# Analytical Formula for the Ratio of Central to Minimum Film Thickness in a Circular EHL Contact

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Material and Methods

#### 2.1. Numerical Calculations

_{a}, y) = p(x

_{b}, y) = p(x, y

_{a}) = p(x, y

_{b}) = 0 and the cavitation condition p(x, y) ≥ 0, film thickness equation

_{film}which is proportional to the α* evaluated up to maximum inlet zone pressure [20]. This pressure is, according to Bair et al. [20], considered to be equal to 3/α*.

_{film}pressure–viscosity parameter is used in the present analytical model of central to minimum film thickness ratio.

#### 2.2. Film Thickness Measurement

## 3. Results and Discussion

^{−4}was required. The ratio of central to minimum film thickness h

_{c}/h

_{min}was evaluated. Contact simulations were done on square grids with different number of points in a range from 129 × 129 to 1025 × 1025. A difference between results on coarse and finer meshes were evaluated and accepted only when data were with relative difference < 1% for M < 750 and < 3% for M ≥ 750 to ensure they were free of mesh density influence. All calculated data of ratios are listed in the Table A1, Table A2 and Table A3.

_{c}/h

_{min}−1 on M parameter for three fixed L values. Figure 3 presents a dependency of film thickness ratio h

_{c}/h

_{min}on L parameter for three fixed M values. Both plots are for α

_{film}= 20.6 GPa

^{−1}. The dependency of the film thickness ratio monotonically rises on M while there is a local maximum in dependency on parameter L. This local maximum is around L = 5 and seems to slightly change with increasing M parameter. The ratio approaches 1 for low L and M values.

^{−1}. It corresponds to the pressure–viscosity coefficient according definition in Equation (8) published by Bair [23]. It was shown that it is able precisely capture relation of viscosity on pressure necessary for film thickness formation in EHL contacts. According Table 3, the values of α

_{film}are not far from α* coefficient. The analytical model fitted to the results is shown in Figure 4 in a form of contour plot on the left side and residuals from fitting in the right plots. Final form of analytical model of film thickness ratio which depends on M and L parameters and α

_{film}pressure–viscosity coefficient is given by Equation (12). Quality parameters of the fits shown in Figure 4 are listed in Table 5; root mean square of the error is 0.03–0.04 which represents about 2% of average film thickness ratio. Equation (12) was plotted as a model in Figure 2, Figure 3, Figure 4 and Figure 5.

_{film}is in GPa

^{−1}. The equation is repeated together with list of assumptions and conditions for which it was obtained in Appendix B.

_{c}/h

_{min}ratio was calculated.

_{c}/h

_{min}ratio which is close to measured value only for 0.5 and 1.2 GPa and small range of speed around 500 mm/s. Present model gives good prediction of trends in all cases. Quantitatively, there is very good agreement for 1.2 GPa and not as good agreement for 0.5 and 0.8 GPa. Average differences between measurement and predictions are listed in Table 6. It shows that present formula predicts the ratio within 11%, while Hamrock–Dowson is off by up to 37%.

_{min}film thickness above 60 nm and h

_{c}film thickness above 140 nm were considered. Measurements were repeated three times for each pressure; ratios were evaluated independently and averaged. Average standard deviation of measured ratios is 0.028 which corresponds to 1.3%. Measurement has another error which comes from the fact that minimum film thickness is evaluated as a global minimum in a contact. Every ball has a certain roughness (3 nm RMS in present measurement); therefore, the evaluated h

_{min}is practically always picked on a peak of roughness. As a result, measured minimum film thickness tends to be systematically evaluated as lower than the true value. Typical in-contact RMS value of roughness was 0.6 nm, therefore, distance of mid plane to highest peaks was about 1.8 nm which makes average impact on film thickness ratio 0.03, i.e., much less then observed difference.

## 4. Conclusions

_{film}

^{0.2}). Isothermal Newtonian numerical simulations were used to compute the film thickness ratio for wide range of M and L parameters and three pressure–viscosity coefficients. A new analytical formula for fitting of the ratio is presented. Final regression of numerical simulations has root mean square error of 0.036, i.e., 1.9%. A comparison with measurement showed good trend agreement and quantitatively smaller difference than Hamrock–Dowson formulas. This new formula together with published formula for central film thickness can be used for minimum thickness prediction.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

a, b, c, d | fitting constants of analytical model, Equation (10) |

E_{r} | reduced elastic modulus, 2/[(1 − υ_{1}^{2})/E_{1}+(1 − υ_{2}^{2})/E_{2}] |

F | load |

G | material parameter, αE’ |

h | film thickness, h_{c} (central), h_{min} (minimum) |

k | contact ellipticity ratio |

M | Moes parameter, G(2U)^{0.25} |

L | Moes parameter, W(2U)^{−0.75} |

P | pressure |

r_{1x},r_{2x} | radii of curvature of surface 1 and 2 in x-direction |

R_{x} | reduced radius of curvature in x-direction, r_{1x}r_{2x}/(r_{1x} + r_{2x}) |

