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Procedía Materials Science 5 (2014) 898 - 907

International Conference on Advances in Manufacturing and Materials Engineering,

AMME 2014

Steady State EHL Line Contact Analysis with Surface Roughness and Linear Piezo-viscosity

Nitesh Talekara'*, Punit Kumarb

aPG Student, Mechanical Department, NIT Kurukshetra-136119, Haryana, India bAssociateProfessor, Mechanical Department, NITKurukshetra-136119, Haryana, India

Abstract

A special form of hydrodynamic lubrication, elastohydrodynamic lubrication (EHL), is generally encountered in mechanical components involving highly stressed non-conformal contact such as gears, cams, roller bearing etc. EHL is characterized with substantial elastic deformation of the interacting surface and piezo-viscous rise in lubricant viscosity which assist largely in evolution of load carrying fluid film.

In the current paper, the effect of surface roughness of different magnitude in EHL line contact on pressure distribution is considered. At the same time graph between film thickness with rolling direction is plotted which gives reader the knowledge of EHL film thickness. Steady state effect of EHL with linear piezo-viscosity response at low pressure is taken into consideration. Full isothermal, Newtonian simulation of the EHL problem is discussed. It investigate the pressure distribution of sinusoidal waviness in the line contact, more precisely it study the effect of linear piezo-viscous fluid on surface roughness of different amplitude in steady state EHL line contact. Graphs plotted gives the clear idea to the designers, the behaviour of surface roughness with several amplitude and wavelength by linear piezo-viscous liquid like 2, 3-Dimethylpentane, L7808 (high temperature lubricant) and water/glycol solution of two water percentage and exponential fluid like santotrac 40 (S 40). The Reynolds equation, which governs the generation of pressure in the lubricated contact, is discretized using finite differences and solved along with the load balance equation using Newton-Raphson technique. A computer code is developed for implementation of solution algorithm. The results are obtained to study the linear and exponential piezo-viscous effect on surface roughness on the EHL characteristics.

© 2014ElsevierLtd.Thisisanopenaccessarticle under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).

Selection and peer-review under responsibility of Organizing Committee of AMME 2014

Keywords: EHL; Surface roughness; Linear piezoviscosity; Line contact.

* Corresponding author. Tel.: +0-976-691-9381; fax:+91-1744-238350. E-mail tfd<irass:kitsniteshtalekar@gmail.com

2211-8128 © 2014 Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

Selection and peer-review under responsibility of Organizing Committee of AMME 2014 doi: 10.1016/j.mspro.2014.07.377

1. Introduction to EHL

Elasto-hydrodynamic lubrication (EHL) is a form of hydrodynamic lubrication, in which due to high contact pressure between non-conformal contacts (such as rolling contact bearings, cams and tappets, gears etc.) elastic deformation of the contacting surfaces take place and viscosity variation with pressure taken into consideration. In EHL, the viscosity of the lubricants, showing exponential pressure viscosity relation, increases significantly with lubricant pressure, due to which load bearing capacity increases. Therefore in a way the EHL film thickness depends on the piezo-viscous relationship. The material which separates these surfaces is called lubricants and the thickness of layer of lubricant is called film thickness. In general the thickness of these films ranges from 1-100 ^m or even thinner and thicker films can be found.

Nomenclature

b half width of hertzian contact, 4R W/2n (m) T

E' effective elastic modulus of roller 1 and 2 (Pa)

h film thickness (m)

