# A Systematic Approach for Energy-Efficient Design of Rolling Bearing Cages

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## Abstract

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## Featured Application

**This study proposes a heat generation model for selecting an energy-efficient design for a rolling bearing cage.**

## Abstract

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Heat Generation Model

#### 2.2. An Energy Efficient Cage Design

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

${K}_{T}$ | Cooling factor |

$n$ | Cage rotation frequency |

${W}_{c}$ | Cage weight |

${f}_{b}$ | Cage with the ring friction coefficient |

${f}_{c}$ | Cage with the rolling element friction coefficient |

${f}_{r}$ | Rolling element with the ring friction coefficient |

${D}_{w}$ | Rolling element diameter |

${d}_{o}$ | Rolling element center diameter |

${r}_{g}$ | Bearing ring groove radius |

${d}_{out}$ | Outer ring diameter |

${d}_{in}$ | Inner ring diameter |

${d}_{bs}$ | Base ring edge diameter |

${h}_{b}$ | Ring thickness |

$k$ | Rolling friction coefficient |

${\alpha}_{i}$ | Bearing contact angle |

${F}_{b}\left(\phi \right)$ | Force of the rolling element with ring interaction |

${F}_{r\mathrm{in}}\left(\phi \right)$ | Radial force component from the inner ring |

${F}_{r\mathrm{out}}\left(\phi \right)$ | Radial force component from the outer ring |

${I}_{r}$ | Second moment of inertia |

$\epsilon $ | Cage eccentricity |

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**Figure 2.**The scheme of force interactions in zone A: 1, 2—outer ring and a rib; 3, 4—inner ring and a rib; 5—rolling body; 6—cage.

**Figure 4.**The temperature gradients diagrams for a typical design of a cylindrical roller bearing (region R, z = 14) and a modernized one (region E, z = 16): curve 1—${F}_{r}=50\mathrm{kN},V=200\mathrm{km}/\mathrm{h};\mathrm{curve}$2—${F}_{r}=30\mathrm{kN},V=100\mathrm{km}/\mathrm{h}$.

**Figure 5.**Graphs of generalized temperature functions T °C depending on the distance L of the passenger car of the Kharkiv-Odesa and Odesa-Kharkiv trains in forward and reverse directions, obtained during a month of research (curve 1—for the typical design of a cylindrical roller bearing; 2—for the modernized design; 3—environment (air); 4—the temperature gradient of the typical design and the environment; 5—the temperature gradient of the modernized design and the environment.

**Table 1.**The temperature gradient in the rolling bearings, caused by structural changes in the cage and corresponding to various operating conditions.

${\mathit{F}}_{\mathit{r}}\mathbf{kN}$ | ${\mathit{F}}_{\mathit{a}}\mathbf{kN}$ | ${\mathit{F}}_{\mathit{c}}{\left(\mathit{\phi}\right)}^{\ast}\mathbf{N}$ | ${{\mathit{F}}^{\prime}}_{\mathit{c}}{\left(\mathit{\phi}\right)}^{\ast}\mathbf{N}$ | ${\mathit{M}}_{\mathit{c}}{}^{\ast}\mathbf{N}\xb7\mathbf{mm}$ | $\mathbf{\Delta}{\mathit{T}}^{\ast}{}^{\mathbf{\circ}}\mathbf{C}$ $(\mathit{V}=27.8)$ | $\mathbf{\Delta}{\mathit{T}}^{\ast}{}^{\mathbf{\circ}}\mathbf{C}$ $(\mathit{V}=55.6)$ | $\mathbf{\Delta}{\mathit{T}}^{\ast}{}^{\mathbf{\circ}}\mathbf{C}$ $(\mathit{V}=83.4)$ |
---|---|---|---|---|---|---|---|

