# Predicting Friction of Tapered Roller Bearings with Detailed Multi-Body Simulation Models

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Multi Body Simulation Model

#### 2.1.1. Contact Calculation

#### Slice Model

#### Cell Model

#### 2.1.2. Damping

#### 2.1.3. Friction

#### Lubricant Friction (Sliding)

#### Lubricant Friction (Rolling)

#### Solid Rolling Friction

#### Material Hysteresis

#### Solid Sliding Friction

#### Mixed Friction

#### Friction in Roller Rib Contact

#### Solid Sliding Friction in Roller Rib Contact

#### Lubricant Friction in Roller Rib Contact

#### Mixed Friction

#### Friction in Roller Cage Contact

#### 2.2. Friction Torque Measurement

## 3. Results

## 4. Discussion

^{®}environment. The highly specialized CFD simulation obtained agrees with the experiment with only insignificant deviations (see Figure 9 on the right side). In addition, the aforementioned simulations with OpenFOAM

^{®}have been validated with respect to lubricant flows in a 32312-A TRB [85,86,87]. Combining the calculated losses from both LaMBDA and CFD simulations, the total bearing losses can be predicted very well (dark blue curve, Figure 9, left). These results show good agreement with the measured total bearing losses (orange). The comparison of the listed calculation methods shows that the state-of-the-art approach predicts the hydraulic losses in a bearing only to a certain extent. With highly specialized CFD simulations, it is possible to determine these losses very precisely [88].

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Bearing data | |

${a}_{\mathrm{p}}$ | Profile parameter |

${c}_{\mathrm{p}}$ | Profile parameter |

${d}_{\mathrm{i}}$ | Inner diameter |

${d}_{\mathrm{p}}$ | Profile parameter |

${d}_{\mathrm{m}}$ | mean rolling bearing diameter |

${d}_{\mathrm{IRL}}$ | Inner ring raceway diameter |

${d}_{\mathrm{ORL}}$ | Outer ring raceway diameter |

${d}_{\mathrm{Pd}}$ | Pitch diameter |

${d}_{\mathrm{RB}}$ | Roller diameter |

${k}_{\mathrm{p}}$ | Profile parameter |

${l}_{\mathrm{RB}}$ | Roller length |

${n}_{\mathrm{RB}}$ | Number of roller |

${r}_{\mathrm{e}}$ | Edge radius |

${A}_{\mathrm{IRL}}$ | effective surface of the inner ring |

${A}_{\mathrm{ORL}}$ | effective surface of the outer ring |

${B}_{\mathrm{ZH}}$ | Surface roughness parameter according to Zhou and Hoepprich |

${C}_{0\mathrm{r}}$ | Basic static load rating, radial |

${C}_{\mathrm{ZH}}$ | Surface roughness parameter according to Zhou and Hoepprich |

