ER fluids can change their behavior from Newtonian type to Bingham type, in which particles form chain like structure. Due to this behavior, ER fluids can endure external pressure or force with handful advantages of simple design, continuous control and fast response time. In Elastohydrodynamic lubrication (EHL) there is high pressure between non-conformal contacts. Because of this high pressure the elastic deformation of contacting surfaces occur. With increase in pressure, viscosity of lubricant increases and if pressure is significantly above atmospheric, the effect of pressure on viscosity is considerably large as compared to the effect of temperature and shear. The load carrying capacity increases because of this increase in viscosity. EHL contacts are generally present in spur gears, cylindrical roller bearings and cams etc. Manuscript received February 20, 2015. Manish Kaushik, Mechanical Department, NIT Kurukshetra, Kurukshetra, India. Punit Kumar, Mechanical Department, NIT Kurukshetra, Kurukshetra, India .

In 1886, theoretical lubrication analysis based on Tower’s journal bearing experiment was published by Reynolds [1]. Since then Reynolds equation is the foundation of hydrodynamic lubrication theory. In 1916, Martin used simplified Reynolds equation to derive an expression for loading capacity but couldn’t find effective film thickness [2]. In the 1930s, lubrication analysis was improved by including the effect of elastic deformation of contact surfaces [3] and effect of pressure on viscosity [4]. Grubin brought a significant breakthrough in understanding the elastohydrodynamic lubrication (EHL) mechanism by publishing his paper on hydrodynamic lubrication of heavily loaded cylindrical surface. Grubin was first to study effect of elastic deformation and viscosity-pressure characteristics simultaneously for non-conformal contact [5].In 1950s and 1960s much attention was given to the numerical solutions without assumptions. Dowson and Higginson published their landmark paper: “A Numerical Solution to the Elastohydrodynamic Problem”. They discovered the inverse solution approach. Difficulties of slow numerical convergence were removed by this approach [6] The procedure of inverse solution came out to be capable of solving heavily loaded cases and producing converged solution within few number of calculation cycles. In 1976 and 1977 Hamrock and Dowson presented a string of papers, investigating the effects of load, material properties, and speed on minimum film thicknesses in elliptical contacts by using straightforward iterative approach [7]. Simultaneous to the above theoretical studies, breakthrough results are produced by experimental investigations also. Kirk and Archard were the first to demonstrate, a measurable lubricant film for point contacts under heavy load, experimentally [8]. Effect of temperature is an important subject as temperature increase results in reduced viscosity which can degrade the EHL performance and may lead to film breakdown and energy loss. Prime studies on the thermal effects in EHL line contacts were shown by Cheng and Sternlicht [9] and by Dowson and Whitaker [10], by considering the energy equation with other EHL equations. Newtonian fluid model gives useful prediction of film thickness but can’t be used in practical EHL situation where very high shear rate is present. To have a crucial understanding of lubrication performance in various tribo-machines it is necessary to associate Non-Newtonian

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On the Role of Smart Lubricants in EHD Lubrication of Rollers under Heavy Loads

effect of lubricant in numerical theory. The following models are so far widely accepted- Johnson and Tevaarwerk [11] and Bair and Winer [12]. Nikolakopoulos and Papadopoulos used Bingham model to describe rheological properties of ER fluids and concluded that the electric field imposed on the ER fluids tends to align the journal in the bearing [13]. Nikolakopoulos and Papadopoulos applied analytical and experimental approach on high speed journal bearings which are lubricated with ER fluids [14]. They concluded that under applied voltage the journal bearing becomes stiff. Peng and Zhu performed numerical analysis on hydrodynamic characteristics of ER fluid flow in journal bearing and showed that ER fluids provide an improvement in load carrying capacity [15]. A. Korenaga et al studied the EHL characteristics of ER fluids by using ball-on-disk type tribo-testing machine to find out the thickness of lubricant film. They found that minimum film thickness increased when electric field was applied across the ER fluid [16]. In this article, modified Bingham model is used to describe ER fluids and EHL characteristics of ER fluids are studied. A numerical procedure is used to solve the Reynolds equation which is based on the modified Bingham model, along with other EHL equations. II. EHL MODEL The equations used for the modelling of EHL problem are given in dimensionless form below. A. Rheological Model Smart fluids are known to follow Bingham plastic model which is quite difficult to incorporate in Reynolds equation in its actual form. Therefore, the present work presents a modified Bingham model given below. 

e      

 0 E  1  e 

 a  /  0 E      

TABLE I.

