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Analysis of grease lubricated isoviscous elastic point contacts-2-3-4
Analysis of grease lubricated isoviscous elastic point contacts-2-3-4
Analysis of grease lubricated isoviscous elastic point contacts-2-3-4
Analysis of grease lubricated isoviscous elastic point contacts-2-3-4
Analysis of grease lubricated isoviscous elastic point contacts-2-3-4
Analysis of grease lubricated isoviscous elastic point contacts-2-3-4
Analysis of grease lubricated isoviscous elastic point contacts-2-3-4
Analysis of grease lubricated isoviscous elastic point contacts-2-3-4
Analysis of grease lubricated isoviscous elastic point contacts-2-3-4
Analysis of grease lubricated isoviscous elastic point contacts-2-3-4
Analysis of grease lubricated isoviscous elastic point contacts-2-3-4
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Analysis of grease lubricated isoviscous elastic point contacts-2-3-4

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  • 1. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 5, July – August (2013), © IAEME 207 ANALYSIS OF GREASE LUBRICATED ISOVISCOUS-ELASTIC POINT CONTACTS Vivek Chacko1 Department of Mechanical Engineering, Saintgits College of Engineering, Kottayam, Kerala, India Bindu Kumar Karthikeyan2 Department of Mechanical Engineering, Government Engineering College, Thiruvananthapuram, Kerala, India ABSTRACT Greases generally behave as shear-thinning or pseudo-plastic fluids- their viscosity reduces under shear i.e. with sufficient shear the viscosity of grease approaches that of the base lubricant. By this behaviour grease may be considered as a plastic fluid. Numerical solution of the modified Reynolds equation for grease for point contacts remains challenging, despite the advent of powerful computational techniques and platforms. Grease lubricated concentrated point contacts are analysed under isothermal conditions. Grease is seen to provide better lubrication in comparison with oil lubricant, thus protracting the life of the contact conjunction. Operating conditions have been defined by dimensionless control parameters. The asymptotic behaviour of grease allows for the development of film thickness equations. From the results of the numerical method, extrapolated equations for grease film thickness in iso-viscous elastic regime lubrication is developed using the dimensionless parameters. Keywords: Grease lubrication, Hydrodynamic, Herschel-Bulkley Model, Moes Parameters, Iso-viscous elastic 1. INTRODUCTION Lubrication theories for oil lubricated contacts have been well documented; however the theory for grease lubrication lags considerably behind because of the complexity of its rheological properties. In practice, approximately 80–90% of rolling element bearings is lubricated with grease. The understanding of the rheological behaviour of lubricating greases is nowadays a decisive factor in the design and optimisation of the tribological systems as well as in the control of their processing. INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN ENGINEERING AND TECHNOLOGY (IJARET) ISSN 0976 - 6480 (Print) ISSN 0976 - 6499 (Online) Volume 4, Issue 5, July – August 2013, pp. 207-217 © IAEME: www.iaeme.com/ijaret.asp Journal Impact Factor (2013): 5.8376 (Calculated by GISI) www.jifactor.com IJARET © I A E M E
  • 2. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 5, July – August (2013), © IAEME 208 Greases are two-phase lubricants composed of a thickener dispersed in base oil. Due to the effects of the thickener, greases are often modelled as a plastic solid. Unlike oil, grease can withstand shear and will not flow until a critical yield stress is reached. Traditionally, these properties of grease have been related to the so-called ‘yield state’ at low strain rates and a shear-thinning behaviour at medium and high strain rates. Balan& Franco (2001) recorded that the typical grease flow curve exhibits constant values of shear stress at low strain rates. The behaviour of grease inside a contact conjunction differs from that for oil so the Reynolds equation used for oil lubricated contacts cannot be used directly for grease lubricated