# An Analytical Approach for Predicting EHL Friction: Usefulness and Limitations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Film Thickness

_{N}is the highest pressure used in this calculation [19].

#### 2.2. Friction Coefficient and Contact Temperature

#### 2.3. Methodology

#### 2.4. Experimentation

_{0}= 7.36 mPa·s, α* = 9.0 GPa

^{−1}at 80 °C; and η

_{0}= 4.78 mPa·s, α* = 8.2 GPa

^{−1}at 100 °C. In addition, according to references [13,14,59], the following values of the power-law exponent, shear modulus and thermal conductivity at 80 °C can be taken for the PAO-6: n = 0.81, G = 0.1 MPa and K

_{L}= 0.15 W/(mK).

## 3. Results and Discussion

#### 3.1. Initial Application of the Models

#### 3.2. Influence of the Piezoviscous Response

_{oo}= 11.804 GPa, K′

_{o}= 11.74, B = 3.811, α

_{V}= 0.0008 K

^{−1}, β

_{K}= 0.008655 K

^{−1}, V

_{∞}

_{R}/V

_{R}= 0.6193, ε = −0.00134 K

^{−1}, and η

_{R}= 0.1803 Pa·s at T

_{R}= 70 °C. On the other hand, experimental measurements of the low-shear viscosity for the PAO-650 are given in reference [66].

^{−1}at 70 °C and 14.0 GPa

^{−1}at 120 °C, whereas for the PAO-650 they are 20.1 GPa

^{−1}at 20 °C and 14.8 GPa

^{−1}at 75 °C. In addition, the variation of the pressure-viscosity coefficient shown in Figure 8 is obtained by fitting Equation (6) to the low-shear viscosity of these lubricants in pressure intervals of 0.2 GPa. The pressure-viscosity coefficient decreases with increasing pressure and moves away from the reciprocal asymptotic isoviscous pressure coefficient. If this variation is not taken into account, there is a general tendency to overestimate the viscosity and consequently also the friction coefficient. This issue becomes quite significant at high Hertz contact pressure, as can be seen in the results for the PAO-6 in Figure 4 and Figure 5.

^{−1}. If we assume a similar variation of this parameter at 80 °C, a rough estimate of the pressure-viscosity coefficient at 80 °C and 1.1 GPa can be obtained for the PAO-6, resulting in 8.3 GPa

^{−1}. The calculations in the Hertzian region using these new values lead to a significantly better agreement with the experimental data than that previously obtained at Hertz pressures of about 1.1 GPa, as demonstrated in Figure 10 and Figure 11.

#### 3.3. Usefulness and Limitations of the Analytical Approach

## 4. Conclusions

- The film thickness formulae employed were obtained by curve fitting data over ranges of operating parameters. Therefore, outside these ranges, deviations in the predictions may be expected. However, the calculation process proposed in the article remains valid for other film thickness formulae, which could be either more general equations or expressions adapted to the range of operating conditions in each case.
- The analytical deduction of friction formulae becomes increasingly difficult as more complex rheological models are considered, such as when using free volume correlations for the low-shear viscosity. To overcome this issue, a simple exponential law can be considered and the values of the pressure-viscosity coefficient can be fitted to the real piezoviscous response.
- Although the use of equations for the central film thickness and the average contact temperature provides very useful information, the analytical approach cannot predict the film thickness and temperature distributions within the EHL contact.
- As a consequence of all the simplifications introduced, less accurate results may be expected in any analytical approach. However, the formulation proposed can capture the essential features of the EHL contacts and exhibits a reasonably good predictive potential.

- The deduction of new film thickness equations applicable in a more general way. They may be obtained from EHL solutions in a broader range of operating conditions by means of curve-fitted regression formulae. Similarly, more general film thickness correction factors for shear-thinning and thermal effects can also be derived.
- The consideration of other effects for an improved formulation, such as transient conditions or starvation. Although they would complicate the development of purely analytical models, these effects could be considered by using machine learning algorithms or semi-analytical approaches, such as those based on the Reynolds-Carreau equations, integrated into the calculation process described in the article.
- A methodology similar to that proposed in the present article may also be applied to other geometries of interest, such as the elliptical contact. To this end, some references and indications are provided in Section 2.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

a | contact half-width (or radius for circular point contact), m |

B | Doolittle parameter |

E_{1}, E_{2} | Young’s modulus of the contacting bodies, Pa |

E’ | $\mathrm{reduced}\text{}\mathrm{Young}\text{\u2019}\mathrm{s}\text{}\mathrm{modulus},\text{}\mathrm{Pa}:\text{}\frac{1}{E\prime}=\frac{1}{2}\left(\frac{1-{\nu}_{1}^{2}}{{E}_{1}}+\frac{1-{\nu}_{2}^{2}}{{E}_{2}}\right)$ |

