ABSTRACT
EHL problems for line and point contact even in the simplest case of Newtonian lubricant are reduced to boundary-value problems for very complex systems of nonlinear integro-differential equations. One of the very first approaches developed for solution of EHL problems for heavily loaded line contacts was the analytical method of Ertel-Grubin [1, 2]. The approximate result of the method is a formula for the lubrication film thickness which, under certain conditions, is in a relatively good agreement with experimentally obtained data. Later, this method was advanced in several studies [3]-[5] to improve the agreement between the analytical and experimental results. For elliptical heavily loaded EHL contacts a semi-analytical approach similar to [1, 2] was developed by Cheng [6]. These studies practically exhausted the advancements in analytical approaches to solution of EHL problems. All of these approaches are based on certain strong assumptions about the solution of the EHL problem validation of which raises a number of questions (see [7]). The major advantage of these methods is that they provide specific approximate analytical formulas for the lubrication film thickness without much of numerical calculations. A different analytical approach to EHL problems for heavily loaded line
contacts based on asymptotic methods was developed in a number of studies presented in the preceding chapters and collected in [7]. These studies are based on perturbation techniques and some asymptotic methods, in particular, on the method of matched asymptotic expansions [8, 9]. These methods do not use any prior assumptions about the problem solution. Some of the results of the asymptotic analysis are a solid understanding of the solution structure, establishing possible lubrication regimes, and analytical structural formulas for the lubrication film thickness. Among other benefits of the asymptotic methods are the following: reduction of the number of the input EHL problem parameters which determine the numerical solution, development of effective numerical approaches for solution of asymptotically valid as well as the original EHL problems for high pressure viscosity coefficients which usually cause numerical solutions to be unstable, and discovering a simple approach to reg-
for Line and Point
ularization of generally unstable solutions in heavily loaded isothermal EHL contacts [7]. It was determined that formulas for the lubrication film thickness are lubrication regime dependent, i.e., contrary to a widespread belief, there are series of formulas for the lubrication film thickness different for distinctly different heavily loaded lubrication regimes. The above analytical analysis is augmented by a stable numerical approach to solution of asymptotically valid equations. The success in getting stable numerical solutions for asymptotically valid equations allowed for the development of a stable numerical approach to solution of the EHL problem in the original (non-asymptotic) formulation. This numerical method is based on the solution structure and general properties of the EHL problem discovered through the EHL problem asymptotic analysis. In particular, the original EHL problem equations for line contacts are transformed to a different much more beneficial for numerical solution form. In addition to that, the grid size necessary for sufficient solution precision is determined based on the EHL problem input parameters (i.e., based on the characteristic size of the inlet zone) and solution structure analyzed asymptotically [7]. Obviously, the initial application of the asymptotic approach requires a
deeper analytical analysis compared to the existing direct numerical methods in application to solution of EHL problems. However, this asymptotic analysis is worth the additional effort and, in essence, is just the extension of the classic asymptotic analysis proposed by Reynolds which led to significant simplification of the full Navier-Stokes equations and derivation of the Reynolds equation. It is clear that the usefulness of the latter analysis is hard to deny. In preceding chapters it became clear that asymptotic methods have some advantages compared to the direct numerical solution of the EHL problems in their original formulations. Some of these specific reasons why the asymptotic methodology applied to heavily loaded line EHL contacts followed by numerical methods applied to asymptotic equations is attractive are collected in Table 12.1. In Table 12.1, the term ”direct numerical methods” is related to the solution methods which make use of only numerical approximation formulas such as various quadrature and finite difference formulas while the essence of the term ”asymptotic methods” is analytical analysis of the problem which is presented below. Advancements in computers led to the development of a variety of numerical
methods such as methods based on Newton-Raphson’s method [11, 12], the multilevel multi-grid methods [13, 14], and methods which use the fast Fourier transform [15, 16] as well as some others. The accessability of computers and the relative simplicity of their use compared to analytical studies led to their domination in practical applications. Let us consider the typical steps involved in numerically based study of
EHL problems with the goal to determine a formula for the lubrication film thickness.