${D}_{\mathrm{a}}$ | Outer diameter |

$\sigma $ | combined standard derivation of surface roughness |

Lubricant parameters | |

${a}_{1}$ | Lubricant dependent parameter according to Dicke |

${a}_{2}$ | Lubricant dependent parameter according to Dicke |

${b}_{1}$ | Lubricant dependent parameter according to Dicke |

${b}_{2}$ | Lubricant dependent parameter according to Dicke |

${h}_{\mathrm{th}}$ | thermal corrected lubricant film height |

${h}_{0}$ | lubricant film height |

${A}_{\mathrm{V}}$ | Lubricant dependent parameter according to Dicke |

$B$ | Lubricant dependent parameter according to Vogel |

${B}_{\mathrm{V}}$ | Lubricant dependent parameter according to Dicke |

$C$ | Lubricant dependent parameter according to Vogel |

${C}_{\mathrm{V}}$ | Lubricant dependent parameter according to Dicke |

${C}_{1}$ | Lubricant dependent parameter according to Gold et al. |

${C}_{2}$ | Lubricant dependent parameter according to Gold et al. |

$K$ | Lubricant dependent parameter according to Vogel |

$\alpha $ | Temperature density coefficient |

${\alpha}_{\mathrm{p}}$ | Pressure-viscosity coefficient |

$\eta $ | Dynamic viscosity of a lubricant |

${\eta}_{0}$ | Lubricant viscosity at ambient pressure |

${\lambda}_{\vartheta}$ | thermal conductivity |

$v$ | kinematic viscosity |

$\rho $ | Lubricant density |

${\tau}_{\mathrm{L}}$ | Limiting shear stress according to Bair and Winer |

${\varphi}_{\vartheta}$ | thermal correction factors |

State variables/states | |

${F}_{\mathrm{a}}$ | Axial load |

${F}_{\mathrm{r}}$ | Radial load |

${n}_{\mathrm{C}}$ | cage speed |

${n}_{\mathrm{IR}}$ | Inner ring speed |

$N$ | Shaft speed |

$\overrightarrow{s}$ | Displacement between two coordinate systems |

$\dot{\overrightarrow{s}}$ | Relative velocity between two coordinate systems |

$\ddot{\overrightarrow{s}}$ | Acceleration between two coordinate systems |

$T$ | Absolute temperature |

${T}_{0}$ | Ambient temperature (20 °C) |

$\vartheta $ | Temperature in °C |

$\overrightarrow{\omega}$ | Angle between two coordinate systems |

$\dot{\overrightarrow{\omega}}$ | Angular velocity between two coordinate systems |

$\ddot{\overrightarrow{\omega}}$ | Angular acceleration between two coordinate systems |

Model input parameters | |

${a}_{\mathrm{cubic}}$ | Parameter defining the coefficient of friction |

${a}_{\mathrm{v}}$ | hysteresis loss factor |

${c}_{\mathrm{r}}$ | rolling friction coefficient |

${e}_{\mathrm{r}}$ | rolling resistance exponent |

$L$ | effective contacting length |

${v}_{\mathrm{s}}$ | Limit of relative velocity for static coefficient of friction |

${v}_{\mathrm{d}}$ | Limit of relative velocity for dynamic coefficient of friction |

${\mu}_{\mathrm{s}}$ | Static coefficient of friction |

${\mu}_{\mathrm{d}}$ | Dynamic coefficient of friction |

Contact state variables | |

$a$ | Axis of the contact ellipse |

$b$ | Hertzian contact width/ axis of the contact ellipse |

$d$ | damping coefficient |

${d}_{\mathrm{max}}$ | maximum damping coefficient |

${f}_{\mathrm{d}}$ | Function describing the coefficient of damping depending on penetration depth |

$h$ | lubricant film height |

${h}_{\mathrm{cubic}}$ | Parameter defining the coefficient of friction |

$\overrightarrow{n}$ | Contact normal vector |

$p$ | Contact pressure |

$\overrightarrow{p}$ | Contact point vector |

${p}_{0}$ | Relative pressure |

${s}_{\mathrm{ZC}}$ | Slippage |

${u}_{\mathrm{av}}$ | Average conveying velocity of the lubricant |

${\overrightarrow{u}}_{\mathrm{rel}}$ | Relative velocity vector in contact point |

${u}_{\mathrm{rel}}$ | Magnitude of relative velocity in contact point |

${\overrightarrow{u}}_{\mathrm{sum}}$ | Sum velocity vector in contact point |

${u}_{\mathrm{sum}}$ | Magnitude of sum velocity in contact point |

${\overrightarrow{v}}_{\mathrm{N}}$ | Velocity vector in contact normal direction |

${v}_{\mathrm{sl}}$ | Effective sliding velocity |

${A}_{\mathrm{Hertz}}$ | Hertzian contact area |

${E}^{\prime}$ | Reduced Young’s modulus of both contacting bodies |

${F}_{\mathrm{N}}$ | Magnitude of contact normal force |

${\overrightarrow{F}}_{\mathrm{N}}$ | Contact normal force |

${\overrightarrow{F}}_{\mathrm{D}}$ | Damping force |

${\overrightarrow{F}}_{\mathrm{T},\mathrm{L},\mathrm{sl}}$ | Force resulting from sliding friction in lubricant |

${\overrightarrow{F}}_{\mathrm{T},\mathrm{S},\mathrm{sl}}$ | Force resulting from sliding friction in solid contact |

${\overrightarrow{F}}_{\mathrm{T}}$ | Traction force |

${\overrightarrow{F}}_{\mathsf{\Sigma}}$ | Contact force |

$G$ | Material parameter |

${M}_{\mathrm{churning},\mathrm{IR}}$ | Churning losses inner ring |

${M}_{\mathrm{churning},\mathrm{OR}}$ | Churning losses outer ring |

${M}_{\mathrm{drag}}$ | Drag losses |

${\overrightarrow{M}}_{\mathrm{D}}$ | Torque from damping force |

${\overrightarrow{M}}_{\mathrm{N}}$ | Torque from contact normal force |

${\overrightarrow{M}}_{\mathrm{T}}$ | Torque from traction force |

${\overrightarrow{M}}_{\mathrm{T},\mathrm{Hys}}$ | Torque resulting from material hysteresis |

