# Numerical Simulation of Rough Thrust Pad Bearing Under Thin-Film Lubrication Using Variable Mesh Density

• Rahul Kumar
• Subrata Kumar Ghosh
• Hasim Khan
Research paper

## Abstract

A numerical simulation of rough slider bearing under thin-film lubrication using variable mesh density has been carried out. The current investigation deals with the effect of deterministic surface roughness patterns, such as triangular, sawtooth, square and sinusoidal roughness patterns and temperature on pressure and film thickness distribution. The nature and shape of roughness and temperature play a significant role in pressure generation, which in turn influences load capacity and frictional coefficient. It has been observed that variable mesh density takes around 25% less number of iterations compared to fixed mesh density. Among all roughness patterns, square roughness dominates the generated pressure. Elastic deformation of greater than 50 nm in bounding surfaces has been found to influence the film formation. Small pressure generated under piezoviscous elastic and thermo-piezoviscous elastic conditions was deficient in causing squeezing effect on lubricant, which suggests that bearing is operating under isoviscous elastic and thermo-viscous elastic conditions instead of piezoviscous elastic and thermo-piezoviscous elastic. Insensitivity in load capacity was observed for a smaller value of film thickness.

## Keywords

Elastic deformation Slider bearing Thin-film lubrication Deterministic surface roughness Variable mesh density (VMD)

## List of symbols

$$\rho$$

Lubricant’s density (kg/m3)

$$h(x)$$

Film thickness (m)

$$p$$

Fluid film pressure (Pa)

$$\mu$$

Lubricant’s viscosity (Pa s)

$$U_{s}$$

Moving surface sliding velocity (m/s)

$$l$$

$$X$$

Non-dimensional width of the pad $$X = {\raise0.7ex\hbox{x} \!\mathord{\left/ {\vphantom {x l}}\right.\kern-0pt} \!\lower0.7ex\hbox{l}}$$

$$P$$

Non-dimensional pressure of the fluid film $$P = {\raise0.7ex\hbox{{ph_{2}^{2} }} \!\mathord{\left/ {\vphantom {{ph_{2}^{2} } {6U_{s} l\mu_{0} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{{6U_{s} l\mu_{0} }}}$$

$$H(X)$$

Non-dimensional film thickness $$H = {\raise0.7ex\hbox{h} \!\mathord{\left/ {\vphantom {h {h_{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{{h_{2} }}}$$

$$\mu^{*}$$

Non-dimensional viscosity $$\mu^{*} = {\raise0.7ex\hbox{\mu } \!\mathord{\left/ {\vphantom {\mu {\mu_{0} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{{\mu_{0} }}}$$

$$\rho^{*}$$

Non-dimensional density $$\rho^{*} = {\raise0.7ex\hbox{\rho } \!\mathord{\left/ {\vphantom {\rho {\rho_{0} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{{\rho_{0} }}}$$

$$h_{1}$$

Film thickness at inlet (m)

$$h_{2}$$

Film thickness at outlet (m)

$$\delta (x)$$

Elastic deformation of the moving surface (m)

$$\delta_{1} (x)$$

Deterministic roughness pattern (m)

$$E_{1}$$

Elastic modulus of moving surface (Pa)

$$E_{2}$$

Elastic modulus of stationary pad (Pa)

$$E^{{\prime }}$$

Equivalent elastic modulus of moving surface (Pa)

$$\vartheta_{1}$$

Poisson’s ratio of moving surface

$$\vartheta_{2}$$

$$A$$

Amplitude of the roughness (nm)

$$\lambda$$

Wavelength of roughness (mm)

Non-dimensional distance between two nodes

$$\delta^{*} (X)$$

Non-dimensional elastic deformation of the moving surface

$$\delta_{1}^{*} (X)$$

Non-dimensional deterministic surface roughness factor

$$k$$

Film thickness ratio $$k = {\raise0.7ex\hbox{{h_{1} }} \!\mathord{\left/ {\vphantom {{h_{1} } {h_{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{{h_{2} }}}$$

$$T_{0}$$

Temperature of the lubricant at inlet (°C)

$$T$$

Temperature of the lubricant (°C)

$$T^{*}$$

Non-dimensional temperature $$T^{*} = {\raise0.7ex\hbox{T} \!\mathord{\left/ {\vphantom {T {T_{0} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{{T_{0} }}}$$

$$\gamma$$

Thermal viscosity coefficient of lubricant (°C−1)

$$\alpha$$

Pressure viscosity coefficient (Pa−1)

$$\beta$$

Coefficient of lubricant thermal expansivity (°C−1)

$$\mu_{0}$$

Viscosity at p = 0 (Pa s)

$$\rho_{0}$$

Density at p = 0 (kg/m3)

$$C_{p}$$

Specific heat of the lubricant (J Kg−1 °C−1)

$$q$$

Mass flow rate (Kg/(m s))

$$q_{c}$$

Couette mass flow rate (Kg/(m s))

$$q_{p}$$

Poiseuille mass flow rate (Kg/(m s))