(1)

DIMENSIONAL PARAMETERS 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).

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)

0E

Yield Stress for ER fluid.

TABLE II.

NOMENCLATURE

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.

B. Reynolds Equation The modified Reynolds equation incorporating the effect of smart lubricant has been derived using perturbation scheme. First, equivalent viscosity (ηe) is introduced and given by

NOMENCLATURE

DIMENSIONAL PARAMETERS

e   / I

(2)

Here, e = Equivalent Viscosity, and I  u / y Now, expansion of velocity in terms of  is given by, u u0 u1

(3)

I  I 0   I1

(4)

u u Where, I  0 , I  1 0 1 y y

Expanding the equivalent viscosity ηe in the region near I0 into a Taylor series:

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

e  o   1

(5)

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

(6)

 I  I 0

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

For Bingham Fluid,

The Momentum Equation is given by, 

p  y x

(7)

e   



 o 1  e  aI /  0 E



Using (2), (4) and (5) and neglecting  , (8)

I 02



Substituting (9) and (8) in (7),



Where,  E  1  ae

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

(10)

(11)

0

y 2

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

u b  u a  y

(13)

Where ua, ub are the velocities of the lower and upper surfaces respectively and h is the film thickness. Substituting (6) into (12) gives 

 ˆ  x

 2 u1 y 2

 x

  h 3 p /  x   12  E 

 I  I0

(21)

   u o   h   x 

(22)

C. Boundary condition Pressure distribution is subjected to following boundary conditions:

(15)

Integrating (14) under the boundary conditions, u1 = 0 at y = 0 and ub = 0 at y = h, one gets

(24)

D. Film Thickness Equation

  0   

(23)

P  0 at X  X in P P  0 at X  X 0 X

H X   H 

 



X2 1 N   Dij Pj 2  j 1

(25)

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



  1   1  Dij   i  j  X ln  i  j  X   1 2    2    (16)

From (3), (9), (13) and (15)

  1   1    i  j  X ln  i  j  X   1 2 2      

y 

 hy  ˆ 2  x

u  ua 

u b  u a  y  y 2  hy  p h

(20)

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

(14)

 Where,     I 0   e 

 a I 0 /  0 E



(12)

x

Integrating (11) under the boundary conditions, u0 = ua at y = 0 and u0 = ub at y = h, one gets u0  ua 



 o  aI 0 eaI 0 /  0 E   0 E



      aI 0 / 0 E e   (9)     I 0   I    0    1  ae I0   

p  0   ˆ

 2u 0

(19)

 

Expanding p ,

0



aI 0e aI 0 /  0 E   0 1  e aI 0 /  0 E  e     I 02  I  I 0

   0    1  I 0   I 1`    0 I 0    1 I 0   0 I 1` 

(18)

2 

x

(17)

E. Density Pressure Relationship The present work uses the following density-pressure

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On the Role of Smart Lubricants in EHD Lubrication of Rollers under Heavy Loads

relationship 9  0.6  10 p      1  0  1  1.7  10 9 p   

(26)

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

 





z 



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

z 





  

 5 . 1  10

9

(ln  0  9 . 67 )

Load Equilibrium Equation G. Load Equilibrium Equation

x 0  PdX   2 xin

Inlet Density of fluid (ρ0)

846 kg/m3

Slide to Roll Ratio (S)

0.5

Rolling Speed (u0)

