Asymptotic analysis, regularisation and stable numerical solutions for heavily loaded line EHL contacts

2013

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The paper is devoted to the development of an asymptotic approach to solution of the steady isothermal problem of elastohydrodynamic lubrication (EHL) for heavily loaded point contacts. It is shown that the whole contact region can be subdivided into three subregions: the central one that is adjacent to the other two regions occupied by the ends of the horseshoe-shaped pressure/gap distribution zone. The central region, in turn, can be subdivided into the Hertzian region and its adjacent inlet and exit zones that, in turn, are adjacent to the inlet and exit boundaries of the contact, respectively. Moreover, in the central region, in the inlet and exit zones of heavily loaded point EHL contact, the EHL problem can be reduced to asymptotically valid equations identical to the ones obtained in the inlet and exit zones of heavily loaded line EHL contacts. The latter means that many of the well-known properties of heavily loaded line EHL contacts are also valid for heavily loaded point EHL contacts. These asymptotically valid equations can be analysed and numerically solved based on the stable methods using a specific regularisation approach that were developed for lubricated line contacts. Cases of pre-critical and over-critical lubrication regimes are considered. The by-product of this asymptotic analysis is an easy analytical derivation of formulas for the lubrication film thickness for pre-critical and over-critical lubrication regimes. The method is validated by the results of some experimental and numerical studies published by a number of researches.

\begin{matrix} center \\ center ({},, r, s, α_{1}, β_{1}) = \frac{1}{r_{0}} ({}, r_{p}, s_{p}, α_{1 p}, β_{1 p}), ({},, q, g) = \frac{1}{N_{0} r_{0}^{1 / 2}} ({}, q_{p}, g_{p}), \\ center ({},, h_{q}, h_{g}) = \frac{1}{h_{\infty}} ({}, h_{italicqp}, h_{italicgp}), ({}, μ_{q} ((), q), μ_{g} ((), g)) = ({}, μ_{qp} ((), q_{p}), μ_{gp} ((), g_{p})), \end{matrix}

and the definitions

A = A_{p} {([], θ N_{0}^{4} ((), y))}^{1 / 5} h_{\infty} (y), H_{0} (y) = H_{00} h_{\infty} (y), r_{0} = {([], θ^{2} N_{0}^{3} ((), y))}^{- 2 / 5}

we will be able to reduce each of the aforementioned systems of asymptotically valid equations to

\begin{matrix} center \\ center A^{3} +1 \frac{\partial}{\partial r} ((), +1 \frac{h_{q}^{3}}{μ_{q}} +1 \frac{\partial q}{\partial r}) = +1 \frac{2}{π} true \int_{0}^{\infty} \frac{q ((),, t, y) - q_{a} ((),, t, y)}{t - r} \end{matrix}

…”

Section: Asymptotic Analysis Of a Heavily Loaded Lubricated Contact Umentioning

confidence: 84%

“…According to , the two systems in and can be rewritten in the equivalent form as follows:

\begin{matrix} center \\ center q ((),, r, y) = \sqrt{2 r} ([], 1 - \frac{1}{2 π} true \int_{0}^{\infty} \frac{\partial}{\partial t} M_{0} (q, h_{q}, μ_{q}, t, y) \frac{d t}{\sqrt{2 t} (t - r)}), \\ center q ((),, r, y) \to q_{a} ((),, r, y) = \sqrt{2 r} + \frac{α_{1} ((), y)}{\sqrt{2 r}}, bold-italic r \to \infty, \\ center A^{3} = π α_{1} / true \int_{0}^{\infty} \frac{\partial}{\partial t} (\frac{h_{q}^{3}}{μ_{q}} \frac{\partial q (t, y)}{\partial t}) \frac{d t}{\sqrt{2 t}}, \\ center h_{q} ((),, r, y) = 1 bold-italic o r bold-italic A ([], h_{q} ((),, r, y) - 1) = \frac{2}{π} true \int_{0}^{\infty} [q (t, y) - q_{a} (t, y)] l n \frac{1}{t - r} d t, \end{matrix}

\begin{matrix} center \\ center g ((),, s, y) = \sqrt{- 2 s} ([], 1 - \frac{1}{2 π} true \int_{- \infty}^{0} \frac{\partial}{\partial t} M_{0} ()) \end{matrix}

…”

Section: Asymptotic Analysis Of a Heavily Loaded Lubricated Contact Umentioning

confidence: 99%

Asymptotic analysis of lubricated heavily loaded point contacts

2013

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“…The numerical results include parametric studies, studies of solution precision and stability and so on. On the basis of the fact that the asymptotically valid equations for the EHL problem with skewed lubricant entrainment can be reduced to equations identical to the asymptotically valid equations for the EHL problem with straight lubricant entrainment analysed and described in detail in here due to the limited space, these studies are not presented.…”