30 | 5 | 110/98 | 99/85 | 578/289 | 0.9/0.5 | 1.1/0.5 | 1.2/0.6 |

10 | 147/133 | 139/120 | 791/400 | 1.3/0.6 | 1.5/0.7 | 1.6/0.8 | |

15 | 182/167 | 175/156 | 987/510 | 1.6/0.8 | 1.8/0.9 | 1.9/1.0 | |

20 | 197/179 | 191/168 | 1073/548 | 1.7/0.9 | 2.0/1.0 | 2.2/1.1 | |

40 | 5 | 143/127 | 134/115 | 766/382 | 1.2/0.6 | 1.4/0.7 | 1.5/0.8 |

10 | 180/161 | 173/150 | 976/491 | 1.6/0.8 | 1.8/0.9 | 1.9/0.9 | |

15 | 215/196 | 210/186 | 1175/604 | 1.9/0.9 | 2.1/1.1 | 2.3/1.2 | |

20 | 230/210 | 226/200 | 1261/648 | 2.0/1.0 | 2.3/1.2 | 2.4/1.3 | |

50 | 5 | 177/155 | 170/143 | 959/471 | 1.5/0.8 | 1.8/0.9 | 1.9/0.9 |

10 | 213/191 | 208/181 | 1164/588 | 1.9/0.9 | 2.1/1.1 | 2.2/1.1 | |

15 | 248/224 | 245/214 | 1363/692 | 2.2/1.1 | 2.5/1.3 | 2.6/1.3 | |

20 | 263/239 | 261/230 | 1449/741 | 2.3/1.2 | 2.7/1.4 | 2.8/1.4 |

Typical Bearing | Modernized Bearing | $\mathit{\%}$ | |||
---|---|---|---|---|---|

$\mathbf{Mean}\mathbf{\Delta}{\mathit{T}}^{\ast}{}^{\mathbf{\circ}}\mathbf{C}$ | Standard Deviation | $\mathbf{Mean}\mathbf{\Delta}{\mathit{T}}^{\ast}{}^{\mathbf{\circ}}\mathbf{C}$ | Standard Deviation | ||

Poltava | 2.96 | 0.43 | 0.92 | 0.12 | 68.9 |

Kremenchuk | 4.12 | 0.45 | 1.97 | 0.16 | 52.2 |

Znamyanka | 4.94 | 0.89 | 3.08 | 0.36 | 37.6 |

Voznesensk | 5.92 | 0.69 | 3.02 | 0.5 | 48.9 |

Odesa | 4.98 | 0.45 | 1.81 | 0.14 | 63.6 |

Voznesensk | 2.12 | 0.29 | 0.81 | 0.11 | 61.8 |

Znamyanka | 3.57 | 0.6 | 2.8 | 0.52 | 21.5 |

Kremenchuk | 5.07 | 0.93 | 2.78 | 0.4 | 45.1 |

Poltava | 6.39 | 0.56 | 3.43 | 0.57 | 46.3 |

Kharkiv | 8.87 | 1.5 | 6.32 | 1.08 | 28.7 |

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**MDPI and ACS Style**

Gaydamaka, A.; Klitnoi, V.; Dobrotvorskiy, S.; Basova, Y.; Matos, D.; Machado, J.
A Systematic Approach for Energy-Efficient Design of Rolling Bearing Cages. *Appl. Sci.* **2023**, *13*, 1144.
https://doi.org/10.3390/app13021144

**AMA Style**

Gaydamaka A, Klitnoi V, Dobrotvorskiy S, Basova Y, Matos D, Machado J.
A Systematic Approach for Energy-Efficient Design of Rolling Bearing Cages. *Applied Sciences*. 2023; 13(2):1144.
https://doi.org/10.3390/app13021144

**Chicago/Turabian Style**

Gaydamaka, Anatoliy, Volodymyr Klitnoi, Sergey Dobrotvorskiy, Yevheniia Basova, Demétrio Matos, and José Machado.
2023. "A Systematic Approach for Energy-Efficient Design of Rolling Bearing Cages" *Applied Sciences* 13, no. 2: 1144.
https://doi.org/10.3390/app13021144