contacts. The flow of grease follows a non-Newtonian behaviour. The numerical solution of the modified Reynolds equation for grease remains challenging despite the advent of powerful computational techniques and platforms. Herschel and Bulkley [1] presented the Herschel–Bulkley (HB) equation as a realistic constitutive model for grease behaviour. Kauzlarich and Greenwood [2] found that most grease behave pseudo-plastically.Kauzlarich and Greenwood [2], Balan& Franco [3], Jonkisz and Freda [4], Zhu and Neng[5],Yoo and Kim [8] have carried out numerical method research in grease lubricated using HB Model. But all the studies described are limited to line contact conjunctions. Numerical model for grease lubrication capable of solving EHL of circular point contacts for isoviscous elastic regime of lubrication has been developed herein. For this a modified Reynolds equation is derived considering grease as a Bingham solid incorporating Herschel Bulkley flow model. Results from the numerical method, with input values taken from Williamson [17] are found to be well comparable to experimental film thickness. The generic nature of the numerical method developed is confirmed. A range of load, speed, material and lubricant viscosity parameters have then been analysed. Extrapolated film thickness equation, incorporating asymptotic behaviour of grease in the iso- viscous elastic (IE) regime of lubrication, is developed herein. The film thickness and corresponding dimensionless parameters are tabulated from the results of the numerical method. The equation for grease lubricant film thickness in IE regime in terms of dimensionless Moes’ Parameters [18] is reported in this paper. 2.MATHEMATICAL MODEL The mathematical model is composed of the Modified Reynolds’ equation for grease flow [9and10], the film thickness equation, the load balance equation, together with the expression of viscosity and density of lubricant. Details of the model and the numerical methods can be seen in refs. [9], [10]. For the numerical analysis grease flow Herschel–Bulkley flow equation is selected, which is described by a three-parameter rheological model as [1]: 0 hn p Dτ τ η= + (1) Assuming no side leakage, modified Reynolds’ equation for grease flow the equation can be simplified to [10]: (2) ( ) ( )3 3 12a a b p p p hh hp p u h uh x x y y x t ρρ ρ ρ η η      ∂∂ ∂ ∂ ∂ ∂ + = − +        ∂ ∂ ∂ ∂ ∂ ∂    
  • 3. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 5, July – August (2013), © IAEME 209 For oil lubricated contacts with no core formation and plug flow: and the plug flow velocities equate to that of the base oil, thus: and . This yields the usual Reynolds’ equation. a. LUBRICANT RHEOLOGY The density dependence on pressure follows the usual relationship given by Dowson and Higginson [11], in dimensionless form, is considered here: (3) The values are , In general, grease is assumed to be fully shear-degraded by its passage through the contact inlet when its structure becomes a large scale tri-dimensional network of discrete spherical soap particles, dispersed in the base oil [12]. In this case the viscosity of the grease would conform to: (4) where is the plastic viscosity of grease, is the base oil viscosity, is a constant and is the volume fraction of soap in oil. The region of contact in rolling element bearings is subjected to high pressures. Therefore, the viscosity of the grease inside the rolling element contact cannot remain constant, and changes with the pressure increase. In order to calculate the variation of viscosity owing to pressure, the Reolands’ equation[9]is used and is expressed as: ( )0ln 1 .2 ln 1 .2 1 2 0 0 0 R P φ φ   + = + +    (5) where P is the gauge pressure and R is the Reolands’ viscosity-pressure index. Elastohydrodynamic lubrication has a characteristically flat film shape, with parallel surfaces at contact. b. THICKNESS OF PLUG To find a solution the thickness of grease plug must be determined at any position in the conjunction, (6) c. ELASTIC FILM SHAPE The elastohydrodynamic film shape is given by the approximate parabolic shape of the contact of a ellipsoidal solid near a flat semi-infinite elastic half-space subjected to a localised 0ph = p bu u= p bv v= , , , . 1 1 . h i j i j h i j P P P P α ρ β ⋅ = + + ⋅ 1 0 5 .8 3 1 0α − = × 9 1 . 6 8 1 0β − = × ( )1p b o Bη η= + Φ pη b oη 0 .2 5B ≈ Φ ,x y 0 22 2 ph p p x y τ =   ∂ ∂  +    ∂ ∂    