R_{y} | reduced radius of curvature in y-direction, r_{1y}r_{2y}/(r_{1y} + r_{2y}) |

u | mean speed |

U | speed parameter, η_{0}u/E’R_{x} |

W | load parameter, F/E’R_{x}^{2} |

x, y | computational domain coordinates (x_{a}, x_{b}, y_{a}, y_{b}—boundaries of the domain) |

α_{0} | initial pressure–viscosity coefficient of lubricant |

α* | reciprocal asymptomatic isoviscous pressure coefficient |

α_{film} | film pressure–viscosity coefficient |

ρ | lubricant density |

ρ_{0} | lubricant density at ambient pressure |

η | lubricant viscosity |

η_{0} | lubricant viscosity at ambient pressure |

## Appendix A. Simulation Data

**Table A1.**Ratios of central film thickness to minimum film thickness in a point contact for α

_{film}= 8.7 GPa

^{−1}.

M | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

2 | 5 | 10 | 20 | 50 | 100 | 200 | 500 | 750 | 1000 | ||

L | 1 | 1.24 | 1.35 | 1.36 | 1.48 | 1.61 | 1.79 | 2.08 | 2.23 | 2.35 | |

2 | 1.24 | 1.34 | 1.38 | 1.53 | 1.70 | 1.90 | 2.24 | 2.41 | 2.55 | ||

5 | 1.23 | 1.31 | 1.40 | 1.59 | 1.78 | 2.01 | 2.41 | 2.63 | 2.77 | ||

7 | 1.23 | 1.30 | 1.40 | 1.60 | 1.79 | 2.03 | 2.43 | 2.64 | 2.81 | ||

10 | 1.23 | 1.29 | 1.40 | 1.59 | 1.78 | 2.02 | 2.42 | 2.63 | 2.80 | ||

15 | 1.14 | 1.21 | 1.28 | 1.38 | 1.56 | 1.74 | 1.97 | 2.36 | 2.57 | 2.74 | |

20 | 1.13 | 1.19 | 1.26 | 1.36 | 1.53 | 1.70 | 1.92 | 2.30 | 2.50 | 2.65 | |

25 | 1.12 | 1.18 | 1.25 | 1.34 | 1.50 | 1.66 | 1.87 | 2.23 | 2.43 | 2.58 | |

30 | 1.10 | 1.17 | 1.24 | 1.32 | 1.48 | 1.63 | 1.83 | 2.17 | 2.36 | 2.50 |

**Table A2.**Ratios of central film thickness to minimum film thickness in a point contact for α

_{film}= 20.6 GPa

^{−1}.

M | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

2 | 5 | 10 | 20 | 50 | 100 | 200 | 500 | 750 | 1000 | ||

L | 1 | 1.25 | 1.37 | 1.39 | 1.52 | 1.67 | 1.87 | 2.20 | 2.38 | 2.50 | |

2 | 1.25 | 1.37 | 1.42 | 1.61 | 1.81 | 2.04 | 2.43 | 2.64 | 2.78 | ||

5 | 1.28 | 1.36 | 1.49 | 1.71 | 1.93 | 2.18 | 2.62 | 2.85 | 3.02 | ||

7 | 1.30 | 1.37 | 1.49 | 1.71 | 1.92 | 2.17 | 2.60 | 2.82 | 3.00 | ||

10 | 1.29 | 1.36 | 1.47 | 1.67 | 1.86 | 2.10 | 2.51 | 2.73 | 2.90 | ||

15 | 1.19 | 1.25 | 1.32 | 1.42 | 1.59 | 1.77 | 1.98 | 2.37 | 2.58 | 2.73 | |

20 | 1.16 | 1.22 | 1.29 | 1.38 | 1.53 | 1.69 | 1.90 | 2.25 | 2.45 | 2.61 | |

25 | 1.14 | 1.20 | 1.26 | 1.34 | 1.49 | 1.64 | 1.83 | 2.17 | 2.36 | 2.51 | |

30 | 1.18 | 1.24 | 1.31 | 1.45 | 1.59 | 1.78 | 2.10 | 2.28 | 2.42 |

**Table A3.**Ratios of central film thickness to minimum film thickness in a point contact for α

_{film}= 32.7 GPa

^{−1}.

M | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

2 | 5 | 10 | 20 | 50 | 100 | 200 | 500 | 750 | 1000 | ||

L | 1 | 1.25 | 1.37 | 1.40 | 1.54 | 1.70 | 1.90 | 2.25 | 2.44 | ||

2 | 1.26 | 1.38 | 1.44 | 1.65 | 1.86 | 2.11 | 2.52 | 2.75 | |||

5 | 1.30 | 1.39 | 1.53 | 1.77 | 1.99 | 2.27 | 2.72 | 2.96 | |||

7 | 1.33 | 1.40 | 1.53 | 1.76 | 1.97 | 2.23 | 2.67 | 2.89 | 3.09 | ||

10 | 1.32 | 1.39 | 1.51 | 1.70 | 1.90 | 2.13 | 2.54 | 2.77 | 2.95 | ||

15 | 1.21 | 1.27 | 1.34 | 1.44 | 1.61 | 1.78 | 1.99 | 2.37 | 2.57 | 2.73 | |

20 | 1.18 | 1.24 | 1.33 | 1.41 | 1.54 | 1.70 | 1.90 | 2.24 | 2.44 | 2.59 | |

25 | 1.35 | 1.51 | 1.64 | 1.83 | 2.15 | 2.37 |

## Appendix B. Film Thickness Ratio Equation and Table with Assumptions

_{film}is in GPa

^{−1}. This equation was obtained under contact assumptions and for the range of conditions listed in the following Table A4.

Isothermal Newtonian (without shear thinning effects) conditions |

Circular point contact |

Smooth surface |

Lubricant rheology governed by Roelands pressure–viscosity and Dowson–Higginson pressure–density relationships |

M parameter from 2 to 1000 |

L parameter from 1 to 30 |

Pressure–viscosity coefficient α_{film} from 8.7 to 32.7 GPa^{−1} |

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**Figure 1.**Present calculation domain compared to operating points of experiments and domain for regression of Hamrock–Dowson formulas.

**Figure 4.**Numerical results of film thickness ratio h

_{c}/h

_{min}fitted by analytical function: (

**a**–

**c**) contours plots; and (

**d**–

**f**) fitting residuals ((

**a**,

**d**) for α

_{film}= 8.7 GPa

^{−1}; (

**b**,

**e**) for α

_{film}= 20.6 GPa

^{−1}; and (

**c**,

**f**) α

_{film}= 32.7 GPa

^{−1}).

**Figure 6.**Comparison of measured h

_{c}/h

_{min}ratios with prediction from Hamrock–Dowson equations (Equations (8) and (9)) and present analytical model given by Equation (12).

TOTM | |
---|---|

Test temperature | 30 ± 0.5 °C |

Ambient viscosity | 0.1517 Pa·s |

Initial pressure–viscosity coefficient α_{0} | 23.9 GPa^{−1} |

Pressure–viscosity coefficient α* | 21.5 GPa^{−1} |

Film pressure–viscosity coefficient α_{film} | 20.9 GPa^{−1} |

Steel | Glass | Sapphire | |
---|---|---|---|

Radius of curvature | 12.7 mm | ∞ | ∞ |

Young modulus | 206 GPa | 81 GPa | 405 GPa |

Poisson’s ratio | 0.3 | 0.209 | 0.25 |

M | L | α_{0} | α* | α_{film} |
---|---|---|---|---|

2, 5, 10, 20, 50, 100, 200, 500, 750, 1000 | 1, 2, 5, 7, 10, 15, 20, 25, 30 | 11, 22, 33 GPa^{−1} | 8.2, 20.3, 32.6 GPa^{−1} | 8.7, 20.6, 32.7 GPa^{−1} |

Comparison 1 | Comparison 2 | Comparison 3 | |
---|---|---|---|

Published result | Venner [18] | Venner [18] | Chevalier [17] |

Present result | numerical results | analytical model | analytical model |

Average difference | 4.4% | 4.3% | 3.5% |

α_{film} | 8.7 GPa^{−1} | 20.6 GPa^{−1} | 32.7 GPa^{−1} |
---|---|---|---|

Root mean square error (RMSE) | 0.031 | 0.038 | 0.039 |

RMSE in % | 1.7% | 2.0% | 2.1% |

**Table 6.**Average difference of measurement and h

_{c}/h

_{min}ratio estimation by Hamrock–Dowson (H&D) and present analytical formula.

Case 1 | Case 2 | Case 3 | |
---|---|---|---|

Measurement pressure | 0.5 GPa | 0.8 GPa | 1.2 GPa |

H&D formulas (Equations (9) and (10)) | 12.9% | 36.5% | 9.7% |

Present formula (Equation (12)) | 8.8% | 10.6% | 2.2% |

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**MDPI and ACS Style**

Sperka, P.; Krupka, I.; Hartl, M.
Analytical Formula for the Ratio of Central to Minimum Film Thickness in a Circular EHL Contact. *Lubricants* **2018**, *6*, 80.
https://doi.org/10.3390/lubricants6030080

**AMA Style**

Sperka P, Krupka I, Hartl M.
Analytical Formula for the Ratio of Central to Minimum Film Thickness in a Circular EHL Contact. *Lubricants*. 2018; 6(3):80.
https://doi.org/10.3390/lubricants6030080

**Chicago/Turabian Style**

Sperka, Petr, Ivan Krupka, and Martin Hartl.
2018. "Analytical Formula for the Ratio of Central to Minimum Film Thickness in a Circular EHL Contact" *Lubricants* 6, no. 3: 80.
https://doi.org/10.3390/lubricants6030080