On the effects of surface roughness in lubrication
Abstract: Tribology is a multidisciplinary field defined as the science and technology of interacting surfaces in relative motion, and embraces the study of friction, wear and lubrication. A typical tribological application is the rolling element bearing. Tribological contacts may also be found in other types of bearings, cam-mechanisms, gearboxes and hydraulic systems. Examples of bearings inside the human body are the operation of the human hip joint and the contact between teeth during chewing. To fully understand the operation of this type of application one has to understand the couplings between the lubricant fluid dynamics, the structural dynamics of the bearing material, the thermodynamical aspects and the resulting chemical reactions. This makes modeling tribological applications an extremely delicate task. Because of the multidisciplinary nature, such theoretical models lead to mathematical descriptions generally in the form of non-linear integro-differential systems of equations. Some of these systems of equations are sufficiently well posed to allow numerical solutions to be carried out, resulting in accurate predictions on performance. In this work, the influence on performance of a surface microscopical nature, the surface roughness, in contact interfaces between different types of machine element components is the subject of study. An example is the non-conformal lubricated contact between one of the rollers and the inner ring in a rolling element bearing. The tribological contact controlling the operation of the human hip joint is also very similar to this. Another example of a non-conformal contact occurs when driving on rainy roads, where the hydrodynamic action of the water separates the tire. To enable investigations of these types of problems, different theoretical models were studied; for the selected model, a numerical solution technique was developed within this project. This model is based on the Reynolds equation coupled with the film thickness equation. The numerical solution technique involves a multilevel technique to facilitate the solution process. Results presented in this thesis, utilizing this approach, study elementary surface features such as ridges and indentations passing each other inside the lubricated conjunction. The Reynolds equation is derived under the assumptions of thin fluid film and creeping flow, and considers in its most general form shear thinning of the lubricant. This type of equation describes the hydrodynamic action of the lubricant flow and may be used when the interfaces consist of either conformal or non-conformal conjunctions. Examples of applications having conformal interfaces are thrust- and journal- bearings or the contact between the eye and a (optical) contact lens. In such types of applications the load carried by the interface is distributed over a fairly large area that under certain circumstances helps to prevent mechanical deformation of the contacting surfaces. Such applications are said to operate in the hydrodynamic lubrication (HL) regime. Lubricant compressibility and cavitation are important aspects and have received some attention. However, the main objective when modeling HL has been to investigate and develop methods that enable the influence of surface roughness to be to be studied efficiently. Homogenization is a rigorous mathematical concept that when applied to a certain problem may be regarded as an averaging technique as well as it provides information about the induced effects of local surface roughness. Homogenization inflicts no restrictions on the surface roughness representation other than the representative part of the chosen surface roughness being assumed periodically distributed and of course the assumptions of thin film flow made through the Reynolds equation. The homogenization process leads to a two sets of equations one for the local scale describing surface roughness, scale and one for the global scale describing application geometry. The unequivocally determined coefficients of the global problem, which may be regarded as flow factors, are obtained through the solution of local problems. This makes homogenization an eminent approach to be used investigating the influence of surface roughness on hydrodynamic performance. In the present work, homogenization has been used to derive computationally feasible forms of problems originating from incompressible and compressible Reynolds type equations that describe stationary and unstationary flows in both cartezian and cylindrical co-ordinates. This technique enables simulations of surface roughness induced effects when considering surface roughness descriptions originating from measurements. Moreover, the application of homogenization facilitates the interpretation of results. Numerical investigations following the homogenization process have been carried out to verify the applicability of homogenization in hydrodynamic lubrication. Homogenization has also been shown here to enable efficient analysis of rough hydrodynamically lubricated problems. Also of note, in connection to the scientific contribution within tribology, collaboration with a group in applied mathematics has lead to the development of novel techniques in that area. These ideas have also been successfully applied, with some results presented in this thesis. 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, the tribo film, or both. This is hereby modeled as the unlubricated frictionless contact between rough surfaces, i.e. a contact mechanical approach. A variational principle was used in which the real area of contact and the contact pressure distribution minimize the total complementary potential energy. The material model is linear elastic-perfectly plastic and the energy dissipation due to plastic deformation is accounted for. The numerics of this contact mechanical approach involve the fast Fourier transformation (FFT) technique in order to facilitate the solution process. Investigation results of the contact mechanics of realistic surfaces are presented in this thesis. In this investigation the variation in the real area of contact, the plasticity index and some surface roughness parameters due to applied load were studied.
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