  • 4. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 5, July – August (2013), © IAEME 210 Hertzian deformation. With an initial gap of the elastic film shape for a spherical solid such as a ball bearing is [13,14]: (7) is the local deformation d. METHOD OF SOLUTION The value of pressure at iteration within a time step is obtained as: (8) where is an under-relaxation factor. In each small step of time, two convergence criteria are required: one for pressures and the other for the contact load, which should attain the value determined by a previous dynamic analysis, for instance load per ball-to-race contact in a ball bearing. For pressure convergence: (9) The error tolerance is usually in the range of . For load convergence the following criterion is used: (10) The error tolerance .If the criterion in (9) is not satisfied the film thickness is relaxed as: (11) where is referred to as a damping factor. 0h ( ) ( ) 2 2 0, , 2 2x y x y h x y h x y R R δ= + + + ( ),x yδ k 1 , , , k k k i j i j i jp p p− = + Ω ∆ Ω ,i jp 1 , , 1 1 , 1 1 n yn x k k i j i j i j pn yn x k i j i j p p p ε − = = = = − ≤ ∑ ∑ ∑ ∑ pε 5 4 5 X 1 0 1 0− − − * wW π ε− ≤ 2 1 0wε − ≈ * * 1 * 0 0 k k H H Wξ π− = + − 7 1 0ξ − ≈
  • 5. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 5, July – August (2013), © IAEME 211 e. ALGORITHM FOR COMPUTER PROGRAM Figure 1 Algorithm for computer program f. VALIDATION OF NUMERICAL METHOD The computer program for the numerical method is executed for values of input conditions in Williamsons’ experimental analysis [16].The entrainment velocity vs. film thickness plot is shown below in Figure 2. Comparing the results of film thickness from the numerical method with those from experiment, a good match is obtained. The numerical results show a slightly higherpredicted film thickness over experimental results at higher entrainment velocities. A maximum difference of 10% in the film thickness is found at aninlet linear velocityof about 1.1 m/s. Viscosity of the lubricant is sensitive to temperature variations. Experimental investigations are prone to temperature variations, in contrast to the isothermal method assumed for numerical analysis. Also squeeze phenomenon is not factored in the current study. The slight difference in film thickness may be attributed to these the differences in considerations.
  • 6. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 5, July – August (2013), © IAEME 212 Figure 2 Overlay of Grease film thickness over experimental OFT (Williamson) g. DIMENSIONLESS PARAMETERS If lubricant is assumed to be incompressible and Roelands’ equation is followed for viscosity calculation inside contact conjunction, then three parameters can be defined to fully express the contact parameters in dimensionless from [17].These are the operating parametersM, L and the curvature ratio,D expressed as: ‫ܯ‬ ൌ ி ா′ோೣ మ ቀ ηబ ௎ ா′ோೣ ቁ షయ ర (12) ‫ܮ‬ ൌ αE′ ቀ ηబ ௎ ா′ோೣ ቁ భ ర (13) For circular point contacts, the ratio of reduced radii, D = Rx/Ry =1. The advantage over the conventional dimensionless parameters of speed, load and material is that the number of control parameters defining the contact conditions can be reduced for the given equation[18]. The load, speed and material parameters and the corresponding Moes’ Parameters have been tabulated in table 1. Regression analysis was used to determine the relation between dimensionless central film thickness and Moes’ parameters. 3. RESULTS AND DISCUSSION Grease lubrication studies for numerical solution of IE regime have not been reported hitherto in open literature. The analysis method is generic and may be valid across applications. Figure shows three dimensional pressure profile for IE regime of grease lubrication in point contact conjunctions. The maximum dimensionless load acting on the ball is W*=7.0088E-06. Centre line film thicknesses are assumed for further calculation since the maximum pressure is known to lie on this axis.
