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# EHL Film Thickness Calculator (Central and Minimum): Line (Cylindircal) Contact

An EHL Film Thickness Calculator allows calculating central and minimum film thicknesses in a full film lubricated line (cylindrical) contact as a function of entrainment speed. The central and minimum film thicknesses are calculated using equations developed by Dowson et al [1,3] and Moes et al [2,4]. The equations used in the calculator are given below the calculator along with the definitions and references. The calculator allows choosing one of the two equations in the calculations.

## Central film thickness formulas

### Dowson et al formula [1]

Dowson and Toyoda developed a following equation for central film thickness calculations [1]:

*** QuickLaTeX cannot compile formula: \begin{equation*} H_c = 3.06 \cdot G^{0.56} \cdot U^{0.69} \cdot W^{-0.1} \end{equation*} *** Error message: Cannot connect to QuickLaTeX server: cURL error 7: Failed connect to www.quicklatex.com:443; Connection timed out Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

where

*** QuickLaTeX cannot compile formula: \begin{equation*} h_c = H_c \cdot R_X \end{equation*} *** Error message: Cannot connect to QuickLaTeX server: cURL error 7: Failed connect to www.quicklatex.com:443; Connection timed out Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

Definition of the dimensionless variables are given below.

Moes developed another central film thickness equation [2]:

*** QuickLaTeX cannot compile formula: \begin{equation*} H_c = ((Hc_{IR}^{7/3} + Hc_{IE}^{-7/3})^{3s/7} + (Hc_{VR}^{-7/2} + Hc_{VE}^{-7/2})^{-2s/7})^{1/s} \end{equation*} *** Error message: Cannot connect to QuickLaTeX server: cURL error 7: Failed connect to www.quicklatex.com:443; Connection timed out Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

where

*** QuickLaTeX cannot compile formula: \begin{eqnarray*} H_{c,IR} &=& 3 / M\\ H_{c,IE} &=& 2.62105 \cdot M^{-1 / 5}\\ H_{c,VR} &=& 1.28666 \cdot L^{2 / 3}\\ H_{c,VE} &=& 1.31106 \cdot M^{-1/8} \cdot L^{3/4}\\ s &=& 1 / 5 \cdot ( 7 + 8 \cdot \exp(-2 \cdot H_{c,IE} / H_{c,IR})) \end{eqnarray*} *** Error message: Cannot connect to QuickLaTeX server: cURL error 7: Failed connect to www.quicklatex.com:443; No route to host Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

with

*** QuickLaTeX cannot compile formula: \begin{equation*} h_c = H_c \cdot (U)^{1/2} \cdot R_X \end{equation*} *** Error message: Cannot connect to QuickLaTeX server: cURL error 7: Failed connect to www.quicklatex.com:443; No route to host Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

## Minimum film thickness formulas

### Dowson et al formulas

For the calculation of minimum film thickness in a cylindrical contact following Dowson [3], the following equations were used:

*** QuickLaTeX cannot compile formula: \begin{equation*} H_{min} = 2.65 \cdot G^{0.54} \cdot U^{0.7} \cdot W^{-0.13} \end{equation*} *** Error message: Cannot connect to QuickLaTeX server: cURL error 7: Failed connect to www.quicklatex.com:443; No route to host Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

With

*** QuickLaTeX cannot compile formula: \begin{equation*} h_{min} = H_{min} \cdot R_X \end{equation*} *** Error message: Cannot connect to QuickLaTeX server: cURL error 7: Failed connect to www.quicklatex.com:443; No route to host Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

### Moes et al formulas

According to Moes, the equations for calculation of the minimum film thickness are as follows[4]:

*** QuickLaTeX cannot compile formula: \begin{equation*} H_{min} = \left([(0.99\cdot M^{-1/8} L^{3/4}*t)^r + (2.05\cdot M^{-1/5})^r]^{s/r} + (2.45 \cdot M^{-1})^s \right)^{1/s} \end{equation*} *** Error message: Cannot connect to QuickLaTeX server: cURL error 7: Failed connect to www.quicklatex.com:443; Connection timed out Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

where

(9)

with

(10)

## Definitions:

Poisson’s ratio dimensionless,

Young’s modulus of elasticity , [Pa],

Equivalent elastic constant , [Pa],

Base oil viscosity (dynamic) , [Pa ],

Pressure-viscosity coefficient , [],

Mean entraining velocity , [] in the equations by Dowson et al. For Moes et al equations, the sum velocity is used: , []

Equivalent radii of curvature in X direction , []

Normal applied load , [N]

Material parameter [1], , dimensionless

Speed parameter [1], , dimensionless

Load parameter [1], , dimensionless

Modified load parameter [3], , dimensionless

Viscosity parameter [3], , dimensionless

## References:

[1] Dowson, D.; Toyoda, S. A central film thickness formula for elastohydrodynamic line contacts. In Proceedings of the 5th Leeds-Lyon Symposium on Tribology, Leeds, UK, September 1978; pp. 60–65.

[2] H. Moes. Lubrication and beyond, University of Twente lecture notes, 2000.

[3] D. Dowson. Elastohydrodynamic and micro-elastohydrodynamic lubrication. WEAR, 190(2), December 1995.

[4] Moes, H. Optimum Similarity Analysis with applications to Elastohydrodynamic Lubrication. Wear 1992, 159, 57–66.