$$Q$$

Non-dimensional mass flow rate $$Q = \frac{q}{{\rho_{0} U_{s} h_{2} }}$$

$$Q_{c}$$

Non-dimensional Couette mass flow rate $$Q_{c} = \frac{{q_{c} }}{{\rho_{0} U_{s} h_{2} }}$$

$$Q_{p}$$

Non-dimensional Poiseuille mass flow rate $$Q_{p} = \frac{{q_{p} }}{{\rho_{0} U_{s} h_{2} }}$$

$$w$$

$$w^{*}$$

Non-dimensional load capacity $$w^{*} = \frac{{wh_{2}^{2} }}{{6\mu_{0} U_{s} l^{2} }}$$

$$f$$

Frictional force (N)

$$F$$

Non-dimensional frictional force

$$F^{*}$$

Non-dimensional coefficient of friction

GPa

Gigapascal

## References

1. Bakolas V (2004) Analysis of rough line contacts operating under mixed elastohydrodynamic lubrication conditions. Lubr Sci 16:153–168.
2. Baudry RA, Kuhn EC, Wise WW (1958) Influence of load and thermal distortion on the design of large thrust bearings. Trans ASME 58:807–818Google Scholar
3. Bennett A, Ettles C (1967) A self-acting parallel surface thrust bearing. Proc Inst Mech Eng Part 3N 182:139–146Google Scholar
5. Cameron A (1960) New theory for parallel surface thrust bearing. Engineering 190:904Google Scholar
6. Cameron A, Robinson CL (1975) Studies in hydrodynamic thrust bearings. I. Theory considering thermal and elastic distortions. Pholos Trans R Soc Lond Ser A Math Phys Sci 278(1283):351–366.
7. Carl TE (1963) An experimental investigation of a cylindrical journal bearing under constant and sinusoidal loading. Proc Inst Mech Eng (Part 3 N) 178:100–119Google Scholar
8. Choo JW, Olver AV, Spikes HA (2007) The influence of transverse roughness in thin film, mixed elastohydrodynamic lubrication. Tribol Int 40:220–232.
9. de Guerin D, Hall LF (1957) Some characteristics of conventional tilting-pad thrust bearings. In: Proceedings of the conference on lubrication and wear. The Institution of Mechanical Engineers, London, pp 142–146Google Scholar
10. Dobrica MD, Fillon M (2005) Reynolds’ model suitability in simulating Rayleigh step bearing thermohydrodynamic problems. Tribol Trans 48:522–530.
11. Dobrica MD, Fillon M (2009) About the validity of Reynolds equation and inertia effects in textured sliders of infinite width. Proc Inst Mech Eng Part J J Eng Tribol 223:69–78.
12. Dowson D (1967) Elastohydrodynamics. Proc Inst Mech Eng 182(6):151–167Google Scholar
13. Dowson D, Higginson GR (1959) A numerical solution to the elasto-hydrodynamic problem. J Mech Eng Sci 1(1):6–15.
14. Ettels CMM, Cameron A (1964) Thermal and elastic distribution in thrust-bearings. In: Proceedings of the lubrication and wear convention. The Institution of Mechanical Engineers, London, pp 60–71Google Scholar
15. Ettles CMM (1980) Size effects in tilting pad thrust bearings. Wear 59(1):231–245.
16. Gananath DT, Sharma SC, Harsha SP, Tyagi MR (2016) A theoretical study of ionic liquid lubricated µ-EHL line contacts considering surface texture. Tribol Int 94:39–51.
17. Grubin AN (1949) Investigation of the contact of machine components. In: Central Scientific Research Institute for Technology and Mechanical Engineers: DSIR translation, Moscow, 337: 30Google Scholar
18. Hemingway EW (1964) The measurement of film thickness in thrust bearing and the deflected shape of parallel surface thrust pads. Proc Inst Mech Eng (Part 1) 180(1):1025–1034Google Scholar
19. Higginson GR (1965) The theoretical effect of elastic deformation of the bearing liner on journal bearing performance. Proc Inst Mech Eng (Part 3B) 180:31–38Google Scholar
20. Higginson G, Dowson D (1977) Elastohydrodynamic lubrication. Pergamon press, OxfordGoogle Scholar
21. Höhn BR, Michaelis K, Kreil O (2006) Influence of surface roughness on pressure distribution and film thickness in EHL-contacts. Tribol Int 39(12):1719–1725.
22. Hooke CJ, Brighton DK, O’Donoghue J (1967) The effect of elastic distortions on the performance of thin shell bearings. Proc Inst Mech Eng (Part 3B) 181:63–69.
23. Houpert LG, Hamrock BJ (1986) Fast Approach for calculating film thickness and pressure in elastohydrodynamically lubricated contacts at high load. Trans ASME J Tribol 108(3):411–420
24. Huang P (2013) Numerical calculation of lubrication, 1st edn. Wiley, Guangzhou, pp 117–124
25. Jain SC, Sinhasan R, Singh DV (1982) Effect of bearing pad deformation on the performance of finite fixed-pad slider bearings. Wear 76:189–198
26. Jin ZM, Dowson D (2005) Elastohydrodynamic lubrication in biological systems. IMechE Part J J Eng Tribol 219:367–380