M0 molecular weight of lubricant (Kg) -q

p pressure (Pa) H

pH maximum Hertzian pressure, E'b/4R (Pa) Hmin

R equivalent radius of contact (m) H0

Uo average rolling speed, (ua + Ub)/2 (m/s) P

ua,ub velocitiesof lowerand uppersurfaces, (m/s) S

v surface displacement (m) U

w applied load per unit length (N/m) ^

a pressure-viscosity coefficient (Pa"1) ^

ao initial pressure-viscosity coefficient (Pa"1) ^

aB Barus pressure viscosity coefficient (Pa"1) ^

r strain rate across the fluid film, du/dy (s"1) ^

Po inletdensityof lubricant(Kg/m3) ^

p lubricant density at the local pressure and

temperature (Kg/m )

shear stress in fluid (Pa) inlet viscosity of the Newtonian fluid (Pa s) first and second Newtonian viscosities (Pa s) apparent viscosity (Pa s) dimensionless film thickness, hR/b2 dimensionless minimum film thickness, hminR/b2 dimensionless offset film thickness, h0R/b2 dimensionless pressure, p/ph slide to roll ratio, (ub-ua)/uo dimensionless speed parameter, -q0 Uo/E'R dimensionless displacement, vR/b dimensionless load parameter, w/E'R dimensionless deviation from the smooth profile dimensionless initial amplitude, a;R/b2 dimensionless material parameter, aE' dimensionless fluid density, p/ p0 dimensionless apparent viscosity, -q/

2. Surface Roughness Effect.

Surface roughness plays an important role in deciding the tribological behaviour of mechanical components (such as gears, cams, roller bearings etc.) operating under full-film and mixed (or partial) elastohydrodynamic lubrication conditions. Surface roughness not only interacts with the hydrodynamics of the fluid film but it also leads to very high local contact pressures near the asperity tip regions where the fluid film is not thick to separate the contacting surfaces completely. This phenomenon is responsible for causing the collapse of the fluid film, which leads to either failure of the lubrication system or change of lubrication from full-film to mixed EHL. Both of these phenomena bring a larger portion of the contacting surfaces under direct contact leading to high coefficient of friction and hence, high wear rates. This causes premature failure of the components. It is, therefore, important to study the minimum film thickness of the lubricant film under a given surface roughness and the corresponding condition under which the component may work safely. Several efforts have been made in this regard.

Several researchers have used this approach to analyse the effect of surface roughness in EHL contacts, such as Chang (1988) and Ai and Zheng (1989). The major drawback of micro-EHL approach is the difficulty of representing realistic roughness patterns which are random in nature. Since the solution is in a numerical form, the high frequency contents of the roughness are filtered out through the discretization process. Therefore, a sufficiently dense grid configuration is necessary to obtain an accurate description of the irregularities features, which significantly increases the computational efforts and storage requirements. Therefore, micro-EHL analysis is mainly used for EHL contacts between a smooth surface and a rough surface with relatively large size irregularities. A few researchers such as Lubrecht (1989) and Kweh (1992) have used this approach to analyse EHL contacts of wavy surfaces. However, their analyses are limited to steady-surface condition and single sinusoidal wave with relatively

long wavelength. Venner and Napel (1992) have used a deterministic description of an actually measured surface roughness profile in the calculations for the analysis of EHL line contacts under Newtonian isothermal conditions. The surface roughness profile is assumed to be stationary with respect to the contact zone. By this approach, they proved that there is a reduction in the minimum film thickness due to surface roughness.

An extensive review of the studies on surface roughness effects in elastohydrodynamic lubrication has been presented above. Although many workers have used sinusoidal roughness, but their analyses are mostly restricted to long wavelength surface roughness, whereas, it is well known that the topography of realistic surfaces contain short wavelength components. Hence, it is required to study the influence of surface roughness with short wavelength (less than 0.2 times the Hertzian half-contact width). Also, most of the workers have studied the behaviour of rough EHL contacts with a fixed amplitude and wavelength of surface roughness. Therefore, it is required to investigate the effect of varying the amplitude and wavelength of surface roughness on the EHL behaviour of rolling/sliding line contacts for various speed-load combinations and slide to roll ratios under steady state conditions.

