# Power Loss Analysis of an Oil-Jet Lubricated Angular Contact Ball Bearing: Theoretical and Experimental Investigations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- -
- -
- -
- -

## 2. Static Model

#### 2.1. REB Geometry

#### 2.2. Distribution of Internal Loading

_{nj}) located at a position ψ

_{j}(Figure 1b). In the present study, the speed range is moderate. Neither centrifugal force nor gyroscopic moment are considered, resulting in similar contact angles and loads on the inner and outer rings. From this hypothesis, the global REB equilibrium can be written as:

## 3. Lubrication Regime in Full Film

#### 3.1. Moes’ Parameters

#### 3.2. Mapping the Lubrication Regime

- -
- The ball lubrication regime may change from one ball to another, due to the load distribution.
- -
- The ball lubrication regime may change according to contact temperature.

## 4. Power Loss Modeling

#### 4.1. Ball/Ring Contact Area

#### 4.2. Sliding

#### 4.3. Hydrodynamic Rolling

- -
- For the IVR regime, hydrodynamic rolling can be estimated using Houpert’s work [30]:

- -

- -

#### 4.4. Drag Losses

#### 4.5. Thermal Model

## 5. Experimental Investigation

#### 5.1. Test Rig and Test Procedure

#### 5.1.1. Limits and Measurements Performed during This Test Campaign

#### 5.1.2. Calibration

#### 5.1.3. Range and Accuracy

#### 5.2. Rolling Element Bearing and Test Matrix

#### 5.2.1. Oil Lubrication

#### 5.2.2. Rotational Speed

#### 5.2.3. Load

#### 5.2.4. Test Matrix

#### 5.3. Power Loss Measurement Results

#### 5.3.1. Evolution with Speed

#### 5.3.2. Evolution with Oil Injection Temperature

#### 5.3.3. Evolution with Oil Flow Rate

#### 5.3.4. Evolution with Axial Preload

#### 5.3.5. Evolution with Radial Load

## 6. Comparison between Measurements and Power Loss Calculations

#### 6.1. Comparison According to Different Parameters

- -
- In Figure 14a, the model shows a very good agreement with experiments. Speed appears to be correctly taken into account in the model. The power loss distribution at 50 °C is shown in Figure 15 for different rotational speeds. Hydrodynamic rolling is the major contributor (about 95% of the losses). Sliding is very small (<1%), as well as drag, especially at low speeds.

- -
- In Figure 14b, the influence of the oil temperature injection is similar for the experiments and for the developed model. It emphasizes that the oil temperature which influences power losses is the one at the contact interface between the balls and the rings. This oil temperature is directly related to the temperature of the balls and the rings, and not to the oil injection one.
- -
- In Figure 14c, the calculated power losses are slightly modified by the oil flow rate. The oil flow rate only changes the drag contribution through the intermediary of the oil volume fraction. This fraction modifies the mixture properties, especially its density. Multiplying the oil flow rate by 1.5 leads to an increase of drag losses equal to 16%. However, as shown in Figure 15, the drag contribution represents only 5% of the total power losses. Therefore, the influence of the oil flow rate is limited. It should be noted that the oil flow is still necessary to avoid starvation effects, which could have an impact on the hydrodynamic rolling contribution.
- -
- In Figure 14d, a good agreement is found between the model and the experiments, especially at high temperatures for a stabilized thermal behavior. One can note a constant increase when the axial preload changes from 225 N to 450 N, whatever the temperature is.

#### 6.2. Study of the Radial Load Influence

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$a$,b | Dimensions of the contact ellipse [m] |

$A$ | Distance between raceway groove curvature centers [m] |

${C}_{0}$ | Static capacity [N] |

${d}_{m}$ | Mean diameter [m] |

$D$ | Rolling element diameter [m] |

${E}^{\prime}$ | Equivalent Young’s modulus = $\frac{2}{\left[\frac{1-{\nu}_{1}^{2}}{{E}_{1}}+\frac{1-{\nu}_{2}^{2}}{{E}_{2}}\right]}$ [Pa] |

$f$ | Osculation = $r/D$ [−] |

${F}_{HR}$ | Hydrodynamic rolling force [N] |

${F}_{r}$ | Radial load [N] |

${F}_{a}$ | Axial load [N] |

${F}_{s\mathrm{l}}$ | Sliding force [N] |

${G}^{\mathrm{*}}$ | Dimensionless materials parameter = $\alpha {E}^{\prime}$ [−] |

