# Three-Dimensional DEM Modelling of Ball Bearing with Lubrication Regime Prediction

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Three Dimensional Modelling of Radial Ball Bearing Using DEM

#### 2.1. Contact Stiffness and Damping Effect

#### 2.2. Fluid Film Thickness and Fluid Parameters

#### 2.3. Space and Time Discretization of the DEM Ball Bearing Model

## 3. Numerical Prediction of Lubrication Regimes in Operating Radial Ball Bearing

#### 3.1. Effect of Radial Load and Diametral Clearance

#### 3.2. Effect of Angular Speed

## 4. Conclusions and Perspectives

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Nomenclature

α_{e} | Restitution coefficient of steel |

δ_{r} | Radial shift |

δ_{n,t} | Normal/tangential relative displacement |

η | Dynamic viscosity of the fluid |

Λ_{f} | Fluid number |

μ | Friction coefficient |

v | Poisson’s ratio of steel |

w | Shaft/inner-ring angular speed |

$\overline{\mathcal{E}}$ | Approximate elliptic integral of second kind |

$\overline{\mathcal{F}}$ | Approximate elliptic integral of first kind |

ψ | Azimuth angle |

δ_{a}_{1,2} | Roughness parameter |

ε | Load parameter |

ξ | Viscosity-pressure coefficient of the lubricant |

a | Restitution coefficient of steel |

C_{n,t} | Normal/tangential viscous damping coefficient |

E | Young’s modulus of steel |

E′ | Effective elastic modulus of steel |

F_{r} | Applied radial load |

G | Shear modulus of steel |

h_{min} | Semi-major axis of the Hertz’s elliptical contact |

J_{r} | Radial integral |

k | Ellipticity parameter |

K_{n,t} | Contact normal/tangential stiffness |

P_{d} | Diametral clearance |

Q_{max} | Maximum normal load at ψ = 0 |

R_{b} | Rolling element radius |

R_{c} | Curvature radius of inner/outer raceway |

R_{i} | Inner ring radius |

R_{o} | Outer ring radius |

R_{z} | Cage element radius |

R_{curve} | Curvature sum radius |

U_{r} | Rolling velocity |

v_{t} | Sliding velocity |

Z | Number of rolling elements |

## Appendix A. Normal Load and Radial Deflection in Radial Ball Bearing

**Figure A1.**(

**a**) Radial deflection ${\delta}_{\psi}$ and (

**b**) normal load ${Q}_{\psi}$ distributions under radial load ${F}_{r}=3000\phantom{\rule{3.33333pt}{0ex}}$N.

**Figure A2.**(

**a**) Radial deflection ${\delta}_{\psi}$ and (

**b**) normal load ${Q}_{\psi}$ distributions under several radial load ${F}_{r}$.

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**Figure 1.**(

**a**) Radial ball bearing of 6208 series; (

**b**) DEM modelling using MULTICOR-3D software; (

**c**) Exploded view.

**Figure 3.**Curvatures in contact in two orthogonal cross sections of a ball bearing: (

**a**) x–z plane ; (

**b**) y–z plane.

**Figure 5.**Schematic representation of Stribeck curve: friction coefficient and fluid film thickness as function of fluid parameter.

**Figure 7.**(

**a**) Imposed shaft angular speed $\omega $ and simulated angular speed ${\omega}_{ball}$ of rolling element and (

**b**) SRR for a single rolling element.

**Figure 8.**(

**a**) Film fluid thickness ${h}_{min}^{out}$ and (

**b**) fluid parameter ${\mathrm{\Lambda}}_{f}^{out}$ with respect to azimuth angle $\psi $ of the ball-outer-raceway contact.

**Figure 9.**(

**a**) Fluid parameter ${\mathrm{\Lambda}}_{f}^{out}$ as function of azimuth angle $\psi $; (

**b**) Fluid parameter ${\mathrm{\Lambda}}_{f}^{out}$ and maximum normal load ${Q}_{max}$ in the loaded zone at $\psi =0$.