  • 7. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 5, July Figure 3: Pressure profiles for different speeds Figure 5: Corresponding film thicknesses Figure 7: Film thickness contour for W*=6.14241E-07 Moes Dimensionless Parameters are introduced for the formation of film thickness equation. Table 1 shows the input parameters along with the corresponding minimum film thicknesses from the results of the numerical method for Iso International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 6499(Online) Volume 4, Issue 5, July – August (2013), © IAEME 213 Pressure profiles for different speeds Figure 3: 3D Pressure Profile for W*=7 Corresponding film thicknesses Figure 6: W*=2.89743E Film thickness contour for Figure 8: W*=7.0088E-06 (for 168.5 rpm) 07 s Parameters are introduced for the formation of film thickness equation. Table 1 shows the input parameters along with the corresponding minimum film thicknesses from the results of the numerical method for Iso-viscous rigid regime lubrication and Moes’ p International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN August (2013), © IAEME Pressure Profile for W*=7.0088E-06 W*=2.89743E-06 06 (for 168.5 rpm) s Parameters are introduced for the formation of film thickness equation. Table 1 shows the input parameters along with the corresponding minimum film thicknesses from viscous rigid regime lubrication and Moes’ parameters.
  • 8. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 5, July – August (2013), © IAEME 214 The variation of material parameters is expected to bring slight variations to the coefficients in the below equation. Using suitable regression analysis, the film thickness equation is arrived at: ݄‫כ‬ ൌ ሺ1.8035‫01ݔ‬ିଶሻ ‫כ‬ ‫ܯ‬ି଴.ହ଴଺଺ ‫כ‬ ‫ܮ‬ିଵ.ଵ଼଺ଽ (14) Table 1 Contact characteristics in terms of Moes' Parameters Sl. No Load Point W (N) Rx (m) E'( Gpa) η0 us ɑɑɑɑ M L hcen(m) h* h* from equation Difference in film thickness (%) n=500 1 1 4.30143 6.70E-03 1.56E+11 0.0202 2.42 1.33E-08 34.28 5.42 2.18E 06 3.25E-04 4.05E-04 24.7 2 11 7.30756 6.70E-03 1.56E+11 0.0202 2.424 1.33E-08 58.23 5.42 1.97E-06 2.94E-04 3.10E-04 5.4 3 21 11.0558 6.70E-03 1.56E+11 0.0202 2.424 1.33E-08 88.10 5.42 1.77E-06 2.65E-04 2.51E-04 -5.2 4 31 15.4373 6.70E-03 1.56E+11 0.0202 2.424 1.33E-08 123.01 5.42 1.59E-06 2.37E-04 2.12E-04 -10.5 5 41 20.2902 6.70E-03 1.56E+11 0.0202 2.424 1.33E-08 161.68 5.42 1.41E-06 2.11E-04 1.85E-04 -12.6 6 51 25.4101 6.70E-03 1.56E+11 0.0202 2.424 1.33E-08 202.48 5.42 1.25E-06 1.87E-04 1.65E-04 -11.9 7 61 30.5624 6.70E-03 1.56E+11 0.0202 2.424 1.33E-08 243.53 5.42 1.11E-06 1.66E-04 1.50E-04 -9.7 8 71 35.4969 6.70E-03 1.56E+11 0.0202 2.424 1.33E-08 282.85 5.42 9.96E-07 1.49E-04 1.39E-04 -6.5 9 81 39.9641 6.70E-03 1.56E+11 0.0202 2.424 1.33E-08 318.45 5.42 8.99E-07 1.34E-04 1.31E-04 -2.5 10 91 43.7322 6.70E-03 1.56E+11 0.0202 2.424 1.33E-08 348.47 5.42 8.25E-07 1.23E-04 1.25E-04 1.5 11 101 46.6012 6.70E-03 1.56E+11 0.0202 2.424 1.33E-08 371.33 5.42 7.72E-07 1.15E-04 1.21E-04 5.0 12 111 48.4172 6.70E-03 1.56E+11 0.0202 2.424 1.33E-08 385.80 5.42 7.40E-07 1.11E-04 1.19E-04 7.5 13 121 49.0815 6.70E-03 1.56E+11 0.0202 2.424 1.33E-08 391.10 5.42 7.28E-07 1.09E-04 1.18E-04 