$\mathrm{H}$ | Dimensionless film thickness [m] |

$K$ | Load-deflection factor [$\mathrm{N}\xb7\mathrm{m}{\mathrm{m}}^{-1.5}]$ |

$N,L$ | Moes parameters, [$\mathrm{r}\mathrm{p}\mathrm{m}$] |

${P}_{d}$ | Diametral clearance [m] |

$P$ | Power losses, [W] |

$Q$ | Oil flow rate [${\mathrm{m}}^{3}\xb7{\mathrm{s}}^{-1}$] |

${Q}_{n}$ | Load on a ball [N] |

$r$ | Raceway groove curvature radius [m] |

${R}_{x}$, ${R}_{y}$ | Equivalent radii [m] |

${U}^{\mathrm{*}}$ | Dimensionless speed =$\frac{{v}_{r}\eta}{{E}^{\prime}{R}_{x}}$ [−] |

${v}_{r}$ | Rolling velocity [$\mathrm{m}\xb7{\mathrm{s}}^{-1}$] |

${v}_{sl}$ | Sliding velocity [$\mathrm{m}\xb7{\mathrm{s}}^{-1}$] |

${W}^{\mathrm{*}}$ | Dimensionless load = $\frac{Q}{{E}^{\prime}{R}_{x}^{2}}$ [−] |

Z | Number of rolling elements [−] |

Subscripts | |

${\alpha}^{0}$ | Free contact angle [rad] |

$\alpha $ | Contact angle [rad] |

${\alpha}_{p}$ | Pressure–viscosity coefficient [$\mathrm{P}{\mathrm{a}}^{-1}$] |

$\beta $ | Coefficient of penetration [−] |

$\gamma $ | Geometrical ratio [−] |

$\lambda $ | Radii ratio = $\frac{{R}_{x}}{{R}_{y}}$ [−] |

${\rho}_{equiv}$ | Curvature sum [${\mathrm{m}}^{-1}$] |

$\delta $ | Deflection or contact deformation [m] |

$\eta $ | Dynamic viscosity [$\mathrm{P}\mathrm{a}\xb7\mathrm{s}$] |

${\mu}_{sl}$ | Friction coefficient [−] |

$\psi $ | Location angle of a ball [rad] |

$\omega $ | Rotational speed [$\mathrm{r}\mathrm{a}\mathrm{d}\xb7{\mathrm{s}}^{-1}$] |