G | shear modulus of the lubricant, Pa |

h | central film thickness, m |

h_{N} | Newtonian central film thickness, m |

K_{1}, K_{2} | thermal conductivity of the contacting bodies, W/(mK) |

K_{L} | thermal conductivity of the lubricant, W/(mK) |

K′_{o} | pressure rate of change of isothermal bulk modulus at p = 0 |

K_{oo} | isothermal bulk modulus at zero absolute temperature and p = 0, Pa |

L_{T} | thermal loading factor |

n | power-law exponent |

p | pressure, Pa |

p_{H} | Hertz (maximum) pressure, Pa |

R | reduced radius of curvature, m: $R={\left(\frac{1}{{R}_{1}}+\frac{1}{{R}_{1}}\right)}^{-1}$ |

R_{1},R_{2} | radii of the contacting surfaces, m |

R_{q} | combined RMS surface roughness, m |

SRR | $\mathrm{slide}-\mathrm{to}-\mathrm{roll}\text{}\mathrm{ratio},\text{}\%:\text{}SRR=\frac{\mathrm{\Delta}u}{{u}_{m}}100$ |

T | absolute temperature in the Tait-Doolittle equation, K |

T_{b} | lubricant bath temperature, °C |

T_{c} | contact temperature, °C |

T_{in} | inlet temperature, °C |

T_{R} | reference temperature in the Tait-Doolittle equation, K |

u_{1}, u_{2} | velocities of the contacting surfaces, m/s |

u_{m} | average velocity or rolling velocity, m/s |

V_{R} | volume at reference temperature and p = 0, m^{3} |

V_{∞R} | occupied volume at reference temperature and p = 0, m^{3} |

W | normal load, N |

W/L | normal load per unit length, N/m |

α | pressure-viscosity coefficient, Pa^{−1} |

α* | reciprocal asymptotic isoviscous pressure coefficient, Pa^{−1} |

α_{V} | thermal expansivity, K^{−1} |

β | temperature-viscosity coefficient, K^{−1} |

β_{k} | temperature coefficient of isothermal bulk modulus at p = 0, K^{−1} |

$\dot{\gamma}$ | shear rate, s^{−1} |

ΔT_{f} | average flash temperature rise, °C |

ΔT_{L} | average temperature rise with respect to the surfaces due to viscous heating, °C |

Δu | sliding velocity, m/s |

ε | occupied volume thermal expansivity, K^{−1} |

η | low-shear viscosity, Pa·s |

η_{0} | low-shear viscosity at p = 0, Pa·s |

η_{G} | generalized viscosity, Pa·s |

η_{R} | low-shear viscosity at reference temperature and p = 0, Pa·s |

λ | specific lubricant film thickness or lambda ratio |

μ | friction (or traction) coefficient |

ν_{1}, ν_{2} | Poisson ratio of the contacting bodies |

ρ_{1}, ρ_{2} | density of the contacting bodies, kg/m^{3} |

σ_{1}, σ_{2} | specific heat of the contacting bodies, J/(kgK) |

τ | shear stress, Pa |

φ_{T} | thermal film thickness reduction factor |

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**Figure 3.**Friction test results and calculated lambda ratio for each test. Ball-on-disc contact lubricated with PAO-6 at T

_{b}= 80 °C and SRR = 50%.

**Figure 4.**Comparison of the models with experimental results. Ball-on-disc contact lubricated with PAO-6 at T

_{b}= 80 °C and SRR = 50%.

**Figure 5.**Comparison of the models with experimental results for the PAO-6 oil at T

_{b}= 80 °C: (

**a**) Ball-on-disc contact at u

_{m}= 2 m/s; (

**b**) Triple contact between steel discs at u

_{m}= 2.5 m/s.