8.5 n=250 1 1 4.30143 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 114.61 3.62 2.05E-06 3.06E-04 3.54E-04 15.9 2 11 7.30756 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 194.70 3.62 1.86E-06 2.77E-04 2.71E-04 -2.2 3 21 11.0558 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 294.57 3.62 1.66E-06 2.48E-04 2.20E-04 -11.6 4 31 15.4373 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 411.31 3.62 1.48E-06 2.20E-04 1.85E-04 -15.8 5 41 20.2902 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 540.61 3.62 1.31E-06 1.95E-04 1.61E-04 -17.2 6 51 25.4101 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 677.02 3.62 1.15E-06 1.72E-04 1.44E-04 -16.3 7 61 30.5624 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 814.30 3.62 1.02E-06 1.52E-04 1.31E-04 -13.6 8 71 35.4969 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 945.77 3.62 9.01E-07 1.35E-04 1.22E-04 -9.6 9 81 39.9641 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 1064.79 3.62 8.06E-07 1.20E-04 1.14E-04 -4.8 10 91 43.7322 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 1165.19 3.62 7.32E-07 1.09E-04 1.09E-04 0.1 11 101 46.6012 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 1241.63 3.62 6.79E-07 1.01E-04 1.06E-04 4.5 12 111 48.4172 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 1290.01 3.62 6.47E-07 9.65E-05 1.04E-04 7.6 13 121 49.0815 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 1307.71 3.62 6.36E-07 9.49E-05 1.03E-04 8.7 n=168.5 1 1 4.30143 6.70E-03 1.56E+11 0.0202 0.792 1.33E-08 79.31 4.10 2.00E-06 2.98E-04 3.69E-04 23.7 2 11 7.30756 6.70E-03 1.56E+11 0.0202 0.792 1.33E-08 134.74 4.10 1.81E-06 2.70E-04 2.82E-04 4.5 3 21 11.0558 6.70E-03 1.56E+11 0.0202 0.792 1.33E-08 203.85 4.10 1.61E-06 2.40E-04 2.29E-04 -4.9 4 31 15.4373 6.70E-03 1.56E+11 0.0202 0.792 1.33E-08 284.64 4.10 1.43E-06 2.14E-04 1.93E-04 -9.7 5 41 20.2902 6.70E-03 1.56E+11 0.0202 0.792 1.33E-08 374.12 4.10 1.27E-06 1.89E-04 1.68E-04 -11.0 6 51 25.4101 6.70E-03 1.56E+11 0.0202 0.792 1.33E-08 468.52 4.10 1.11E-06 1.66E-04 1.50E-04 -9.7 7 61 30.5624 6.70E-03 1.56E+11 0.0202 0.792 1.33E-08 563.52 4.10 9.78E-07 1.46E-04 1.37E-04 -6.4 8 71 35.4969 6.70E-03 1.56E+11 0.0202 0.792 1.33E-08 654.51 4.10 8.63E-07 1.29E-04 1.27E-04 -1.7 9 81 39.9641 6.70E-03 1.56E+11 0.0202 0.792 1.33E-08 736.87 4.10 7.69E-07 1.15E-04 1.19E-04 3.9 10 91 43.7322 6.70E-03 1.56E+11 0.0202 0.792 1.33E-08 806.35 4.10 6.95E-07 1.04E-04 1.14E-04 9.8 11 101 46.6012 6.70E-03 1.56E+11 0.0202 0.792 1.33E-08 859.25 4.10 6.43E-07 9.60E-05 1.10E-04 15.0 12 111 48.4172 6.70E-03 1.56E+11 0.0202 0.792 1.33E-08 892.74 4.10 6.12E-07 9.13E-05 1.08E-04 18.5 13 121 49.0815 6.70E-03 1.56E+11 0.0202 0.792 1.33E-08 904.98 4.10 6.01E-07 8.97E-05 1.07E-04 19.8 n=100 1 1 4.30143 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 114.61 3.62 1.96E-06 2.92E-04 3.54E-04 21.2 2 11 7.30756 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 194.70 3.62 1.77E-06 2.64E-04 2.71E-04 2.6 3 21 11.0558 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 294.57 3.62 1.57E-06 2.35E-04 2.20E-04 -6.6 4 31 15.4373 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 411.31 3.62 1.40E-06 2.09E-04 1.85E-04 -11.3 5 41 20.2902 