i | Refers to inner ring |

o | Refers to outer ring |

j | Refers to ball number |

## References

- Zhao, Y.; Zi, Y.; Chen, Z.; Zhang, M.; Zhu, Y.; Yin, J. Power loss investigation of ball bearings considering rolling-sliding contacts. Int. J. Mech. Sci.
**2023**, 250, 108318. [Google Scholar] [CrossRef] - Zhou, S.; Singh, A.; Kahraman, A.; Hong, I.; Vedera, K. Power Loss Studies for Rolling Element Bearings Subject to Combined Radial and Axial Loading. SAE Tech. Pap.
**2023**, 1, 461. [Google Scholar] [CrossRef] - Wingertszahn, P.; Koch, O.; Maccioni, L.; Concli, F.; Sauer, B. Predicting Friction of Tapered Roller Bearings with Detailed Multi-Body Simulation Models. Lubricants
**2023**, 11, 369. [Google Scholar] [CrossRef] - Maccioni, L.; Rüth, L.; Koch, O.; Concli, F. Load-independent power losses of full-flooded lubricated tapered roller bearings: Numerical and experimental investigation of the effect of operating temperature and housing walls distances. Tribol. Trans.
**2023**, 66, 1078–1094. [Google Scholar] [CrossRef] - Yilmaz, M.; Lohner, T.; Michaelis, K.; Stahl, K. Bearing Power Losses with Water-Containing Gear Fluids. Lubricants
**2020**, 8, 5. [Google Scholar] [CrossRef] - Darul, L.; Touret, T.; Changenet, C.; Ville, F. Power Losses of Oil-Jet Lubricated Ball Bearings with Limited Applied Load: Part 1—Theoretical Analysis. Tribol. Trans.
**2023**, 66, 801–808. [Google Scholar] [CrossRef] - Jones, A.B. Ball Motion and Sliding Friction in Ball Bearings. ASME J. Basic Eng.
**1959**, 81, 1–12. [Google Scholar] [CrossRef] - Heathcote, H.L. The ball bearing: In the making, under test and on service. Proc. Inst. Automob. Eng.
**1920**, 15, 569–702. [Google Scholar] [CrossRef] - Poritsky, H.; Hewlett, C.W., Jr.; Coleman, R.E. Sliding Friction of Ball Bearings of the Pivot Type. ASME J. Appl. Mech.
**1947**, 14, 261–268. [Google Scholar] [CrossRef] - Johnson, K.L. The Influence of Elastic Deformation Upon the Motion of a Ball Rolling between Two Surfaces. Proc. Inst. Mech. Eng.
**1959**, 173, 795–810. [Google Scholar] [CrossRef] - Tevaarwerk, J.L.; Johnson, K.L. The Influence of Fluid Rheology on the Performance of Traction Drives. ASME J. Technol.
**1979**, 101, 266–273. [Google Scholar] [CrossRef] - Houpert, L. Piezoviscous-Rigid Rolling and Sliding Traction Forces, Application: The Rolling Element–Cage Pocket Contact. ASME J. Tribol.
**1987**, 109, 363–370. [Google Scholar] [CrossRef] - Zhou, R.S.; Hoeprich, M.R. Torque of Tapered Roller Bearings. ASME J. Tribol.
**1991**, 113, 590–597. [Google Scholar] [CrossRef] - Biboulet, N.; Houpert, L. Hydrodynamic Force and Moment in Pure Rolling Lubricated Contacts. Part 1: Line Contacts. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol.
**2010**, 224, 765–775. [Google Scholar] [CrossRef] - Pouly, F.; Changenet, C.; Ville, F.; Velex, P.; Damiens, B. Power Loss Predictions in High-Speed Rolling Element Bearings Using Thermal Networks. Tribol. Trans.
**2010**, 53, 957–967. [Google Scholar] [CrossRef] - Nelias, D.; Sainsot, P.; Flamand, L. Power Loss of Gearbox Ball Bearing under Axial and Radial Loads. Tribol. Trans.
**1994**, 37, 83–90. [Google Scholar] [CrossRef] - Parker, R.J. Comparison of Predicted and Experimental Thermal Performance of Angular Contact Ball Bearings; NASA Technical Paper; NASA: Washington, DC, USA, 1984; Volume 2275, pp. 1–20. [Google Scholar]
- Palmgren, A. Les Roulements: Description, Théorie, Applications, 2nd ed.; SKF Group: Göteborg, Sweden, 1967; 241p. [Google Scholar]
- Harris, T.A. Rolling Bearing Analysis, 3rd ed.; John Wiley and Sons Inc.: New York, NY, USA, 1991; ISBN 0 471 51349 0. [Google Scholar]
- SKF Group. Rolling Bearings; SKF Group: Göteborg, Sweden, 2013; 1375p. [Google Scholar]
- de Cadier de Veauce, F.; Darul, L.; Marchesse, Y.; Touret, T.; Changenet, C.; Ville, F.; Amar, L.; Fossier, C. Power Losses of Oil-Jet Lubricated Ball Bearings With Limited Applied Load: Part 2—Experiments and Model Validation. Tribol. Trans.
**2023**, 66, 822–831. [Google Scholar] [CrossRef] - Dindar, A.; Hong, I.; Garg, A.; Kahraman, A. A Methodology to Measure Power Losses of Rolling Element Bearings under Combined Radial and Axial Loading Conditions. Tribol. Trans.
**2022**, 65, 137–152. [Google Scholar] [CrossRef] - Brossier, P.; Niel, D.; Changenet, C.; Ville, F.; Belmonte, J. Experimental and Numerical Investigations on Rolling Element Bearing Thermal Behaviour. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol.
**2021**, 235, 842–853. [Google Scholar] [CrossRef] - Kerrouche, R.; Dadouche, A.; Mamou, M.; Boukraa, S. Power Loss Estimation and Thermal Analysis of an Aero-Engine Cylindrical Roller Bearing. Tribol. Trans.
**2021**, 64, 1079–1094. [Google Scholar] [CrossRef] - Popescu, A.; Olaru, D.N. Influence of lubricant on the friction in an angular contact ball bearing under low load conditions. IOP Conf. Ser. Mater. Sci. Eng.
**2020**, 724, 12040. [Google Scholar] [CrossRef] - Bălan, M.R.D.; Houpert, L.; Tufescu, A.; Olaru, D.N. Rolling Friction Torque in Ball-Race Contacts Operating in Mixed Lubrication Conditions. Lubricants
**2015**, 3, 222–243. [Google Scholar] [CrossRef] - Moes, H. Optimum similarity analysis with applications to elastohydrodynamic lubrication. Wear
**1992**, 59, 57–66. [Google Scholar] [CrossRef] - Dowson, D.; Higginson, G. Elasto-Hydrodynamic Lubrication, SI ed.; Pergamon Press Ltd.: Oxford, UK, 1977; ISBN 9781483181899. [Google Scholar]
- Coulomb, C.A. Théorie des Machines Simples; Bachelier: Paris, France, 1821; 387p. [Google Scholar]
- Houpert, L. Hydrodynamic Load Calculation in Rolling Element Bearings. Tribol. Trans.
**2016**, 59, 538–559. [Google Scholar] [CrossRef] - Biboulet, N.; Houpert, L. Hydrodynamic Force and Moment in Pure Rolling Lubricated Contacts. Part 2: Point Contacts. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol.
**2010**, 224, 777–787. [Google Scholar] [CrossRef]

