: LICENTIATE T H E SI S Rough Surface Elastohydrodynamic Lubrication and Contact Mechanics Andreas Almqvist Luleå University of Technology Department of Applied Physics and Mechanical Engineering, Division of Machine Elements :|: -|: - -- ⁄ -- 2004:35 ROUGH SURFACE ELASTOHYDRODYNAMIC LUBRICATION AND CONTACT MECHANICS ANDREAS ALMQVIST Luleå University of Technology Department of Applied Physics and Mechanical Engineering, Division of Machine Elements 2004 : 35| ISSN : 1402 − 1757|ISRN:LTU-LIC--04/35--SE Cover figure: A modeled surface topography pressed against a rigid plane, assuming linear elastic surface material. The theory describing the contact mechanics tool used to produce this result is given in Chapter 3 Title page figure: Elementary surface features passing each other inside the EHD lubricated conjunction, see Fig. 6.5 for details. ROUGH SURFACE ELASTOHYDRODYNAMIC LUBRICATION AND CONTACT MECHANICS Copyright c Andreas Almqvist (2004). This document is freely available at http://epubl.ltu.se/1402-1757/2004/35 or by contacting Andreas Almqvist, [email protected] The document may be freely distributed in its original form including the current author’s name. None of the content may be changed or excluded without permissions from the author. ISSN: 1402-1757 ISRN: LTU-LIC--04/35--SE This document was typeset in LATEX2ε . Abstract In the field of tribology, there are numerous theoretical models that may be described mathematically in the form of integro-differential systems of equations. Some of these systems of equations are sufficiently well posed to allow for numerical solutions to be car- ried out resulting in accurate predictions. This work has focused on the contact between rough surfaces with or without a separating lubricant film. The objective was to investi- gate how surface topography influences contact conditions. For this purpose two different numerical methods were developed and used. For the lubricated contact between rough surfaces the Reynolds equation were used as a basis. This equation is derived under the assumptions of thin fluid film and creeping flow. In highly loaded, lubricated, non-conformal contacts of surfaces after running-in, the load concentration no longer results in plastic deformations, however large elastic de- formations will be apparent. It is the interaction between the hydrodynamic action of the lubricant and the elastic deformations of the surfaces that, in certain applications, enable the lubricant film to fully separate the surfaces. This is commonly referred to as full film elastohydrodynamic (EHD) lubrication. Typical machine elements that operate in the full film EHD lubrication (FL) regime include rolling element bearings, cams and gears. Unfortunately, a cost effective way of machining engineering surfaces seldom results in a surface topography that influence contact conditions in the same way as a surface after running-in. Such topographies may prevent the lubricant from fully separating the surfaces because of deteriorated hydrodynamic action. In this case the applied load is carried in part by the lubricant and in part by surface asperities and/or surface active lubricant additives. This could also be the case in lubricant starved contacts, which is a common situation in not only grease lubricated contacts but also in many liquid lubricated contacts, such as high speed operating rolling element bearings. The load sharing between the highly compressed lubricant and the surface and/or surface active lubricant additives is the reason why this lubrication regime is most commonly referred to as mixed EHD lubrication (ML). Machine elements that while running operate in the FL regime may experience a transition into the ML regime at stops or due to altered operating conditions. It is not possible to simulate direct contact between the surfaces using a numerical method based on Reynolds equation. A parameter study, of elementary surface features passing each other inside the EHD lubricated conjunction, was performed. The results obtained, even though no direct contact could be simulated, does indicate that a transition from the FL to the ML regime would occur for certain combinations of the varied parameters. At start-ups, the contact in a rolling element bearing could be both starved and drained from lubricant. In this case the hydrodynamic action becomes negligible in terms of load carrying capacity. The load is carried exclusively by surface asperities and/or surface ac- i tive lubricant additives. This regime is referred to as boundary lubrication (BL). Operation conditions could also make both FL and ML impossible to achieve, for example, in the case in a low rpm operating rolling element bearing. The BL regime is in this work mod- eled as the unlubricated frictionless