**Figure 9.**Influence of the pressure-viscosity coefficient on the prediction of the friction coefficient: (

**a**) Steel-steel line contact lubricated with PAO-100 at T

_{b}= 70 °C, p

_{H}= 1 GPa and SRR = 100%; (

**b**) Steel ball on glass disc, lubricated with PAO-650 at T

_{b}= 75 °C, p

_{H}= 0.53 GPa and u

_{m}= 0.03, 0.13 and 0.26 m/s. (For more details, see references [65,66]).

**Figure 10.**New prediction for the results at highest pressure in Figure 4. Steel ball on steel disc contact lubricated with PAO-6 at T

_{b}= 80 °C, SRR = 50% and W = 50 N (p

_{H}= 1.12 GPa).

**Figure 11.**New prediction for the results at highest pressures in Figure 5: (

**a**) Steel ball on steel disc contact lubricated with PAO-6 at T

_{b}= 80 °C, u

_{m}= 2 m/s and W = 50 N (p

_{H}= 1.12 GPa); (

**b**) Triple contact between steel discs lubricated with PAO-6 at T

_{b}= 80 °C, u

_{m}= 2.5 m/s and W = 150 N (p

_{H}= 1.06 GPa).

**Figure 12.**Friction calculated with the methodology proposed in the article (µ

_{F}) versus numerical integration of the Tait-Doolittle equation (µ

_{TD}) for the PAO-100, Δu = 1 m/s and h = 80 nm. Percentage deviations calculated as the absolute value of 100(µ

_{F}− µ

_{TD})/µ

_{TD}.

**Table 1.**Classical central film thickness formulae and Hertzian contact parameters. The film thickness formula for each type of geometry is expressed both as a function of non-dimensional groups and in terms of dimensional parameters.

Type of Geometry | Newtonian Film Thickness | Hertzian Parameters | |
---|---|---|---|

Circular contact | $\frac{{h}_{N}}{R}=1.90{\left(\frac{{\eta}_{0}{u}_{m}}{E\prime R}\right)}^{0.67}{\left(\alpha *E\prime \right)}^{0.53}{\left(\frac{W}{E\prime {R}^{2}}\right)}^{-0.067}$ $\mathrm{Equivalently}:\text{}{h}_{N}=1.55\alpha {*}^{0.53}{\left({\eta}_{0}{u}_{m}\right)}^{0.67}E{\prime}^{0.061}{R}^{0.33}{p}_{H}^{-0.201}$ | ${p}_{H}=\frac{3W}{2\pi {a}^{2}}$ | $a={\left(\frac{3WR}{2E\prime}\right)}^{1/3}$ |

Line contact | $\frac{{h}_{N}}{R}=2.922{\left(\frac{{\eta}_{0}{u}_{m}}{E\prime R}\right)}^{0.692}{\left(\alpha *E\prime \right)}^{0.47}{\left(\frac{W/L}{E\prime R}\right)}^{-0.166}$ $\mathrm{Equivalently}:\text{}{h}_{N}=2.154\alpha {*}^{0.47}{\left({\eta}_{0}{u}_{m}\right)}^{0.692}E{\prime}^{0.110}{R}^{0.308}{p}_{H}^{-0.332}$ | ${p}_{H}=\frac{2(W/L)}{\pi a}$ | $a={\left(\frac{8(W/L)R}{\pi E\prime}\right)}^{1/2}$ |

Type of Geometry | Equation |
---|---|

Circular contact | ${\left[{\eta}_{0}({T}_{in})\right]}^{0.67}{\left[\alpha *({T}_{in})\right]}^{0.53}={\phi}_{T}{\left[{\eta}_{0}({T}_{b})\right]}^{0.67}{\left[\alpha *({T}_{b})\right]}^{0.53}$ |

Line contact | ${\left[{\eta}_{0}({T}_{in})\right]}^{0.692}{\left[\alpha *({T}_{in})\right]}^{0.47}={\phi}_{T}{\left[{\eta}_{0}({T}_{b})\right]}^{0.692}{\left[\alpha *({T}_{b})\right]}^{0.47}$ |

