# An Analysis of the Influence of Surface Roughness and Clearance on the Dynamic Behavior of Deep Groove Ball Bearings Using Artificial Neural Networks

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{r}), raceway radius ratio (R

_{i}/R

_{e}), surface roughness of the raceway of the outer ring (R

_{ae}), raceway waviness of the outer ring (W

_{te max}), deviation from the circularity of the raceway of the outer ring (W

_{te}), surface roughness of the raceway of the inner ring (R

_{ai}), raceway waviness of the inner ring (W

_{ti max}), and deviation from the circularity of the raceway of the inner ring (W

_{ti}).

_{a ekv}):

_{t max ekv}):

_{t ekv}):

#### 2.1. Analysis of Measured Data

#### 2.2. Application of the Neural Network

#### 2.3. Organization of Data Set

#### 2.4. Data Pre-Processing and Defining Datasets for Training, Validation and Testing

^{2}) and the average prediction error expressed as a percentage. The evaluation was calculated for each band separately. The overall evaluation was calculated as the cumulative value of the Pearson’s correlation coefficient and the separate coefficients of determination, and the prediction fault was calculated as the mean value for all three bands. If the model is absolutely correct, then the values of the coefficients will be 1 in each band, that is, their cumulative value will be 3, and the mean value of the prediction fault will be zero. The model whose cumulative values of the coefficients are closest to the maximum, and at the same time, has a minimum mean value of the prediction fault, is considered the best. When selecting the model to be used for the analysis of the dynamic behavior of the bearing, the priority in the evaluation is assigned to the networks that have the smallest average prediction fault, and as a secondary evaluation, the cumulative values of the coefficients are used.

#### 2.5. Analyzed Models of Artificial Neural Networks

#### 2.6. Method of Training Artificial Neural Networks

#### 2.7. Description of the Training Algorithm

#### 2.8. Selection of Artificial Neural Network Models

## 3. Results and Discussion

#### 3.1. Prediction of Quality Classes of Bearing

#### 3.2. Influence of the Surface Roughness of the Outer Ring

#### 3.3. Influence of the Surface Roughness of the Inner Ring

#### 3.4. The Influence of Equivalent Surface Roughness and Radial Clearance

## 4. Conclusions

- The adopted models are capable of predicting the quality class for new ball bearings and can reduce time required for quality control in bearing production;
- The increase in roughness on the outer raceway causes a significant increase in the vibration level in the medium-frequency band (300–1800 Hz) and a moderate increase in the low-frequency band (50–300 Hz), whereas the change in vibration level in the high-frequency band is negligibly small;
- An increase in surface roughness on the raceway of the inner ring has a negligible effect on the amplitude of the vibration velocity in the low-frequency band, and causes a moderate increase in the medium and high band. The growth in the newly introduced parameter of the equivalent roughness of the raceway affects the moderate growth in the amplitudes of the vibration velocity in the low-frequency band. In the medium-frequency band, the model predicts global minimum vibration velocities at an equivalent roughness amplitude of 0.1 µm. In the high-frequency band, there is a slight decrease in the velocity of vibrations with an increase in the amplitude of the equivalent roughness;
- The neural network model predicted that the minimum vibration level is obtained in all frequency bands if the radial clearance has amplitude of around 20 µm and the equivalent roughness has an amplitude of around 0.05 µm.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Linear correlation coefficients of bearing parameters and frequency bands (

**a**) low-frequency band; (

**b**) medium-frequency band; (

**c**) high-frequency band.

**Figure 6.**Linear correlation coefficients of bearing parameters and frequency bands (equivalent technological parameters): (

**a**) low-frequency band; (

**b**) medium-frequency band; and (

**c**) high-frequency band.

**Figure 8.**Structure of I/O parameters of the artificial neural network; input parameters are considered together (equivalent technological parameters).

**Figure 11.**Process of training in a group and process of training algorithm (the red sample is out of training).

**Figure 12.**Dependence of the RMS value of the vibration velocity on the surface roughness of the raceway of the outer ring.

**Figure 13.**Dependence of the RMS value of the amplitude of the vibration velocity on the amplitude of the surface roughness of the raceway of the inner ring.

**Figure 14.**Mutual influence of radial clearance and equivalent roughness amplitude on RMS values of vibration velocity amplitude.

Bearing Characteristics | Minimum Value x _{min} | Maximum Value x _{max} | Mean Value $\overline{\mathit{x}}$ | Standard Deviation s |
---|---|---|---|---|

Gr, [µm] | 6 | 51 | 24.03 | 14.54 |

Ri/Re | 0.95 | 0.98 | 0.96 | 0.01 |

Rae, [µm] | 0.07 | 0.28 | 0.13 | 0.04 |

Rai, [µm] | 0.07 | 0.56 | 0.26 | 0.1 |

Wte max, [µm] | 0.1 | 1.97 | 0.8 | 0.32 |

Wti max, [µm] | 0.25 | 0.66 | 0.39 | 0.13 |

Wte, [µm] | 1.5 | 17.82 | 5.01 | 2.86 |

Wti, [µm] | 0.4 | 2.63 | 0.92 | 0.41 |

Bearing Characteristics | Minimum Value x _{min} | Maximum Value x _{max} | Mean Value $\overline{\mathit{x}}$ | Standard Deviation s |
---|---|---|---|---|

