# Study on Cage Wear of Railway Traction Motor Bearings Based on Analysis of Rolling Element Motion

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Target Bearing

#### 2.2. Method for Measuring Contact Force between Roller and Cage

#### 2.3. Method for Dynamic Analysis for Determining Impulse

- I.
- Only the translational velocity of the roller changes when the roller and the cage make contact (the angular velocity of the cage ${\omega}_{c}$ is constant at a geometrically determined value.).
- II.
- In the no-load zone, the roller contacts the outer ring raceway due to centrifugal force and does not contact the inner ring raceway.
- III.
- The rotational speed of rollers is assumed to be constant at a theoretical value (The angular velocity of the roller ${\omega}_{r}$ is constant at a geometrically determined value.).

- (1)
- The roller within the no-load zone is located in the front of the cage pocket. This is because the roller accelerates the cage within the loading zone.
- (2)
- The orbital velocity of the roller is decelerated by ${F}_{o}$ and the tangential force of gravity. This causes the roller to move to the rear of the cage pocket. ${F}_{o}$ is generated by the oil film between the roller and the outer ring.
- (3)
- The roller contacts the rear of the cage pocket. ${F}_{dec}$ is generated by the contact.
- (4)
- The roller bounces and moves to the front of the cage pocket.

- (5)
- When the relative velocity between the roller and the cage is zero, ${F}_{o}$ and the tangential forces of gravity act on the cage through the roller.

## 3. Results and Discussion

#### 3.1. Comparison of Measurement and Dynamic Analysis

- (a)
- Force to accelerate the cage ${F}_{acc}$

- (b)
- Force to decelerate the cage ${F}_{dec}$

#### 3.2. Determining Wear Modes

## 4. Conclusions

- The volume of cage wear is considered to be proportional to the impulse caused by contact, so a method was proposed to calculate this impulse. In this method, a model consisting only of a roller and a cage was constructed, and the movement of the roller relative to the cage was calculated. Using this method, the impulse caused by the contact was determined and compared with the measured results. The calculated results of the impulse were in close agreement with the measured values. Based on this, the analysis model in this paper is reasonable.
- The experiment was conducted to determine the wear mode of the cage. The results showed that the volume of cage wear is equal when the sum of the impulse is the same, regardless of the magnitude and frequency of the contact forces. In other words, the assumption that cage wear is proportional to the impulse is valid.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${r}_{r}$ | radius of roller (m) |

${r}_{p}$ | pitch circle radius of roller (m) |

${r}_{o}$ | inner radius of outer ring (m) |

$m$ | mass of roller (kg) |

${F}_{c}$ | contact force between roller and cage (N) |

${F}_{acc}$ | ${F}_{c}$ to accelerate cage (N) |

${F}_{dec}$ | ${F}_{c}$ to decelerate cage (N) |

${n}_{i}$ | rotational speed of inner ring (rpm) |

${V}_{w}$ | volume of cage wear (m^{3}) |

$k$ | coefficient of wear |

$W$ | contact force of two objects (N) |

${L}_{s}$ | sliding distance (m) |

$H$ | hardness of softer material in two contact materials |

$K$ | coefficient of impact wear |

${e}_{k}$ | kinetic energy at impact (J) |

$n$ | coefficient of impact wear |

$t$ | time (s) |

${I}_{c}$ | impulse caused by ${F}_{c}$ (=$\int {F}_{c}dt$) (N·s) |

${I}_{acc}$ | impulse caused by ${F}_{acc}$ (=$\int {F}_{acc}dt$) (N·s) |

${I}_{dec}$ | impulse caused by ${F}_{dec}$ (=$\int {F}_{dec}dt$) (N·s) |

${l}_{r}$ | position of roller in cage pocket (m) |

${F}_{o}$ | pressure force exerted by oil film between roller and outer ring (N) |

