The line contact problem of elastohydrodynamic lubrication - I. Asymptotic structure for low speeds

Abstract
This paper reports the first formal asymptotic solution to the line contact problem of elastohydrodynamic lubrication (EHL), a fundamental problem describing the elastic deformation of lubricated rolling elements such as roller bearings, gear teeth and other contacts of similar geometry. The asymptotic régime considered is that of small λ , a dimensionless parameter proportional to rolling speed, viscosity and the elastic modulus. The solution is shown to possess four regions: a zone where the lubricating film is both thin and slowly narrowing and which is closely related to the contact area that occurs in the absence of lubricant, an upstream inlet zone of low pressure, and two thin layers on either side of the contact zone. The solutions in the first two just-mentioned zones are given by simple analytical expressions. The solutions in the two thin layers are obtained from two universal functions obtained by Bissett & Spence ( Proc. R. Soc. Lond . A 424, 409 (1989)). Although these two functions, related to the local film thickness, are obtained by numerical techniques by Bissett & Spence, it should be emphasized that all cases in the asymptotic régime considered are hereby solved definitively without recourse to further computation. Although some features of this structure have been suggested by other solution approaches, generally, these are numerical or ad hoc approximations. See the texts by Johnson ( Contact Mechanics , pp. 328 (1985)) and Dowson & Higginson ( Elasto-hydrodynamic lubrication (1977)), this work provides a formal mathematical basis for understanding most of the principal features of EHL. The solution provides a simple formula for minimum film thickness and displays the sharp narrowing of the lubricating film in the thin layer near the exit. In the basic asymptotic solution provided here, the dimensionless pressure-viscosity coefficient, α , is assumed to be O (1), and in this parameter régime, no pressure spike will occur. By comparing with the work of Hooke ( J. mech. Engng Sci . 19(4), 149 (1977)), we can show that an incipient pressure spike occurs when α becomes as large as O ( λ -1/5 ). However, asymptotic solutions in this latter parameter régime require new numerical solutions for each case of interest and are not pursued here.

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