R_{a ekv} | 0.04 | 0.14 | 0.08 | 0.02 |

W_{t max ekv} | 0.16 | 0.47 | 0.25 | 0.06 |

W_{t ekv} | 0.36 | 1.78 | 0.74 | 0.26 |

Analyzed Artificial Neural Network Models | |||||
---|---|---|---|---|---|

Technological Parameters Separately | Technological Parameters Equivalent | ||||

Training algorithm | Training algorithm | ||||

Levenberg– Marquardt | Bayesian Regularization | Scaled Conjugate Gradient | Levenberg– Marquardt | Bayesian Regularization | Scaled Conjugate Gradient |

ANN architecture | ANN architecture | ||||

One hidden layer (Number of neurons from 1 to 30) | One hidden layer (Number of neurons from 1 to 30) | ||||

Two hidden layers (Number of neurons from 1 to 30) | Two hidden layers (Number of neurons from 1 to 30) | ||||

Three hidden layers (Number of neurons from 1 to 30) | Three hidden layers (Number of neurons from 1 to 30) |

Bearing 6006 | RMS of Vibration Velocity, µm/s Class Q7 | RMS of Vibration Velocity, µm/s Class Q6 | RMS of Vibration Velocity, µm/s Class Q5 |
---|---|---|---|

Low-frequency band | 224 | 112 | 71 |

Medium-frequency band | 160 | 80 | 80 |

High-frequency band | 450 | 224 | 112 |

**Table 5.**Experimental and predicted RMS values of vibration velocity for testing samples, measured quality class and predicted quality class based on surface roughness of the outer ring raceway.

Test Sample 1 Rae 0.09 µm | Test Sample 2 Rae 0.072 µm | Test Sample 3 Rae 0.125 µm | Test Sample 4 Rae 0.168 µm | |||||
---|---|---|---|---|---|---|---|---|

Measured RMS of Vibration Velocity, µm/s, Class Quality | Predicted RMS of Vibration Velocity, µm/s, Class Quality | Measured RMS of Vibration Velocity, µm/s, Class Quality | Predicted RMS of Vibration Velocity, µm/s, Class Quality | Measured RMS of Vibration Velocity, µm/s, Class Quality | Predicted RMS of Vibration Velocity, µm/s, Class Quality | Measured RMS of Vibration Velocity, µm/s, Class Quality | Predicted RMS of Vibration Velocity, µm/s, Class Quality | |

Low-frequency band | 80 Q6 | 105 Q6 | 60 Q6 | 103 Q6 | 73 Q6 | 108 Q6 | 97 Q6 | 83 Q6 |

Medium-frequency band | 56 Q5 | 62 Q5 | 57 Q5 | 61 Q5 | 83 Q6 | 77 Q5 | 82 Q6 | 110 Q6 |

High-frequency band | 121 Q6 | 119 Q6 | 107 Q5 | 118 Q6 | 115 Q6 | 118 Q6 | 132 Q6 | 117 Q6 |

**Table 6.**Experimental and predicted RMS values of vibration velocity for testing samples, measured quality class and predicted quality class based on the surface roughness of the inner ring raceway.

Test Sample 1 Rai 0.073 µm | Test Sample 2 Rai 0.157 µm | Test Sample 3 Rai 0.179 µm | Test Sample 4 Rai 0.271 µm | |||||
---|---|---|---|---|---|---|---|---|

Measured RMS of Vibration Velocity, µm/s, Class Quality | Predicted RMS of Vibration Velocity, µm/s, Class Quality | Measured RMS of Vibration Velocity, µm/s, Class Quality | Predicted RMS of Vibration Velocity, µm/s, Class Quality | Measured RMS of Vibration Velocity, µm/s, Class Quality | Predicted RMS of Vibration Velocity, µm/s, Class Quality | Measured RMS of Vibration Velocity, µm/s, Class Quality | Predicted RMS of Vibration Velocity, µm/s, Class Quality | |

Low-frequency band | 80 Q6 | 105 Q6 | 60 Q6 | 110 Q6 | 73 Q6 | 111 Q6 | 97 Q6 | 111 Q6 |

Medium-frequency band | 56 Q5 | 71 Q5 | 57 Q5 | 71 Q5 | 83 Q6 | 78 Q5 | 82 Q6 | 84 Q6 |

High-frequency band | 121 Q6 | 113 Q6 | 107 Q5 | 115 Q6 | 115 Q6 | 116 Q6 | 132 Q6 | 117 Q6 |

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**MDPI and ACS Style**

Knežević, I.; Rackov, M.; Kanović, Ž.; Buljević, A.; Antić, A.; Tica, M.; Živković, A.
An Analysis of the Influence of Surface Roughness and Clearance on the Dynamic Behavior of Deep Groove Ball Bearings Using Artificial Neural Networks. *Materials* **2023**, *16*, 3529.
https://doi.org/10.3390/ma16093529

**AMA Style**

Knežević I, Rackov M, Kanović Ž, Buljević A, Antić A, Tica M, Živković A.
An Analysis of the Influence of Surface Roughness and Clearance on the Dynamic Behavior of Deep Groove Ball Bearings Using Artificial Neural Networks. *Materials*. 2023; 16(9):3529.
https://doi.org/10.3390/ma16093529

**Chicago/Turabian Style**

Knežević, Ivan, Milan Rackov, Željko Kanović, Anja Buljević, Aco Antić, Milan Tica, and Aleksandar Živković.
2023. "An Analysis of the Influence of Surface Roughness and Clearance on the Dynamic Behavior of Deep Groove Ball Bearings Using Artificial Neural Networks" *Materials* 16, no. 9: 3529.
https://doi.org/10.3390/ma16093529