$N$ | normal force between roller and outer ring (N) |

$V$ | rolling velocity (m/s) |

$\gamma $ | ${r}_{r}/{r}_{p}$ |

${\omega}_{i}$ | angular velocity of inner ring (rad/s) |

${\omega}_{r}$ | angular velocity of roller (=$\left(1-\gamma \right){\omega}_{i}/2$)) (rad/s) |

${\omega}_{c}$ | angular velocity of cage (=$\left(1/\gamma -\gamma \right){\omega}_{i}/2$) (rad/s) |

$c$ | constant in Equation (2) |

$v$ | relative velocity of roller and cage before contact (m/s) |

${v}^{\prime}$ | relative velocity of roller and cage after contact (m/s) |

$e$ | coefficient of restitution between roller and cage (=$\left|{v}^{\prime}/v\right|$) |

$\varphi $ | half angle of load zone (°) |

$\theta $ | angle of rotation (°) |

$g$ | acceleration of gravity (m/s^{2}) |

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**Figure 6.**Impulse by contact force between roller and cage. (

**a**) by force to accelerate the cage (

**b**) by force to decelerate the cage (2000 rpm, Forward, Data for 100 cage rotations).

**Figure 7.**Impulse by contact force between roller and cage at each rotational speed [21]. (

**a**) By force to accelerate the cage; (

**b**) by force to decelerate the cage (average and standard deviation for 100 cage rotations).

**Figure 10.**Impulse by contact force between roller and cage at each rotational speed. (

**a**) Calculation results; (

**b**) measurement results.

Bearing Type | Cylindrical Roller Bearing | |
---|---|---|

Bearing number | NU214 | |

Inner diameter | 70 mm | |

Outer diameter | 125 mm | |

Width | 24 mm | |

Radial clearance | 0.090–0.125 mm | |

Pitch circle diameter of roller $2{r}_{p}$ | 97.5 mm | |

Number of rollers | 16 | |

Roller diameter $2{r}_{r}$ | 13 mm | |

Length of roller | 13 mm | |

Mass of roller $m$ | 13 g | |

Cage guide | By rollers | |

Material | Race rings Rollers | Bearing steel |

Cage | High-strength brass | |

Basic dynamic load rating | 83,500 N |

Rotational speed of inner ring ${n}_{i}$ | 500, 1000, 2000, 3000, 4000 rpm |

Direction of rotation | Forward, reverse |

Radial load | 970 N |

Lubricant | Lithium complex soap grease |

Rotational speed of inner ring ${n}_{i}$ | 6000 rpm |

Rotational direction of inner ring | Forward |

Radial load | 922 N |

Lubricant | Lithium complex soap grease 0.1 g |

Data for This Paper | Data for Reference [27] | |
---|---|---|

Mass (Kg) | 0.013 | 0.00055 |

Impact velocity (m/s) | 0.112 | 2.16 |

Kinetic energy $\mathrm{of}\mathrm{impact}{e}_{k}$ (J) | 8.15 × 10^{−5} | 1.28 × 10^{−3} |

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## Share and Cite

**MDPI and ACS Style**

Suzuki, D.; Takahashi, K.; Itoigawa, F.; Maegawa, S.
Study on Cage Wear of Railway Traction Motor Bearings Based on Analysis of Rolling Element Motion. *Machines* **2023**, *11*, 594.
https://doi.org/10.3390/machines11060594

**AMA Style**

Suzuki D, Takahashi K, Itoigawa F, Maegawa S.
Study on Cage Wear of Railway Traction Motor Bearings Based on Analysis of Rolling Element Motion. *Machines*. 2023; 11(6):594.
https://doi.org/10.3390/machines11060594

**Chicago/Turabian Style**

Suzuki, Daisuke, Ken Takahashi, Fumihiro Itoigawa, and Satoru Maegawa.
2023. "Study on Cage Wear of Railway Traction Motor Bearings Based on Analysis of Rolling Element Motion" *Machines* 11, no. 6: 594.
https://doi.org/10.3390/machines11060594