27. Johnson KL (1970) Regimes of elastohydrodynamic lubrication. J Mech Eng Sci 12(1):9–16
28. Kaneta M, Todoroki H, Nishikawa H (1992) Tribology of flexible seals reciprocating motion. Trans ASME J Tribol 144:290–296Google Scholar
29. Kotia A, Ghosh SK (2015) Experimental analysis for rheological properties of aluminium oxide (Al2O3)/gear oil (SAE EP-90) nanolubricant used in HEMM. Ind Lubr Tribol 67(6):600–605.
30. Krupka I, Koutny D, Hartl M (2008) Behavior of real roughness features within mixed lubricated non-conformal contacts. Tribol Int 41:1153–1160
31. Krupka I, Sperks P, Hartl M et al (2010) Effect of real longitudinal surface roughness on lubrication film formation within line elastohydrodynamic contact. Tribol Int 43:2384–2389.
32. Kumar P, Jain S, Ray S (2001) Study of surface roughness effects in elastohydrodynamic lubrication of rolling line contacts using a deterministic model. Tribol Int 34(10):713–722.
33. Kumar R, Azam MdS, Ghosh SK, Khan H (2017) Effect of surface roughness and deformation on Rayleigh step bearing under thin film lubrication. Ind Lubr Tribol 69(6):1016–1032.
34. Lia X, Guo F, Yang S, Wong PL (2012) Measurement of Load Carrying capacity of Thin Lubricating Films. ASMEJ Tribol 134:044501–1-5.
35. Lundberg J (1995) Influence of surface roughness on normal-sliding lubrication. Tribol Int 28(5):317–322.
36. Malvano R, Vatta F (1986) Elastohydrodynamic lubrication in a plane slider bearing. Meccanica 21(3):134–139.
37. Martin HM (1916) Lubrication of gear teeth. Eng Lond 102:119–121Google Scholar
38. Mc Callion H, Yousif F, Lloyd T (1970) The analysis of thermal effects in a full journal bearing. Trans ASME J Lubr Technol 92:578–587
39. Nakamura T, Lakawathana P, Matsubara T et al (1999) Isoviscous-EHL mechanism of parallel slide-way with oil groove. J Jpn Soc of Tribollogists 44(4):258–264Google Scholar
40. O’Donoghue J, Brioghton DK, Hooke CJ (1967) The effect of elastic distortions on journal bearing performance. Trans ASME J Lubr Technol 89(4):409
41. Osterle F, Sabiel E (1958) The effect of bearing deformation in slider bearing lubrication. ASLE Trans 1(1):213–216.
42. Patir N, Cheng HS (1978) An average flow model for determining the effects of three-dimensional roughness on partial hydrodynamic lubrication. ASME. J Tribol 100:12–17Google Scholar
43. Prakash J, Peeken H (1985) The combined effect of surface roughness and elastic deformation in the hydrodynamic slider bearing problem. ASLE Trans 28(1):69–74.
44. Ramanaiah G, Sljndarammal A (1982) Effect of bearing deformation on the characteristics of a slider bearing. Wear 78:273–278
45. Rhode SM, Oh KP (1975) A thermoelastohydrodynamic analysis of a finite slider bearing. Trans ASME J Lubr Technol 97:450–460
46. Robinson CL, Cameron A (1975) Studies in hydrodynamic thrust bearings. II. Comparison of calculated and measured performance of tilting pads by means of interferometry. Pholos Trans R Soc Lond Ser A Math Phys Sci 278(1283):367–384.
47. Roelands C, Vlugter J, Watermann H (1963) The viscosity-temperature-pressure relationship of lubricating oils and its correlation with chemical constitution. ASME J Basic Eng 85:601–610
48. Shyu SH, Hsu WC (2018) A numerical study on the negligibility of cross-film pressure variation in infinitely wide plane slider bearing, Rayleigh step bearing and micro-grooved parallel slider bearing. Int J Mech Sci 137:315–323.
49. Shyu SH, Talmage M, Carpino M (2000) Comparison of lubrication models for plane slider bearings. Tribol Trans 43(1):74–81.
50. Sudeep U, Pandey RK, Tondon N (2013) Effects of surface texturing on friction and vibration behaviors of sliding lubricated concentrated point contacts under linear reciprocating motion. Tribol Int 62:198–207.
51. Xiao L, Rosen B, Amini N et al (2003) A study on the effect of surface topography on rough friction in roller contact. Wear 254(11):1162–1169.
52. Yagi K, Sugimura J (2013) Elastic deformation in thin film hydrodynamic lubrication. Tribol Int 59:170–180.
53. Yagi K, Kyogoku K, Nakahara T (2006) Experimental investigation of effects of slip ratio on elastohydrodynamic lubrication film related to temperature distribution in oil films. Proc IMech E Part J J Eng Tribol 220(4):353–363
54. Yagi K, Baba J, Abe T et al. (2010) Behaviour of oil in thin film lubrication with moving textures. In: Proceedings of KAST Tribology Conference, Tokyo, Japan, pp 373–374Google Scholar