**Figure 3.**Mapping the lubrication regime. (

**a**) Moes − Deep groove ball bearing (DGBB) − Fr = 1 kN (

**b**) Moes − Angular contact ball bearing (ACBB) − Fa = 400 N. (

**c**) Regime evolution of each ball for DGBB. (

**d**) Regime evolution of each ball for ACBB.

**Figure 9.**(

**a**) Evolution of power losses for 4 different speeds with respect to time. (

**b**) Evolution of power losses for 4 different speeds with respect to outer ring (OR)–inner ring (IR) mean temperature.

**Figure 10.**Evolution of power losses for 2 different oil injection temperatures with respect to OR-IR mean temperature.

**Figure 11.**Evolution of power losses for 2 different oil flow rates with respect to OR-IR mean temperature.

**Figure 12.**Evolution of power losses for different axial preloads with respect to OR-IR mean temperature: (

**a**) 4800 rpm, (

**b**) 6500 rpm.

**Figure 13.**Evolution of power losses for different radial loads with respect to OR-IR mean temperature: (

**a**) 4800 rpm, (

**b**) 6500 rpm.

**Figure 14.**Comparison between model and experiments for the different parameters investigated. (

**a**) Influence of speed (

**b**) Influence of oil injection temperature. (

**c**) Influence of oil flow rate. (

**d**) Influence of axial preload.

**Figure 16.**Load distribution for 3 load cases. (

**a**) Radial load = 50 N. (

**b**) Radial load = 650 N. (

**c**) Radial load = 1500 N.

**Figure 17.**Regime lubrication of each ball for 3 load cases. (

**a**) Radial load = 50 N. (

**b**) Radial load = 650 N. (

**c**) Radial load = 1500 N.

**Figure 18.**Comparison between model and experiments for three different radial loads, speed = 4800 rpm.

**Figure 19.**Hydrodynamic rolling distribution, at 50 °C, for three different radial loads, speed = 4800 rpm.

**Figure 20.**Comparison between model and experiments for two different radial loads, speed = 6500 rpm.

Characteristics | DGBB | ACBB |
---|---|---|

Reference | 6210Z | 7210 |

Mean diameter (mm) | 70 | 70 |

Outer diameter (mm) | 90 | 90 |

Bore diameter (mm) | 50 | 50 |

Free contact angle (°) | 0 | 40 |

Width (mm) | 20 | 20 |

Number of balls (−) | 10 | 15 |

Characteristics | Value |
---|---|

Young modulus (GPa) | 210 |

Poisson coefficient (−) | 0.3 |

Characteristics | Value |
---|---|

Kinematic viscosity at 40 °C (cSt) | 36.6 |

Kinematic viscosity at 100 °C (cSt) | 7.8 |

Density at 15 °C (kg.m^{−3}) | 864.6 |

Pressure–viscosity coefficient (Pa^{−1}) | $20\times {10}^{-9}$ |

Test n° | Speed (rpm) | Oil Inj. T° (°C) | Oil Flow Rate (L/h) | Axial Preload (N) | Radial Load (N) |
---|---|---|---|---|---|

1 | Four points | 30 °C | 10 | 450 | 650 |

2 | 6500 | Two points | 10 | 450 | 650 |

3 | 8070 | 30 °C | Two points | 450 | 650 |

4 | 4800 | 30 °C | 10 | Three points | 0 |

5 | 6500 | 30 °C | 10 | Two points | 650 |

6 | 4800 | 30 °C | 10 | 450 | Three Points |

7 | 6500 | 30 °C | 10 | 450 | Two Points |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Darul, L.; Touret, T.; Changenet, C.; Ville, F.
Power Loss Analysis of an Oil-Jet Lubricated Angular Contact Ball Bearing: Theoretical and Experimental Investigations. *Lubricants* **2024**, *12*, 14.
https://doi.org/10.3390/lubricants12010014

**AMA Style**

Darul L, Touret T, Changenet C, Ville F.
Power Loss Analysis of an Oil-Jet Lubricated Angular Contact Ball Bearing: Theoretical and Experimental Investigations. *Lubricants*. 2024; 12(1):14.
https://doi.org/10.3390/lubricants12010014

**Chicago/Turabian Style**

Darul, Lionel, Thomas Touret, Christophe Changenet, and Fabrice Ville.
2024. "Power Loss Analysis of an Oil-Jet Lubricated Angular Contact Ball Bearing: Theoretical and Experimental Investigations" *Lubricants* 12, no. 1: 14.
https://doi.org/10.3390/lubricants12010014