How to better predict the minimum film thickness of elastohydrodynamic contacts?

Jean-David Wheeler

January, 11 2021

Let’s say you have the choice between an almost infinite lifespan device, and a device that undergoes wear and failure within a few minutes or hours of running. In most cases, you would choose the former and discard the latter. Let’s say that this device is a rolling element bearing and that its lifespan highly depends on the lubricant film separating the rolling elements from the rings: you would make sure that the film is sufficiently thick to limit and maybe avoid the potentially damaging solid to solid contacts. The work from Sperka et al. [1] I would like to introduce to you today provides a new tool to help you achieve such a goal.

To control the film separation between the bodies, it is mandatory to accurately predict its minimum. However, as the loaded solids are lubricated and in relative motion, the behaviour of the contact is impossible to predict through analytical methods: advanced numerical analysis is necessary. Because such analysis requires computation power and time, semi-analytical formulae have been derived from numerical analysis and proposed over the last 50 years (see Figure 2). Whereas the formulae shape is based on physical considerations, their coefficients are obtained by fitting the formulae to the numerical analysis results. You would think that after almost 5 decades these formulae would provide a prediction accuracy that can compete with the standard model of particle physics? Not always. Whereas the central film thickness ( $h_c$ in Figure 1) is generally well predicted, the minimum film thickness ( $h_m = min(h_{m,c} ; h_{m,l-} ; h_{m,l+})$ in Figure 1) can be overestimated to a great extent [2].

Figure 2: non-exhaustive list of publications proposing film thickness semi-analytical prediction formulae for different configurations: circular and elliptical with entrainment velocity either along its large semi-axis or its small semi-axis. Blue frames: $h_c$ and $h_m$ formulae; purple frames: $h_c/h_m$ formulae

For this reason, Sperka et al. [1] proposed a formula for the ratio of central to minimum film thickness. Their formula is based on a numerical analysis applied to a very wide set of operating conditions. Moreover, the proposed formula is compared to a ratio obtained from Hamrock & Dowson [3] formulae and measurements performed on a very accurate test rig. A similar approach was adopted by Chevalier [4], but no semi-analytical formula was proposed. Instead, a table of ratios was made available. The semi-analytical formula from [1] reads:

$h_c/h_m = 1 + 0.1\times \alpha_{film}^{0.128} \times M^{0.38} - \sqrt{M} \times \left( \frac{\alpha_{film}^{0.2}\times ln(L)-3}{22.7}\right)^2$

with the different parameters presented in Table 1 (with $\alpha_{film}$ in $GPa^{-1}$ ).

Figure 3: Comparison of the ratios from Hamrock and Dowson [3], Chevalier [4] and Sperka et al. [1]

The ratios of the previously mentioned publications are presented in Figure 3. The first noticeable conclusion of this comparison is that the $h_c/h_m$ ratio is not precisely predicted by the most widely spread Hamrock & Dowson formulae. Besides, Chevalier [4] and Sperka et al. [1] seem to provide similar predictions which confirms both of their independent approaches: in [1], a quantitative comparison is made, and the difference is below 4%. However, Sperka et al. [1] proposed a proper semi-analytical formula. At last, they noticed an influence of the piezo-viscosity that could not be accounted for with the dimensionless parameters $M$ and $L$ and the classical power laws: therefore, the formula is enhanced by such influence.

To summarize, Sperka et al. [4] proposed a formula to predict the $h_c/h_m$ ratio of elastohydrodynamic contacts. Together with another formula to estimate $h_c$ , it is possible to improve the accuracy of the $h_m$ semi-analytical predictions. While hoping that the authors will one day extend their work to elliptical contacts, this work certainly further improves the prediction toolbox available to the mechanical engineers.

Online EHL film thickness calculator

An online film thickness calculator is available on tribonet for line and elliptical (point) contacts. The calculators allow calculating central and minimum film thicknesses using various equations. Exact equations are described on the calculator’s page. Here is the calculator for elliptical contact:

Variable	Unit	Description
$a$	$m$	contact radius
$E_1, E_2$	$Pa$	Young modulii of solids 1 and 2
$E′$	$Pa$	reduced modulus of elasticity $2/E'=(1-\nu_1^2)/E1+(1-\nu_2^2)/E_2$
$G$	–	dimensionless material parameter (Hamrock & Dowson) $=\alpha \times E'$
$h_c$	$m$	central film thickness
$h_m$	$m$	minimum film thickness
$L$	–	dimensionless material parameter (Moes) $=G(2U)^{0.25}$
$M$	–	dimensionless load parameter (Moes) for point contact $=W/(2U)^{0.75}$
$R_x$	$m$	reduced radius of curvature in the entrainment direction
$u_e$	$m/s$	mean entrainment velocity
$u_1, u_2$	$m/s$	velocity in the entrainment direction of surfaces 1 and 2
$U$	–	dimensionless speed parameter (Hamrock & Dowson) $=\mu \times u_e/(E' \times R_x)$
$w$	$N$	normal load
$W$	–	dimensionless load parameter (Hamrock & Dowson) $= w/(E' \times R_x^2)$
$\alpha^*$	$Pa^{-1}$	reciprocal asymptotic isoviscous pressure, according to Blok [5]
$\alpha_{film}$	$GPa^{-1}$	$= \alpha^* \times (1-e^{-3})$
$\mu$	$Pa\times s$	lubricant dynamic viscosity
$\rho$	$kg/m^3$	lubricant density

Table 1: nomenclature

Bibliography

P. Sperka, I. Krupka and M. Hartl, Analytical Formula for the Ratio of Central to Minimum Film Thickness in a Circular EHL Contact, 2018, Lubricants, 6, 80
J.D. Wheeler, P. Vergne, N. Fillot, D. Philippon, On the relevance of analytical film thickness EHD equations for isothermal point contacts: Qualitative or quantitative predictions?, 2016, Friction, 4(4), 369-379
Hamrock B J, Dowson D. Isothermal elastohydrodynamic lubrication of point contacts Part III – Fully flooded results. Trans ASME J Lubr Technol 99(2): 264–276 (1977)
Chevalier F. Modélisation des conditions d’alimentation dans les contacts élastohydrodynamiques ponctuels. (in French). PhD thesis, INSA de Lyon, France, 1996.
Blok H. Inverse problems in hydrodynamic lubrication and design directives for lubricated flexible surfaces. In Proceedings of the International Symposium on Lubrication and wear, Houston, 1963: 1–151

Jean-David Wheeler

After a PhD thesis with SKF at the INSA de Lyon - LaMCoS dedicated to the lubrication of large size roller bearings, Jean-David joined SIMTEC (https://www.simtecsolution.fr/fr/) to continue helping industries to develop their processes and products.