3. Pressure Viscosity Coefficient

If precise descriptions of the pressure dependence of viscosity were required for every film thickness, the classical film thickness formulas would not have been very useful. Instead, a single parameter known as the pressure viscosity coefficient has been employed to quantify the piezo-viscous response along the inlet zone. This is a powerful concept, which should have provided a well accepted and unique definition. There have been, however, many definitions leading to a large range of possible values from a single viscosity pressure data. The pressure viscosity coefficient, a, was originally based upon the exponential Bridgman Equation for relatively high pressure

H = jua exp( ap) (1)

It is rare for liquids other than ionic liquids to follow this law at low pressures, with the exception of low temperature, and a conventional pressure viscosity coefficient is often employed, given by

d (id ju) ~| (2)

dP J P=D

The uncertainties in reported values of ao for liquids that show Bridgman response at high pressure are large, as great as one half of the reported value and Newtonian liquids with the same measured values of ^o and ao do not necessarily generate similar film thicknesses. Therefore ao is a poor choice for a coefficient for general piezo-viscous response.

4. Liquids with linear pressure viscosity response at low pressures

In EHL analysis, researchers focus on piezo-viscous relationships involving an exponential function of lubricant pressure. Also, the classical film thickness equations developed for EHL line and point contacts are based on such pressure-viscosity relationships. Recently, P. Kumar (2010) addressed the practical situations involving high operating temperatures, low viscosity lubricants and water/glycol solutions where the lubricant piezo-viscous response is more closely described by a linear relationship. There is a substantial difference between the piezo-viscosities represented by a simple exponential and a linear relationship. As pointed out in, the inlet viscosity of pure hydrocarbon, 2-3-dimethylpentane may be considered linear at ordinary temperatures and similarly, a linear fit can be obtained for octane and toluene as well. The aircraft engine lubricants like L7808 and L23699 used in turbine bearings operating at very high temperatures are known to exhibit linear response at low pressure with some accuracy is given. Similar behaviour can be observed for water/glycol solutions which have been reported to produce thinner films as compared to conventional lubricants for the same operating conditions.

linear

exponential

O.O 0.2 0.4 0.6 0.8 1.0 1.2

Non-Dimensional Pressure

Fig. 1. Comparison of viscosity for linear and exponential pressure viscosity relationship.

Full EHL line contact simulations for smooth surfaces are carried out under fully flooded condition to obtain central and minimum film thickness equations pertaining to lubricants with linear piezo-viscous response. Current analysis based upon the assumptions of steady state isothermal condition and Newtonian fluid model. A major drop in the sensitivity of pressure viscosity coefficient is observed.

The earliest measurements of the pressure dependence of viscosity found linear response for low viscosity liquids. Barus described the effect by

M = (! + asP) (3)

Liquids which show linear piezo-viscous response at low pressure may be expected to show faster than linear and even exponential behaviors at some higher pressure. The Doolittle free volume model employing the Tait equation of state can reproduce with some accuracy the linear response at low pressure and high temperature for another jet oil meeting specification L23699 as presented in detail in. It may be the case that, for the prediction of Newtonian film thickness, two definitions of the pressure-viscosity coefficient are required; one when the integral of Blok does converge for pressures that exist within the contact, a "Bridgman liquid", and another when the integral does not converge, a "Barus liquid". For the purpose of comparison, the dimensionless groups of Moes (2009) and of Hamrock and Dowson (1977) will be used without further comment as to the appropriateness of this selection; shear dependent viscosity and variable compressibility are ignored in these groups. The definition of pressure-viscosity coefficient employed here is the Barus coefficient, aB, from Eq. (3).

5. Mathematical Model

5.1. Reynolds Equation

The Reynolds equation applicable to any generalized Newtonian constitutive equation for the case of EHL line contacts is

^ ( pp^ 1 = (" 1 + U 2 ) (" 2 ~ " 1 ) $

d x \ d x j 2 0 x 2 d x

p\H - 2f

Where, p o, Fl, f 2 are the integral functions defined as

F0 = JI dy , Fl = J y~dy , F2 = J ^ | y - f V

The left hand side term in the Reynolds Equation is referred to as the Poisseuille term and the terms on the right hand side denote the Couette term that generates the hydrodynamic pressure in the gap.