contact between rough surfaces, i.e., a dry contact approach. A variational principle was used in which the real area of contact and contact pressure distribution are those which minimize the total complementary energy. A lin- ear elastic-perfectly plastic deformation model in which energy dissipation due to plastic deformation is accounted for was used. The dry contact method was applied to the contact between four different profiles and a plane. The variation in the real area of contact, the plasticity index and some surface roughness parameters due to applied load were investigated. The surface roughness pa- rameters of the profiles differed significantly. Contents I The Thesis 1 1 Introduction 3 1.1 Elastohydrodynamic lubrication . .................... 3 1.2 Lubrication regimes . ........................... 4 1.3 Surface topography . ........................... 5 1.4 Objectives . ............................... 5 1.5 Outline of this thesis . ........................... 7 2 Full Film EHD Lubrication 9 2.1 Deterministic roughness models . .................... 9 2.2 Governing equations . ........................... 11 2.3 The Block-Jacobi method . .................... 15 2.3.1 Ordinary Jacobi ........................... 15 2.3.2 Coupled systems of equations . ................ 16 2.4 A brief overview of the multilevel technique ................ 17 2.4.1 Grid levels . ........................... 17 2.4.2 Intergrid transfer operators . .................... 18 2.4.3 Two level solver for the Poisson equation . ............ 19 2.4.4 Two-dimensional functional operators . ............ 20 2.5 Dimensionless formulation . .................... 21 2.6 Discrete formulation . ........................... 23 2.7 Solution method ............................... 24 3 Dry Elasto-Plastic Contact 27 3.1 Statistical roughness models . .................... 27 3.2 Deterministic roughness models . .................... 28 3.3 Numerical solution techniques . .................... 28 3.4 Governing equations . ........................... 29 3.5 Spectral analysis . ........................... 30 3.6 Dimensionless formulation . .................... 31 3.7 Discrete formulation . ........................... 32 3.8 Solution method ............................... 33 iii 4 Surface Characterization 35 4.1 Theoretical model . ........................... 35 4.2 Measured topographies ........................... 35 4.3 Parameter study ............................... 36 4.4 Conclusions . ............................... 39 5 Reynolds vs. CFD 41 5.1 The CFD approach . ........................... 41 5.2 CFD - Governing equations . .................... 42 5.3 The model problem . ........................... 42 5.3.1 Interpolation of solution data .................... 43 5.3.2 Error estimation . .................... 44 5.4 The results of the comparison . .................... 46 5.5 Discussion and concluding remarks .................... 46 6 Simulations of Rough FL 51 6.1 The different overtaking situations . .................... 51 6.2 The Dent-Ridge overtaking . .................... 54 6.3 Conclusions . ............................... 58 7 Simulations of the dry contact 61 7.1 Varying the applied load . .................... 62 7.2 Conclusions . ............................... 64 8 Concluding Remarks 65 9 Future Work 67 II Appended Papers 69 A 71 A.1 Introduction . ............................... 74 A.2 Governing equations . ........................... 74 A.2.1 Boundary conditions and cavitation treatment . ........ 76 A.3 Numerics . ............................... 76 A.3.1 The numerics for the CFD approach ................ 77 A.3.2 The numerics for the Reynolds approach . ............ 77 A.3.3 Error estimation . .................... 78 A.3.4 Interpolation of solution data .................... 79 A.4 Results . ................................... 80 A.5 Discussion . ............................... 81 A.6 Conclusions . ............................... 84 B 87 B.1 Introduction . ............................... 90 B.2 Theory . ................................... 91 B.2.1 Equations . ........................... 91 B.2.2 Numerics . ........................... 92 B.2.3 Error estimation . .................... 93 B.3 Results and discussion ........................... 94 B.4 Conclusions . ............................... 98 C 103 C.1 Introduction . ............................... 106 C.2 Theory . ................................... 107 C.3 Surface characterization ........................... 108 C.4 Results . ................................... 109 C.5 Conclusions . ............................... 112 Preface This licentiate thesis comprises the results from numerical simulations of both lubricated and unlubricated contacts, specifically on the influence of surface topography