0.1 m/s

Domain, X

-4≤X≤1.5

Grid Size, ∆X

0.02

IV. RESULTS AND DISCUSSIONS

Figs. 1 and 2 show the variation of pressure distribution and film shape respectively, at different values of maximum (27) Hertzian pressure (pH) and yield stress (0E). In this case, the variation of yield stress has only marginal effect on pressure distribution while the film thickness is substantially enhanced due to increase in yield stress. Figs. 3 and 4 show the variation of central film thickness (hc) and minimum film thickness (hmin) with respect to maximum Hertzian pressure (pH) at different yield stress respectively. It can be seen that central and minimum film thickness decrease with increase in pH; however, an increase in the value of yield stress (0E) is found to cause significant enhancements in the values of central and minimum film thickness. (28) 2.5e+9

3 p =1 GPa and 0E =10 Pa H

The integral is calculated using Simpson's rule and it can be written in the following form:



j 2

W   C j Pj  Where, X 3  C j  4 X 3 2 X 3 

2.0e+9

3 p =2 GPa and 0E =10 Pa H

6 p =2 GPa and 0E=10 Pa

0

(29)

Pressure, p (Pa)

6 p =1 GPa and 0E =10 Pa

j 1

1.5e+9

1.0e+9

5.0e+8

j  2,4,6... j  3,5,7...

0.0 -2

-1

Fig. 1. Comparison of pressure distributions at different values of maximum Hertzian pressure (pH) and yield stress (0E)

The present analysis employs the modified Bingham model to describe the rheological behavior of ER fluids. The solution domain ranges from X= –4 to 1.5 with a grid size of ∆X =0.02. First, an initial guess for offset film thickness (H0) and pressure distribution (P) is made. Using finite differences method, the Reynolds equation is discretized and solved along with the load balance equation subject to the prescribed boundary conditions using Newton-Raphson technique. The corrected offset film thickness and pressure distribution serve as inputs for the next iteration and the above procedure is repeated until a relative accuracy of 10-5 is achieved. TABLE III.

INPUT PARAMETERS

Maximum Hertzian Pressure (pH)

1-2 GPa

Pressure-Viscosity Coefficient ()

15-20 GPa-1 2.2*10 Pa

Equivalent Radius (R)

0.02 m

Yield Stress (om)

pH =1 GPa and 0E =103 Pa pH =1 GPa and 0E =106 Pa pH =2 GPa and 0E =103 Pa

0.15

pH =2 GPa and 0E =106 Pa 0.10

0.05

0.00 -1.0

Equivalent Elastic Modulus (E)

0.20

Film Thickness, H

III. SOLUTION PROCEDURE

-0.5

0.0

0.5

1.0

Fig. 2. Comparison of film shapes at different values of maximum Hertzian pressure (pH) and yield stress (0E) .

10 -10 Pa

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

1e-4

1e-4 at 

at  =105 Pascals

=105 Pascals

at  =5*105 Pascals Central Film Thickness, log(Hc/R)

Central film thickness, log(Hc/R)

at  = 5*105 Pascals

1e-5

1e-6 14

1e-6 0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Pressure-Viscosity Coefficient, (GPa)

Maximum Hertzian Pressure, pH (GPa)

Fig. 3. Comparison of central film thickness (hc) vs maximum Hertzian pressure (pH) at different yield stress (0E)

-1

Fig. 5. Comparison of central film thickness (hc) vs pressure-viscosity coefficient  at different yield stress (0E)

1e-4 at  =105 Pascals at  =5*105 Pascals

1e-4 Minimum Film Thickness, log(Hmin/R)

at  =105 Pascals

Minimum Film Thickness, log(Hmin/R)

at  =5*105 Pascals

1e-5

1e-6 14

Pressure-Viscosity Coefficient, (GPa)

1e-6 0.8

1.0

1.2

1.4

1.6

1.8

2.0

Fig. 6. Comparison of minimum film thickness pressure-viscosity coefficient  at different yield stress (0E)

2.2

Maximum Hertzian Pressure, pH (GPa)

Fig. 4. Comparison of minimum film thickness (hmin) vs maximum Hertzian pressure (pH) at different yield stress (0E)

Variation of central film thickness and minimum film thickness with respect to pressure-viscosity coefficient (α) at different yield stress (0E) are shown in Fig. 5 and Fig. 6. The slopes of the curves in these figures clearly indicate that the sensitivity of EHL film thickness to pressure-viscosity coefficient (α) increases with increasing yield stress.

