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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 89
Edited by: M. Papadrakakis and B.H.V. Topping
Paper 19

Stabilized Finite Elements for Elastohydrodynamic Lubrication Problems

W. Habchi1, D. Eyheramendy2, P. Vergne1 and G. Morales-Espejel3

1LaMCoS, INSA-Lyon, CNRS UMR5259, France
2LMA, Ecole Centrale Marseille, CNRS UPR7051, France
3SKF Engineering and Research Center, Nieuwegein, The Netherlands

Full Bibliographic Reference for this paper
W. Habchi, D. Eyheramendy, P. Vergne, G. Morales-Espejel, "Stabilized Finite Elements for Elastohydrodynamic Lubrication Problems", in M. Papadrakakis, B.H.V. Topping, (Editors), "Proceedings of the Sixth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 19, 2008. doi:10.4203/ccp.89.19
Keywords: finite elements, elastohydrodynamic lubrication, artificial diffusion, full-system approach.

Nowadays, roller bearings are essential components in any mechanical system that includes rotating parts. In order to further improve the functioning of these components, they are lubricated. It is important for the good functioning of a bearing to have a reliable tool to estimate minimum film thickness and frictional contact losses.

In this work, we are mainly interested in a particular lubrication regime known as elastohydrodynamic (EHD). In this regime, the contacting bodies are separated by a full lubricant film. The pressure generated in the film is high enough to induce a significant elastic deformation of the contacting bodies. Therefore a strong coupling between hydrodynamic and elastic effects is involved. Here we present a finite element full-system approach to model the lubricant flow in such contacts. The elastic problem is modelled by applying the classical linear elasticity equations to a structure with large enough dimensions compared to the contact size. As for the hydrodynamic problem, the Reynolds [1] equation for thin film Newtonian flows is solved on a part of the boundary of the structure corresponding to the contact area. This equation is a simplified version of the Navier-Stokes equations dedicated to thin film flows where the pressure gradient across the film thickness can be neglected. It is highly non-linear since the rheological properties of lubricants show a high dependence on pressure. The two problems are solved simultaneously by means of a Newton-like procedure. This leads to high convergence rates of the solution especially when compared to semi-system approach based models where the two problems are solved separately and an iterative procedure is established between them. At the exit of the contact, a free boundary problem arises. It is handled by applying the penalty method. For highly loaded contacts, the standard Galerkin solution of the Reynolds equation exhibits an oscillatory behaviour. This is explained by writing the Reynolds equation as a convection-diffusion equation as a function of pressure. When the load is highly increased, this equation becomes convection dominated and requires streamline upwinding techniques (Streamline Upwind Petrov-Galerkin [2] / Galerkin Least Squares [3]) in order to stabilize the solution.

Finally, the non-Newtonian behaviour of the lubricant is taken into account along with the heat generation that takes place in the lubricant film due to both shear forces and the lubricant's compressibility.

Reynolds, O., "On the Theory of the Lubrication and its Application to Mr Beauchamp Tower's Experiments, Including an Experimental Determination of the Viscosity of Olive Oil", Philos. Trans. R. Soc., 177, pp.157-234, 1886. doi:10.1098/rstl.1886.0005
Brooks, A.N., and Hughes, T.J.R., "Streamline-Upwind/Petrov-Galerkin Formulations for Convective Dominated Flows with Particular Emphasis on the Incompressible Navier-Stokes Equations", Comp. Meth. Appl. Mech. Engnrg., 32, pp.199-259, 1982. doi:10.1016/0045-7825(82)90071-8
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