Geometry | Formula |
---|---|

Point contact | $\mu =3{\left(\frac{{\eta}_{0}\mathrm{\Delta}u}{h}\right)}^{n}\frac{{G}^{1-n}}{{p}_{H}}\left[\frac{\mathrm{exp}(n\alpha {p}_{H})}{n\alpha {p}_{H}}\left(1-\frac{1}{n\alpha {p}_{H}}\right)+\frac{1}{{\left(n\alpha {p}_{H}\right)}^{2}}\right]$ |

Line contact | $\mu ={\left(\frac{{\eta}_{0}\mathrm{\Delta}u}{h}\right)}^{n}\frac{{G}^{1-n}}{{p}_{H}}\left[\frac{4}{\pi}+\frac{n\alpha {p}_{H}}{3}\left(\frac{2+\sqrt{3}}{2}\mathrm{exp}\left(n\alpha {p}_{H}\sqrt{\frac{1}{2}-\frac{\sqrt{3}}{4}}\right)+\mathrm{exp}\left(n\alpha {p}_{H}\frac{\sqrt{2}}{2}\right)+\frac{2-\sqrt{3}}{2}\mathrm{exp}\left(n\alpha {p}_{H}\sqrt{\frac{1}{2}+\frac{\sqrt{3}}{4}}\right)\right)\right]$ |

Geometry | Average Temperature Rise Formulae | |
---|---|---|

Circular contact | $\mathrm{\Delta}{T}_{f}=\frac{0.37\mu W\mathrm{\Delta}u/a}{\sqrt{{K}_{1}(1.74{K}_{1}+a{u}_{1}{\rho}_{1}{\sigma}_{1})}+\sqrt{{K}_{2}(1.74{K}_{2}+a{u}_{2}{\rho}_{2}{\sigma}_{2})}}$ | $\mathrm{\Delta}{T}_{L}=\frac{\mu Wh\mathrm{\Delta}u}{8\pi {a}^{2}{K}_{L}}$ |

Line Contact | $\mathrm{\Delta}{T}_{f}=\frac{0.53\mu (W/L)\mathrm{\Delta}u}{\sqrt{{K}_{1}(0.90{K}_{1}+a{u}_{1}{\rho}_{1}{\sigma}_{1})}+\sqrt{{K}_{2}(0.90{K}_{2}+a{u}_{2}{\rho}_{2}{\sigma}_{2})}}$ | $\mathrm{\Delta}{T}_{L}=\frac{\mu (W/L)h\mathrm{\Delta}u}{16a{K}_{L}}$ |

**Table 5.**Properties of the tested materials [13].

Property | Steel | Copper |
---|---|---|

Young’s modulus, GPa | 210 | 117 |

Poisson ratio | 0.30 | 0.34 |

Thermal conductivity, W/(mK) | 41 | 385 |

Density, kg/m^{3} | 7850 | 8913 |

Specific heat, J/(kgK) | 418 | 398 |

Test Rig and Type of Contact | Materials | Load, N | Hertz Pressure, GPa |
---|---|---|---|

MTM, ball-on-disc | steel-copper | 20 | 0.67 |

steel-steel | 20 | 0.83 | |

steel-steel | 50 | 1.12 | |

MPR, triple-disc | steel-steel | 100 | 0.86 |

steel-steel | 150 | 1.06 |

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**MDPI and ACS Style**

Echávarri Otero, J.; de la Guerra Ochoa, E.; Chacón Tanarro, E.; Franco Martínez, F.; Contreras Urgiles, R.W.
An Analytical Approach for Predicting EHL Friction: Usefulness and Limitations. *Lubricants* **2022**, *10*, 141.
https://doi.org/10.3390/lubricants10070141

**AMA Style**

Echávarri Otero J, de la Guerra Ochoa E, Chacón Tanarro E, Franco Martínez F, Contreras Urgiles RW.
An Analytical Approach for Predicting EHL Friction: Usefulness and Limitations. *Lubricants*. 2022; 10(7):141.
https://doi.org/10.3390/lubricants10070141

**Chicago/Turabian Style**

Echávarri Otero, Javier, Eduardo de la Guerra Ochoa, Enrique Chacón Tanarro, Francisco Franco Martínez, and Rafael Wilmer Contreras Urgiles.
2022. "An Analytical Approach for Predicting EHL Friction: Usefulness and Limitations" *Lubricants* 10, no. 7: 141.
https://doi.org/10.3390/lubricants10070141