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 540.61 3.62 1.23E-06 1.84E-04 1.61E-04 -12.3 6 51 25.4101 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 677.02 3.62 1.08E-06 1.61E-04 1.44E-04 -10.8 7 61 30.5624 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 814.30 3.62 9.45E-07 1.41E-04 1.31E-04 -7.1 8 71 35.4969 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 945.77 3.62 8.32E-07 1.24E-04 1.22E-04 -2.1 9 81 39.9641 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 1064.79 3.62 7.38E-07 1.10E-04 1.14E-04 4.0 10 91 43.7322 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 1165.19 3.62 6.65E-07 9.93E-05 1.09E-04 10.1 11 101 46.6012 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 1241.63 3.62 6.14E-07 9.17E-05 1.06E-04 15.6 12 111 48.4172 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 1290.01 3.62 5.83E-07 8.69E-05 1.04E-04 19.5 13 121 49.0815 6.70E-03 1.56E+11 0.0202 0.485 1.33E-08 1307.71 3.62 5.71E-07 8.53E-05 1.03E-04 21.0
  • 9. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 5, July – August (2013), © IAEME 215 4. CONCLUSION Grease is the preferred lubricant in most rolling-element bearings. The study of grease is desired in optimizing bearing performance. Grease behaves as pseudo-plastic fluid which makes it different from oil, which is a fluid. This necessitates an original study of the performance of grease starting from modified Reynolds equation, considering it as a Bingham plastic, based on which the above numerical model is developed. The analysis based on the numerical model is considered, and the above assumptions yield results which are well-comparable to established experimental data. Grease is found to offer better film thickness under same conditions than base oil. Also grease lubricant has a longer service life in comparison to oil lubricants. Better lubricant film thickness would in turn translate to extended life of the contact conjunction. The result from the numerical method is consolidated in terms of the dimensionless Moes Parameters. By method of regression analysis, the equation for film thickness of grease lubricated circular point contact under iso-viscous elastic regimes of EHL lubrication has been presented. APPENDIX Table 2 Lubricant and Geometrical Parameters
  • 10. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 5, July – August (2013), © IAEME 216 NOMENCLATURE REFERENCES [1] Herschel W.H and Bulkley R, “Measurement of consistency as applied to rubber benzene solutions”, Proc. Annual meeting – ASTM, 1926,621-633. [2] Kauzlarich JJ, Greenwood JA. ,"Elastohydrodynamic lubrication with Herschel–Bulkley model greases".ASLE Transactions, 1972, 15:269 –277. [3] Balan C, Franco JM. “Influence of the geometry on the transient and steady flow of lubricating greases”.Tribology Transactions 2001, 44:53 –58. [4] Jonkisz W, Krzeminski-Freda H. Pressure distribution and shape of an elastohydrodynamic grease films. Wear 1979, 55:81– 89. [5] Zhu WS, Neng YT. “A theoretical and experimental study of EHL lubricated with grease”. Transactions of the ASME 1988; 110:38– 42. [6] Jonkisz W, Krzeminski-Freda H. The properties of elastohydrodynamic grease films. Wear 1982; 77:277–285. [7] Yoo