The Reynolds Equation (4) is expressed in dimensionless form as

1 2 ) dxy ' 2 dX

pH\\- 2 A-

Where x = x / b is the non-dimensional x-coordinate along the film; p - p / p h is the non-dimensional pressure, p H being the maximum Hertzian pressure; -p - p /pa is the non-dimensional fluid density, p o being the inlet density of the lubricant; K = U (E '/p H ){R/b )3 is a dimensionless constant and S = (w2 - ui )/u0 is the

slide to roll ratio defined as the ratio of sliding velocity to average rolling velocity. Hence, S = 0 represents the condition of pure rolling.

The integral functions in dimensionless form are

F0 = dY F, = f = dY and F2 = f = | Y - ^ | dY (7)

0 7 , o 7 o 7 l F0 )

Where Y = y / h is the non-dimensional j-coordinate across the film and ^q = 77 / is the non-dimensional effective viscosity, ¡u0 being the inlet fluid viscosity.

5.2. Boundary Conditions Inlet boundary condition

P = 0 at X = Xin (8)

Outlet boundary condition

0 at x = Xo (9)

5.3. Film Thickness Equation

The film thickness Equation in non-dimensional form is given by

H (X )= H 0 + + 8 -W (10)

Where, 5 = 5 ■ R / b1 is the non-dimensional elastic deformation and ^ denotes the undeformed waviness geometry given by

S =-— |'P In (X - S 0 )2 dS 0

^ = A COS

2n *X - > {A! b )

5.4. Velocity and Velocity Gradient

The fluid velocity distribution for the generalized Newtonian flow along the rolling direction given by Equation is given below in dimensionless form

"= ^=(1 " D+^+ A° ( ^ " iI ^ (12)

Where u 0 = (w , + u 2 )/2 istheaveragerollingspeed and

A = l±f2 ^L, G0 = f 1 , G, = f ^dY (13)

0 u I * ) âx 0 J w i W

The non-dimensional velocity gradient ^M across the fluid film is obtained by differentiating Equation (12),

-r = âû_ = + _ (14)

7 WF, W I Fo J

5.5. Density-Pressure Relationship

P - P0 f 1 + (15)

^ 1 + 1.7 X 10 p )

Where, p0 = density at atmospheric pressure

5.6. Viscosity-Pressure Relationship

For the derivation of EHL film thickness equations for lubricants with linear piezo-viscosity, the Barus linear piezo-viscous Equation (3) is used which is

A = Ao (l + "BP ) (16)

Where, a B =linear pressure viscosity coefficient.

Roelands Equation (1963) given by

M = Ma exP [(in + 9.67 )x {- 1 + (l + 5.1 x 10 ^9 p )* }J (17)

5.1 x 10"9(in ¿u0 + 9.67 )

5.7. Load Equilibrium Equation

The pressure generated in the fluid film must balance the external applied load. The equation in non-dimensional form given as

¡PdX = *

5.8. Coefficient ofFriction

Coefficient of friction ( cof ) is taken as the ratio of total shear force of the lubricant at the surface boundaries and the applied normal load as expressed in non-dimensional form as

cof = U

x" {rvX

Table 1. Viscosities in mPa s for three liquids

Pressure Water/glycol at 40 C L7808 S40

MPa 45% Water 55% Water 165 C 220 C 50 C

0.1 51.1 48.2 1.22 0.709 12.47 45.1 180

50 59.5 54.3 2.1 1.371

100 68.5 62 3.21 1.95

200 88.4 75.2 6.22 3.41

Table 2. Input parameters

Inlet density of fluid, po 846 Kg/m3

Equivalent elastic modulus ofdisks, E' 2.24*10" Pa

Equivalent radius of the disks, R 0.01 m

Pressure-viscosity coefficient, a 12-20*10 9 Pa1

Slide to roll ratio, S 0

The domain, X -4<X< 1.5

Grid size, AX 0.01375

Inlet viscosity, ^o 0.04 Pa-s

Amplitude of waviness, A 0.1-0.4

Wavelength of waviness, X 0.12

Velocity, U 1011

Max. Hertzian pressure, Ph 1 Gpa

6. Results

The EHL simulation results are presented and the effect of different input parameters is studied here. Figure 2-7 has been shown here represent the pressure and fluid film thickness profile under steady state condition for both linear and exponential pressure-viscosity relationships.