-1

(hmin)

V. CONCLUSIONS The present study investigates the effect of ER fluid on EHL performance. In numerical analysis modified Bingham model is used to depict the behavior of ER fluid. With increase in yield stress, central and minimum film thickness increase. Load carrying capacity is also improved with yield stress. These fluids can deal with tribology problems and can be used in EHL contacts to increase EHL performance. References REFERENCES [1]

[2] [3] [4] [5]

Reynolds, O. (1886). “On the Theory of Lubrication and Its Application to Mr. Beauchamp Tower’s Experiments, Including an Experimental Determination of the Viscosity of Olive Oil”, Philos. Trans. R. Soc.London, 177, 157–234. Martin, H. M (1916). “Lubrication of Gear Teeth”, Engineering, 102, 119-121. Meldahl, A. (1941), “Contribution to the Theory of the Lubrication of Gears and of the Stressing of the Lubricated Flanks of Gear Teeth”, Brown Boveri Rev., 28, 374–382. Gatcombe, E. K. (1945). “Lubrication Characteristics of Involute Spur Gears A Theoretical Investigation”, Trans. ASME, 67, 177–185. Grubin, A. N. (1949). “Fundamentals of the Hydrodynamic Theory of Lubrication of Heavily Loaded Cylindrical Surfaces”, Book No. 30,

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On the Role of Smart Lubricants in EHD Lubrication of Rollers under Heavy Loads

[6] [7] [8] [9]

[10]

[11] [12]

[13] [14] [15] [16]

Central Scientific Research Institute for Technology and Mechanical Engineering, Moscow(DSIR Translation), 115–166. Dowson, D., and Higginson, G. R. (1959). “A Numerical Solution to the Elastohydrodynamic Problem”, J. Eng. Sci., 1, 6–15. Hamrock, B. J. and Dowson, D. (1976). “Isothermal Elastohydrodynamic Lubrication of Point Contacts”, Part 1—Theoretical Formulation”, ASME J. Lubr. Technol., 98, 223–229. Archard, J. F. and Kirk, M.T.(1961). “Lubrication at Point Contacts”, Proc. R.Soc. London, Ser. A, 261, 532–550. Cheng, H. S. and Sternlicht, B. (1964). “A Numerical Solution for the Pressure, Temperature, and Film Thickness Between Two Infinitely Long, Lubricated Rolling and Sliding Cylinders, Under Heavy Loads”, ASME J. Basic Eng., 87, 695–707. Dowson, D. and Whitaker, A. V. (1966). “A Numerical Procedure for the Solution of the Elastohydrodynamic Problem of Rolling and Sliding Contacts Lubricated by a Newtonian Fluid”, Proc. Inst. Mech. Eng., 180(3B), 119–134. Johnson, K. L. and Tevaarwerk, J. L. (1977). “Shear Behaviour of EHD Oil Films”, Proc. R. Soc. London, Ser. A, 356, 215–236. Bair, S. and Winer, W. O. (1979). “Rheological Response of Lubricants in EHD Contacts”, Proceedings of the Fifth Leeds-Lyon Symposium on Tribology, Mechanical Engineering Publications, Bury St. Edmunds, UK, 162–169. Nikolakopoulos P. G. and Papadopoulos C. A.(1997). “Controllable Misaligned Journal Bearings, Lubricated with Smart Fluids”, Journal of Intelligent Material Systems and Structures. Nikolakopoulos P. G. and Papadopoulos C.A. (1998). “Controllable high speed journal bearings, lubricated with electrorheological fluids: an analytical and experimental approach”, Tribol Int, 225–34. Peng, J. and Zhu (2006). “Effects of electric field on hydrodynamic characteristics of finite-length ER journal bearings”, Tribology International, 533–540. A. Korenaga, T. Yoshioka, H. Mizutani and K. Kikuchi (1999), “Elastohydrodynamic Lubrication Characteristics of Electrorheological Fluids”, Elsevier Science.