J, Kim K. (1997), Numerical analysis of grease thermalelastohydrodynamic lubrication problems using the Herschel–Bulkleymodel. Tribology International; 30:401– 408. [8] Roelands, C (1966), “Correlational aspects of viscosity-Temperature-Pressure relationship of Lubricating Oils”, PhD thesis, Delft University, Delft (VRBGroeningen, The Netherlands). [9] B.K Karthikeyan, “Tribodynamics of high speed precision spindle bearings” PhD thesis, Loughborough University, UK, December 2008. E’ Effective Elastic Modulus of contacting material GPa h o Initial film thickness (un-deformed) m H Film thickness m h een Centre filmthickness of contact m h * Dimensionless Film thickness (=W/E ’ Rx 2 ) - h p Plug flow thickness m n h HerschelBulkley index - P Pressure Pa Rx Radius of ball in entrainment direction m U Speed of entraining motion in x direction m/s u b Speed of entraining motion of the base oil m/s u p Speed of entraining of the plug flow in x direction m/s u s Speed parameter used in Moes Parameter m/s v Velocity of side-leakage in y direction m/s v b Base oilside-leakage velocity m/s v p Plug flow velocity in y direction m/s W Applied load N W i Load on 'i' th ball N W 0 Initial load N Z 0 Initial axialdeflection m Pressure- Viscosity coefficient of lubricant Pa -1 s δ0 Initial deflection m η0 Lubricant viscosity before entrainment Pa.s ηp Plastic viscosity Pa.s ρ Lubricant density kg/m 3 τ Shear Stress N/m 2 τ0 Yield shear stress N/m 2
  • 11. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 5, July – August (2013), © IAEME 217 [10] B K Karthikeyan, M Teodorescu, H Rahnejat and S J Rothberg, "Thermoelastohydrodynamics of grease-lubricated concentrated point contacts", ProcInstnMechEngrsPart C: Journal of Mechanical Engineering ScienceVolume 224, Number 3 / 2010, 683-695. [11] Dowson, D. and Higginson, G. R., “A numerical solution to the elastohydrodynamic problem", J. Mech. Engng. Sci., 1959, 1(1), 6–15. [12] Mansot, J. L., Terech, P. and Martin, J. M., “Structural investigation of lubricating greases”, Colloids and Surfaces, 1989, 39, 321. [13] Cameron, A., The Principles of Lubrication, Longman Press, London, 1966. [14] Gohar, R. and Rahnejat, H., Fundamentals of Tribology, Imperial College Press, London, 2008. [15] Wada, S., Hayashi H., Haga K, Kawakami, Y., Okjims M., "Elastohydrodynamic lubrication of a Bingham solid", Bull. JSME, 1977, 20, 110 –115. [16] Williamson B.P., “An optical study of grease rheology in Elastohydrodynamic point contact under fully flooded and starvation conditions”, J. Engrng Trib. Part J, ProcInstnMechEngrs, Vol 209:63-74 [17] Moes, H, “Optimum Similarity Analysis with application to Elastohydrodynamic lubrication”, Wear 159, 57-66, 1992 [18] Nijenbanning,G., Venner, C.H., Moes,H., “Film thickness in elastohydrodynamically lubricated elliptic contacts”, Wear 176:217-229, 1994. [19] Rajneesh Kakar, Kanwaljeet Kaur and K. C. Gupta, “Viscoelastic Modeling of Aortic Excessive Enlargement of an Artery”, International Journal of Mechanical Engineering & Technology (IJMET), Volume 4, Issue 2, 2013, pp. 479 - 493, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359.

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