6.1 Effect ofa Variation

Figure 5, Compares the pressure generation for linear and exponential piezo-viscosity at a=12GPa"1 and a =20GPa . There is not too much variation as value of a varies for both relations, but for exponential, pressure start increasing near inlet at higher a value .There is much pressure spikes in exponential fluid relate to linear fluid. For exponential, as a decreases maximum pressure increase and at higher a there is pressure spikes as shown in figure 5.

6.2 EffectofAmplitudeVariation

Figure 3 and figure 4 shows the comparison of Pressure variation for linear and exponential relation respectively, with amplitude variation. As amplitude increase the value pressure increase and due to increase in pressure film thickness decreases. During the start of cycle, at higher amplitude value of minimum film thickness is greater than the lower amplitude and after some time value of minimum film thickness for higher amplitude become less than lower amplitude and after half cycle time minimum film thickness increases and becomes greater than the lower amplitude.

6.3 Effect of linear and exponential lubricant onpressure distribution

Figure 2, shows the comparison of pressure variation for linear and exponential piezo-viscous fluid. The maximum pressure (P) in case of exponential is 1.375789 and for linear 1.179704 for the condition given. So maximum pressure for exponential is greater than linear and interesting point to be noted here is that the minimum film thickness for exponential case is 124.16 nm and for linear is 84.74361 nm i.e. greater in exponential though its pressure is greater and it can be seen in figure 2. The value of film thickness for exponential fluid is greater than the linear relation.

6.4 Effect oflinear and exponential lubricant on film thickness

Figure 6, shows the comparison of film thickness variation for linear and exponential piezo-viscous fluid. For condition, A=0.2, PH= 109, X=0.12, a=15GPa_1 and U=10_1 , minimum film thickness in case of exponential is 124.16 nm and for linear is 84.74361 nm, central film thickness in case of exponential is 156.124 nm and for linear is 102.042 nm. So the minimum film thickness for exponential is found to be greater than linear as shown in figure 6. At the same time maximum pressure (P) found in case of exponential is 1.375789 and for linear is 1.179704, in this case also pressure for exponential is greater than linear. Figure 6 is enlarged and plotted again by taking X values from -lto lto get clear view and shown in figure 7.

Fig. 2. Comparison of pressure for linear and exponential pressure viscosity relation at Ph=lGPa, U=1041, A=0.2 and a =15GPa4

Fig. 3. Comparison of pressure profile for amplitude variation A=0.1, A=0.4, at >,=0.12 a=15GPa4, U=1041, PH=109 Pa for linear relation

Fig. 4. Comparison of pressure profile for amplitude variation Amplitude=0.1 and Amplitude=0.4, at >=0.12 a=15GPa4, U=1041, PH=109 Pa for exponential relation

■ o=12 and Linear

■ o=20 and Linear

- o=12 and Exponential o=20 and Exponential

Fig. 5. Comparison of pressure for a variation at a=12GPa4 and a=20GPa4 at U=1041, A=0.2, >=0.12 and PH=lGPa for linear and exponential relation

Fig. 6. Comparison of minimum film thickness profile for A=0.2, Ph=109 Pa, >,=0.12, a=15GPa4, U=1041 for linear and exponential relation

Fig. 7. Comparison of minimum film thickness profile for A=0.2, Ph=109 Pa, >=0.12, a=15GPa U=1041 for linear and exponential relation

7. Conclusion

Full isothermal, Newtonian simulations of the EHL problem reveal film-forming, which is quite similar to that of exponential liquid. The problem of waviness amplitude in an EHL line contact at steady state, Newtonian fluid and piezo-viscous effect has been studied in this paper by means of the amplitude and wavelength of the film thickness oscillation in the centre of contact by taking domain of -4 < X < 1.5 and gride size AX=0.01375. The minimum film thickness, Central film thickness and pressure variation is found to be greater in case of exponential fluid with linear fluid at certain condition. Further work is required to analyze the change in amplitude and wavelength behaviour of surface roughness under general rolling/sliding contact.