${\overrightarrow{M}}_{\mathrm{T},\mathrm{L},\mathrm{r}}$ | Torque resulting from rolling friction in lubricant |

${\overrightarrow{M}}_{\mathrm{T},\mathrm{L},\mathrm{sl}}$ | Torque resulting from sliding friction in lubricant |

${\overrightarrow{M}}_{\mathrm{T},\mathrm{S},\mathrm{r}}$ | Torque resulting from rolling friction in solid contact |

${\overrightarrow{M}}_{\mathrm{T},\mathrm{S},\mathrm{sl}}$ | Torque resulting from sliding friction in solid contact |

${\overrightarrow{M}}_{\mathsf{\Sigma}}$ | Torque from contact force |

$Q$ | load imposed on one slice/cell of the rolling element |

${Q}_{\mathrm{S}}$ | proportion of the normal force transmitted at solid contacts |

${R}^{\prime}$ | Reduced radii of the contacting bodies |

${R}_{\mathrm{x}}$ | Reduced radius in x direction |

${R}_{\mathrm{y}}$ | Reduced radius in y direction |

$U$ | Velocity parameter |

$W$ | Load parameter |

$\dot{\gamma}$ | Shear gradient |

$\delta $ | Penetration depth |

${\delta}_{\mathrm{cubic}}$ | Parameter defining the coefficient of friction |

${\delta}_{\mathrm{max}}$ | Penetration above which maximum damping coefficient is reached |

$\mu $ | Coefficient of friction |

${\tau}_{\mathrm{EHL}}$ | Shear stresses of the lubricant |

$\varphi $ | Solid load-bearing ratio |

${\Gamma}_{\mathrm{ZC}}$ | Thermal load parameter |

$\Delta t$ | Duration of a time step |

$\Lambda $ | Lubricant film thickness parameter |

## Appendix A

Parameter | Variable | Value | Unit |
---|---|---|---|

Temperature parameter ^{1} | K | 0.062 | mPa s |

Temperature parameter ^{1} | B | 1021.7 | °C |

Temperature parameter ^{1} | C | 101.5517 | °C |

Pressure parameter ^{1} | a_{1} | 327.7918 | bar |

Pressure parameter ^{1} | a_{2} | 2.9862 | bar/°C |

Pressure parameter ^{1} | b_{1} | 4.419·10^{−3} | - |

Pressure parameter ^{1} | b_{2} | 3.0115·10^{−4} | 1/°C |

Density at 15 °C | ρ | 887.6 | kg/m³ |

Thermal conductivity | λ | 0.134 | W/(m K) |

Temperature density coefficient | α | −6·10^{−4} | g/(ml K) |

^{1}The lubricant data use in the MBS model in this work have been measured by [89].

## Appendix B

Parameter | Variable | Value | Unit |
---|---|---|---|

Basic static load rating, radial | C_{0r} | 94,000 | N |

Inner diameter | d_{i} | 40 | mm |

Outer diameter | D_{a} | 80 | mm |

Pitch diameter | d_{Pd} | 60 | mm |

Roller diameter | d_{RB} | 10 | mm |

Roller length | l_{RB} | 17 | mm |

Number of rollers | n_{RB} | 17 | - |

Profile parameter | a_{p} | 0.0005 | - |

Profile parameter | c_{p} | 16.2 | mm |

Profile parameter | d_{p} | 0.0 | mm |

Profile parameter | k_{p} | 1.0 | mm |

Edge radius | r_{e} | 0.7 | mm |

Combined standard derivation of roughness | σ_{Raceway} | 0.1 | μm |

σ_{Rib} | 0.1 | μm | |

Mixed friction parameters for raceway contact according to Zhou and Hoeprich [61,73] | B_{ZH} | 2.1 | |

C_{ZH} | 0.85 | ||

Mixed friction parameters for rib contact according to Zhou and Hoeprich [61,73] | B_{ZH} | 2.1 | |

C_{ZH} | 0.85 |

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**Figure 1.**Flow chart of contact force calculation between rolling elements and ring raceways [26].

**Figure 2.**Friction calculation in roller raceway contact of the TRB MBS model [26].

**Figure 3.**Scematic view and CAD model of the friction torque test bench at MEGT [42].

**Figure 4.**Graphical illustration of the geometry parameters of the TRB used in this study. On the left the sectional view of the TRB and on the right a rolling element is shown. The dotted lines represent the symmetry lines of the bearing and the rolling element.