**Figure 10.**(

**a**) Fluid parameter ${\mathrm{\Lambda}}_{f}^{out}$ as a function of azimuth angle $\psi $; (

**b**) Fluid parameter ${\mathrm{\Lambda}}_{f}^{out}$ and load parameter $\epsilon $ as function of normal load ${Q}_{max}$ at $\psi =0$.

**Figure 11.**Fluid parameter as function of azimuth angle $\psi $: (

**a**) ${\mathrm{\Lambda}}_{f}^{out}$ for ball-outer-raceway contact; (

**b**) ${\mathrm{\Lambda}}_{f}^{inn}$ for ball-inner-raceway contact.

**Figure 12.**(

**a**) Fluid parameter and maximum contact pressure at $\psi =0$ as function of angular speed $\omega $; (

**b**) Minimum fluid film thickness and maximum normal load at $\psi =0$ as function of angular speed $\omega $.

Component | Ball | Inner Ring | Outer Ring | Raceway | Cage |
---|---|---|---|---|---|

Radius | ${R}_{b}$ | ${R}_{i}$ | ${R}_{o}$ | ${R}_{c}$ | ${R}_{z}$ |

Dimension (mm) | 6.3 | 24.0 | 36.6 | 6.552 | 4.19 |

${\mathit{C}}_{\mathit{n}}\left({\mathbf{\Lambda}}_{\mathit{f}}\right)$ | ${\mathit{C}}_{\mathit{hyst}}$ | ${\left(\frac{1}{{\mathit{C}}_{\mathit{hyst}}}+\frac{1}{{\mathit{C}}_{\mathit{fluid}}}\right)}^{-1}$ | ${\mathit{C}}_{\mathit{fluid}}$ |
---|---|---|---|

${h}_{min}$ | $<{\sigma}_{a}$ | $\sim {\sigma}_{a}$ | $>>{\sigma}_{a}$ |

${\mathrm{\Lambda}}_{f}$ | $\le 1$ | ∈]1,3] | ∈]3,5] |

$\mu \left({\mathrm{\Lambda}}_{f}\right)$ | high | moderate | low |

**Table 3.**Lubrication regime in the loaded zone at $\psi =0$ of radial ball bearing with zero clearance $\left({P}_{d}=0,\phantom{\rule{3.33333pt}{0ex}}\epsilon =0.5\right)$.

Angular Speed $\left(\mathbf{rad}\xb7{\mathbf{s}}^{-1}\right)$ | Lubrication Regime | |
---|---|---|

Inner Contact | Outer Contact | |

$\omega \le 100$ | ${\mathrm{\Lambda}}_{f}<1$ | |

$100<\omega \le 1100$ | $1<{\mathrm{\Lambda}}_{f}<3$ | $1<{\mathrm{\Lambda}}_{f}<4$ |

$1100<\omega \le 1700$ | $3<{\mathrm{\Lambda}}_{f}\le 4$ | $4<{\mathrm{\Lambda}}_{f}<5$ |

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**MDPI and ACS Style**

Guessasma, M.; Machado, C.
Three-Dimensional DEM Modelling of Ball Bearing with Lubrication Regime Prediction. *Lubricants* **2018**, *6*, 46.
https://doi.org/10.3390/lubricants6020046

**AMA Style**

Guessasma M, Machado C.
Three-Dimensional DEM Modelling of Ball Bearing with Lubrication Regime Prediction. *Lubricants*. 2018; 6(2):46.
https://doi.org/10.3390/lubricants6020046

**Chicago/Turabian Style**

Guessasma, Mohamed, and Charles Machado.
2018. "Three-Dimensional DEM Modelling of Ball Bearing with Lubrication Regime Prediction" *Lubricants* 6, no. 2: 46.
https://doi.org/10.3390/lubricants6020046