Acknowledgement

Nitesh Talekar is thankful to NIT Kurukshetra, Haryana, India-136119, for the financial support in the form of Scholarship.

References

Chang L, Cusano C and Conry TF. Effects of Lubricant Rheology and Kinematic Conditions Micro-Elastohydrodynamic Lubrication. ASME Journal ofTribology 1988;111:344.

Ai X and Zheng L. A General Model for Microelastohydrodynamic Lubrication and its Full Numerical Solution. ASME Journal of Tribology 1989;111:569-576.

Kweh CC., Patching M. J., Evans H. P. and Snidle R. W. Simulation of Elastohydrodynamic Contacts Between Rough Surfaces. ASME Journal ofTribology 1992;114:412-419.

Lubrecht AA, ten Napel WE and Bosma R. The Influence of Longitudinal and Transverse Roughness on Elastohydrodynamic Lubrication of Circular Contacts. ASME Journal ofTribology 1989; 110: 421-426.

Venner CH, and Napel WE ten. Surface Roughness Effects in EHL Line Contacts. ASME Journal ofTribology 1992; 114: 616-622.

Barns C. Note on the dependence of viscosity on pressure and temperature. Proc Am Acad Arts Sci 1891-1892;27:13-8.

Barns C. Isothermals, isopiestics and isometrics relative to viscosity. Am J Sci Third Ser 1893;XLV(266):87-96.

Harris KR, Woolf LA. Temperature and pressure dependence of the viscosity of the ionic liquid l-Butyl-3-methylimidazolium hexafluorophosphate. J Chem Eng Data 2005;50:1777-82.

Bair S, Liu Y, Wang QJ. The Pressure-viscosity coefficient for newtonian EHL film thickness with general piezoviscous response. ASME J Tribol 2006;123(3):624-31.

Blok H. Inverse problems in hydrodynamic lubrication and design directives for lubricated flexible surfaces. In: Muster D, Sternlicht B, editors.

Johnson KL. Regimes of elastohydrodynamic lubrication. J Mech Eng Sci 1970;12(1):9-16.

Van Leeuwen H. The determination of the pressure-viscosity coefficient of a lubricant through an accurate film thickness formula and accurate film thickness measurements. Proc InstumMech Eng J 2009;223: 1143-1163.

Bair S. High-pressure rheology for quantitative elastohydrodynamics. Amsterdam: Elsevier Science 2007:123-71.

Hamrock BJ, Dowson D. Isothermal elastohydrodynamic lubrication of point contacts. Part III-fully flooded results. J Lubr Techno 1977;99:264— 76.

Dowson D, Higginson GR. Elasto-hydrodynamic lubrication, the fundamentals of roller and gear lubrication. Oxford: Pergamon Press 1966.

Hamrock BJ, Dowson D. Ball bearing lubrication the elastohydrodynamics of elliptical contacts. NewYork, NYJohn Wiley and Sons 1981.

Kumar P, Bair S, Krupkal, Hartl M . Newtonian quantitative elastohydrodynamic film thickness with linear piezoviscosity. Tribology International 2010;43:2159-65.

Harris KR, Malhotra R, Woolf LA. Temperature and density dependence of the viscosity of octane and toluene. Journal of Chemical and Engineering Data 1997;42:1254-60.

Spikes HA. Wear and fatigue problems in connection with water-based hydraulic fluids. Journal of Synthetic Lubrication 1987;4(2):115-35.

P Anuradha, Punit Kumar, EHL line contact central and minimum film thickness equations for lubricants with linear piezo-viscous behaviour. Tribology International 2011; 44: 2157-60.