**Figure 5.**Comparison of the measured and LaMBDA-calculated frictional torque of a TRB of type 32216 under purely axial load of 6 kN and oil bath lubrication with reference oil FVA No. 3.

**Figure 6.**Comparison of the measured and LaMBDA-calculated frictional torque of a TRB of type 32216 under combined axial load of 6.5 kN and radial load of 6 kN with oil bath lubrication with reference oil FVA No. 3.

**Figure 7.**Comparison of the measured and LaMBDA-calculated frictional torque of a TRB of type 32216 under combined load and oil bath lubrication with reference oil FVA No. 3 under variation of the radial load at 50 °C.

**Figure 8.**Load distribution of the TRB type 32216 used. Left: with purely axial load of 6 kN, right: with combined load of 6 kN axial and 16 kN radial.

**Figure 9.**Comparison of the simulated frictional torque of a TRB of type 32208 under oil sump lubrication with reference oil FVA No. 3 at 50 °C and an axial load of 1 kN with the measured frictional torque on a single-bearing test rig. Simulated frictional torque with LaMBDA and experiment (left), experiment and CFD simulation [84] (right). Bearing data is provided in Table A2 in Appendix B) and lubricant data in Table A1 in Appendix A).

**Table 1.**Test setup for friction torque measurement of a TRB type 32216 at pure constant axial load.

Parameter | Variable | Value | Unit |
---|---|---|---|

Axial load | F_{a} | 6 | kN |

Radial load | F_{r} | 0 | kN |

Temperature | $\vartheta $ | 42 and 50 | °C |

Shaft speed | N | 500–4000 | rpm |

Lubrication | Oil bath half roller height | ||

Lubricant | Reference oil FVA3 |

**Table 2.**Test setup for friction torque measurement of a TRB type 32216 at combined, constant axial and radial load.

Parameter | Variable | Value | Unit |
---|---|---|---|

Axial load | F_{a} | 6 | kN |

Radial load | F_{r} | 6.5 | kN |

Temperature | $\vartheta $ | 42 and 50 | °C |

Shaft speed | N | 500–4000 | rpm |

Lubrication | Oil bath half roller height | ||

Lubricant | Reference oil FVA3 |

Parameter | Variable | Value | Unit |
---|---|---|---|

Axial load | F_{a} | 6.5 | kN |

Radial load | F_{r} | 1–15 | kN |

Temperature | $\vartheta $ | 50 | °C |

Shaft speed | N | 2000 | rpm |

Lubrication | Oil bath half roller height | ||

Lubricant | Reference oil FVA3 |

Parameter | Variable | Value | Unit |
---|---|---|---|

Basic static load rating, radial | C_{0r} | 260,000 | N |

Inner diameter | d_{i} | 80 | mm |

Outer diameter | D_{a} | 140 | mm |

Pitch diameter | d_{Pd} | 108.5 | mm |

Roller diameter | d_{RB} | 17 | mm |

Roller length | l_{RB} | 22.7 | mm |

Number of roller | n_{RB} | 16 | - |

Profile parameter | a_{p} | 0.0005 | - |

Profile parameter | c_{p} | 20.7 | mm |

Profile parameter | d_{p} | 0.0 | mm |

Profile parameter | k_{p} | 2.0 | mm |

Edge radius | r_{e} | 1.0 | mm |

Combined standard derivation of roughness | σ_{Raceway} | 0.16 | μm |

σ_{Rib} | 0.24 | μm | |

Mixed friction parameters for raceway contact according to Zhou and Hoeprich [61,73] | B_{ZH} | 2.32 | |

C_{ZH} | 0.97 | ||

Mixed friction parameters for rib contact according to Zhou and Hoeprich [61,73] | B_{ZH} | 1.90 | |

C_{ZH} | 0.99 |

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## Share and Cite

**MDPI and ACS Style**

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.
https://doi.org/10.3390/lubricants11090369

**AMA Style**

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(9):369.
https://doi.org/10.3390/lubricants11090369

**Chicago/Turabian Style**

Wingertszahn, Patrick, Oliver Koch, Lorenzo Maccioni, Franco Concli, and Bernd Sauer.
2023. "Predicting Friction of Tapered Roller Bearings with Detailed Multi-Body Simulation Models" *Lubricants* 11, no. 9: 369.
https://doi.org/10.3390/lubricants11090369