1:8 Svenska Mekanikdagar 2007 30
Homogenization Applied in Rough Surface
Hydrodynamic Lubrication
Andreas Almqvist
1&
Peter Wall
21
Division of Machine Elements Andreas.Almqvist@ltu.se 2
Department of Mathematics Lule˚a University of Technology It is clear surface roughness influence on
lubri-cation of many types of machine elements. It is, however, a delicate problem to tell how much a small change in the surface machining pro-cess will affect the performance of the lubrica-ted interface, in terms of load carrying capacity, frictional losses, etc. This work regards mode-ling of the effects caused by the surface rough-ness in hydrodynamically lubricated conjunction by means of the, linear incompressible and iso-viscous Reynolds equation; ∇ · (h3
ε(x)∇pε(x)) = Λ∇·(e1hε(x)) where pεis the pressure. For small values of ε, the function hεoscillates rapidly, im-plying a time consuming numerical solution. For sufficiently small values of ε it is even
impossib-le. In these situations the concept of homogeni-zation constitutes a valuable tool. In ref.1it has been shown that a branch within homogeniza-tion devoted to finding upper and lower boundsa is also applicable to the problem studied here. In ref.2
these ideas were utilized to perform nu-merical simulations of rough hydrodynamically lubricated contacts. Figure 4 illustrates conver-gence (left) and the preciseness of the the bounds (right). Similar to the work in ref.2
the objective of this work is to demonstrate how the ideas of homogenization may be used to efficient analyze the influence of surface roughness on hydrodyna-mic performance.
a
The ideas of finding bounds have proved to be useful in different physical applications, such as composite engineering. x (m) H y d ro d y n a m ic p re ss u re (P a ) ε = 1/22 ε = 1/23 ε = 1/24 Hom. 0.05 0.06 0.07 0 0.02 0.04 0.06 0.08 0.1 5 5.5 6 6.5×10 7 0 1 2 3 4 5 6 7×10 7 x (m) H y d ro d y n a m ic p re ss u re (P a ) p− Hom. p+ 0.0565 0.057 0.0575 0 0.02 0.04 0.06 0.08 0.1 5.966 5.968 5.97 5.972×10 7 0 1 2 3 4 5 6 7×10 7
Figur 4: A series of pressure solutions pε solutions and the corresponding homogenized
pressure solution (left). Lower and upper bounds pressure solutions compared with the ho-mogenized solution (right).
1D. Lukkassen, A. Meidell, and P. Wall. Bounds on the effective behavior of a homogenized
reynold-type equation. Research Report, No. 3, ISSN 1400-4003, Department of Mathematics, Lule˚a University of Technology, Accepted for publication in Journal of Function Spaces and Applications, 2006.
2A. Almqvist, D. Lukkassen, A. Meidell, and P. Wall. New concepts of homogenization applied in rough
surface hydrodynamic lubrication. International Journal of Engineering Science, In Press, Corrected Proof, Available online 6 December 2006.