Section: Asymptotic Analysis Of a Heavily Loaded Lubricated Contact Umentioning

confidence: 99%

“…It has been shown that by a simple transformation, the derived equations can be reduced to equations identical to the corresponding equations for pre‐critical lubrication regimes in the inlet and exit zones of line heavily loaded contacts. The latter equations have been validated through the comparison with the direct numerical solutions of the original EHL equations in . For over‐critical lubrication regimes, another approach has been used, which led to formulas for the lubrication film thickness for starved and fully flooded conditions.…”

Section: Closurementioning

confidence: 99%

“…For over‐critical lubrication regimes, another approach has been used, which led to formulas for the lubrication film thickness for starved and fully flooded conditions. The asymptotic approaches to EHL problems offer certain distinct advantages in comparison with the direct numerical solutions of EHL problem such as reduction of the number of problem input parameters and the approach to regularization of numerical solutions . Each of the analytically derived formulas for the central H 00 and exit H 0 lubrication film thicknesses contains just one coefficient of proportionality, which can be found by numerically solving the corresponding asymptotically valid equations in the inlet zone or by just curve fitting the experimentally obtained values of the lubrication film thickness.…”

Section: Closurementioning

confidence: 99%

See 1 more Smart Citation

Asymptotic analysis of lubricated heavily loaded point contacts with skewed direction of entrained lubricant

2013

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The paper is devoted to the application of the early developed asymptotic approach to solution of the steady isothermal problem of elastohydrodynamic lubrication (EHL) for heavily loaded point contacts with skewed direction of entrained lubricant. It is shown that the whole contact region can be subdivided into three subregions: the central one, which is far away from the other two regions occupied by the ends of the horseshoe-shaped pressure/gap distribution. The central region, in turn, can be subdivided into the Hertzian region and two adjacent boundary layers -the inlet and exit zones. Moreover, in the central region in the inlet and exit zones, the EHL problem can be reduced to asymptotically valid equations identical to the ones obtained in the inlet and exit zones of heavily loaded line EHL contacts. These equations can be analytically analysed and numerically solved on the basis of the stable methods using a specific regularization approach, which were developed for lubricated line contacts. Cases of pre-critical and over-critical lubrication regimes are considered. The by-product of this asymptotic analysis is an easy analytical derivation of formulas for the lubrication film thickness for pre-critical and over-critical starved and fully flooded lubrication regimes. The latter allows for simple analysis of the film thickness as a function of the contact eccentricity and the direction of the entrained lubricant at the inlet in the contact.

Asymptotic analysis, regularisation and stable numerical solutions for heavily loaded line EHL contacts

2010

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An analytical analysis of a line thermal EHL contact under heavily loaded conditions is proposed. The analysis is based on regular matched asymptotic expansions. In particular, a case of relatively small ratio of rolling and sliding frictional stresses is considered. The problem is reduced to an analogue of the isothermal EHL problem by eliminating lubricant temperature from the problem formulation. The further asymptotic and numerical analysis is very similar to the case of the isothermal EHL problem. THERMAL EHL PROBLEMS FOR HEAVILY LOADED CONTACTSThis is an extension of Part 1 of this paper 1 on the case of thermal EHL (TEHL) problems. We will consider a special case of a TEHL problem for Newtonian lubricant when the contact surface temperatures are known and equal. We will develop a powerful perturbation technique for simplifi cation and analysis of the problem. This approach will allow us to reduce the whole thermal EHL problem (which includes the energy equation for lubricant) to an EHL problem for a modifi ed Reynolds equation, in a certain sense similar to the classic isothermal one. The latter problem can be solved numerically or studied by the appropriate asymptotic methods. Historically, this asymptotic approach was fi rst developed by Kudish 2-6 for the analysis of isothermal and thermal EHL problem for Newtonian and non-Newtonian fl uid lubricants and greases. Formulation and analysis of a TEHL problem for Newtonian fl uid in heavily loaded contactsLet us consider the behaviour of a Newtonian fl uid in a TEHL contact of two elastic cylinders separated by a lubrication fi lm and loaded with a given normal force acting along their center line. The LUBRICATION SCIENCE Using the conditions for our special case, we obtain T w10 = T w20 = T w0 and T w11 = T w21 = 0, where T w0 is a given function. It is easy to see that the solution of equation (23) has the form 296 I. I. KUDISH