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, semianalytical 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 (
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 semianalytical formula was proposed. Instead, a table of ratios was made available. The semianalytical formula from [1] reads:
with the different parameters presented in Table 1 (with
The ratios of the previously mentioned publications are presented in Figure 3. The first noticeable conclusion of this comparison is that the
To summarize, Sperka et al. [4] proposed a formula to predict the
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 
contact radius  
Young modulii of solids 1 and 2  
reduced modulus of elasticity 

–  dimensionless material parameter
(Hamrock & Dowson) 

central film thickness  
minimum film thickness  
–  dimensionless material parameter (Moes) 

–  dimensionless load parameter (Moes) for point contact 

reduced radius of curvature in the entrainment direction  
mean entrainment velocity  
velocity in the entrainment direction of surfaces 1 and 2  
–  dimensionless speed parameter (Hamrock & Dowson) 

normal load  
–  dimensionless load parameter (Hamrock & Dowson) 

reciprocal asymptotic isoviscous pressure, according to Blok [5]  
lubricant dynamic viscosity  
lubricant density 
Table 1: nomenclature
Bibliography
[1] 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
[2] 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), 369379
[3] 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)
[4] Chevalier F. Modélisation des conditions d’alimentation dans les contacts élastohydrodynamiques ponctuels. (in French). PhD thesis, INSA de Lyon, France, 1996.
[5] 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