# Using Machine Learning Methods for Predicting Cage Performance Criteria in an Angular Contact Ball Bearing

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Methodology

#### 2.2. Calculation of Bearing Cage Dynamics

#### 2.3. Simulation Plan for the Cage Geometry and Bearing Load

#### 2.4. Features and Targets for Machine Learning

#### 2.5. Regression Algorithms and Hyperparameter Optimization

## 3. Results

#### 3.1. Dynamics Simulation Results

#### 3.2. Preprocessing of Calculation Results and Data Analysis

#### 3.3. Evaluating Optimization and Regression Results

## 4. Discussion

## 5. Summary and Conclusions

- The cage geometry has a significant influence on the resulting cage dynamics. The occurrence of unstable cage movements can be significantly reduced by changing the geometry of the cage.
- The influence of the geoemtric parameters is non-linear and characterized by strong alternating effects and can therefore hardly be assigned to single parameters.
- There is a low correlation between the axial movement of the cage and the influencing factors such as bearing load and cage geometry. The reason for this is that the contact forces acting on the cage point mostly in radial or circumferential direction. The forces acting on the cage are influenced by the parameters such as cage mass, cage speed, etc.
- In this study, all regression algorithms achieved acceptable values for the coefficient of determination in the range of ${R}^{2}\in [0.75\dots 0.94]$ for the target variables except for the quantile distance of the normalized axial center of mass coordinate of the cage. Therefore, the models appear to be suitable to compare the performance (dynamics, friction) of different cages.
- The use of machine learning algorithms allows prediction even for new data sets of the analyzed bearing for which no dynamics simulation has been performed. The duration of the prediction is less than one second, while the computation time for a simulation is about 10 hours.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ANN | Artificial neural network |

CDI | Cage Dynamics Indicator |

EA | Evolutionary Algorithm |

FE | Finite element |

MSE | Mean squared error |

RF | Random Forest |

med | Median |

qd | Quantile Distance |

## Appendix A

**Table A1.**Parameters of the machine learning models, that were not optimized. The literature reference for each model is the implementation of the algorithms.

Random Forest [26] | XGBoost [27] | ANN [25] | |||
---|---|---|---|---|---|

Parameter | Value | Parameter | Value | Parameter | Value |

Split criterion | MSE | Importance type | gain | Weight initializer | HeNormal/ Glorot |

Minimum samples for split | 2 | Objective | MSE | Slope coefficient for Leaky ReLu activation | 0.01 |

Use out of bag score | False | Subsampling | False | Bias initializer | None |

Bootstraping when building trees | True | Booster | gbtree | Regularizer | None |

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**Figure 1.**Procedure in this study, starting with the dynamics simulation and the design of experiments, followed by the creation of a database, the training of the machine learning models, and analysis of the predictions.

**Figure 2.**(

**a**) Cross-section of the angular contact ball bearing as typically used in machine tools. (

**b**) Exploded view of the three-dimensional dynamics simulation model consisting of two raceways, 19 rolling elements, and the outer ring-guided window cage.

**Figure 3.**(

**a**) Cross-section of the angular contact ball bearing cage. (

**b**) Three-dimensional view of cage pocket and a rolling element. The blue area represents the geometry of the cage pocket and shows an exemplary shape defined by four parameters ${c}_{0},\phantom{\rule{3.33333pt}{0ex}}{c}_{1},\phantom{\rule{3.33333pt}{0ex}}{c}_{2},$ and $\phantom{\rule{3.33333pt}{0ex}}{c}_{3}$.

**Figure 5.**Dynamic behavior of a cage for different operating conditions: an unstable (

**a**–

**c**), stable (

**d**–

**f**), and circling (

**g**–

**i**) cage motion. The three-dimensional deformation of the cages, the center of gravity trajectory, and the amplitude spectrum of the node displacement are illustrated.

**Figure 6.**Dynamic behavior of three different cage variants for the same loading conditions. The three-dimensional deformation of the cages (

**a**,

**d**,

**g**), the center of gravity trajectory (

**b**,

**e**,

**h**), and the amplitude spectrum of the node displacement (

**c**,

**f**,

**i**) are illustrated.

**Figure 7.**Overview of the results of the dynamics simulation. (

**a**) Number of motion types “unstable”, “stable”, and “circling” for each cage variant. (

**b**) Motion type as a function of cage geometry and bearing load. (

**c**) Number of motion types for each bearing load in the experimental design.

**Figure 8.**Distribution of target regression variables (

**a**) med($\Omega $) and qd($\Omega $) as well as (

**b**) qd(${F}_{e}$) and qd($\tilde{n}$) in the database. The data sets marked in red are identified as outliers using the LOF approach and not considered for training the regression models.

**Figure 9.**Matrix with correlation coefficients for determining the relationship between (

**a**) the input and output parameters and (

**b**) the output parameters among each other.

**Figure 10.**Coefficient of determination ${R}^{2}$ for the target variables of the regression algorithms Random Forest, XGBoost, and Neural Network for training (

**a**) and test data (

**b**).

**Figure 11.**Scatter plot of the target parameters for training (

**blue**) and test (

**red**) data sets and the predicted values by the neural network. The colored area represents the range where 90% of the errors for the test data sets are located.

**Figure 12.**Distribution of $\overline{{R}^{2}}$ values for all target variables (

**a**) and without the normalized ${\tilde{x}}_{\mathrm{c}}$-coordinate (

**b**) for a 10-fold cross-validation. Besides the minimum and maximum, the distribution of the values is also illustrated.

**Table 1.**Influencing parameters on the cage dynamics that have been investigated in research papers.

Group | Parameter |
---|---|

Bearing and cage properties | Internal clearance [13] |

Rolling element size [13] | |

Rolling element profile [13] | |

Pocket clearance [13,14,15] | |

Guidance clearance [14,15] | |

Pocket shape [9] | |

Bearing load | Load ratio [14,16] |

Rotational speed [7,13,14,15,16] | |

External vibrations [8] | |

Friction | Coefficient cage/rolling elements [7,14,15,17] |

Coefficient cage/raceway [17] | |

Rolling element/raceway traction [18] | |

Lubrication | Viscosity [8,19] |

Temperature [8,19] | |

Oil injection [8] |

Group | Property | Value |
---|---|---|

Integration | Integration method | Runge–Kutta |

Output time step | 0.0001 s | |

Calculation time | 1.0 s | |

Cage properties | Cage guidance type | rib |

Cage material | fibre reinforced phenolic resin | |

Bearing properties | Inner diameter | 90 mm |

Outer diameter | 140 mm | |

Pitch diameter ${d}_{\mathrm{p}}$ | 115 mm | |

Rolling elements | 19 balls | |

Rolling element diameter | 15.875 mm | |

Ring and ball material | 100 Cr6 | |

Contact angle $\alpha $ | 15° | |

Static load capacity ${C}_{0,\mathrm{r}}$ | 51 kN |

Parameter | Minimum | Maximum |
---|---|---|

Load ratio R | 0.25 | 10 |

Equivalent dynamic bearing load P in N | 1000 | 10,000 |

Torque on inner ring ${T}_{\mathrm{z}}$ in Nm | 10 | 50 |

Rotational speed inner ring ${n}_{\mathrm{i}}$ in rpm | 1000 | 9000 |

Friction coefficient rolling element/cage ${\mu}_{\mathrm{c}}$ | 0.05 | 0.35 |

Pocket shape parameters ${c}_{0},\phantom{\rule{0.277778em}{0ex}}{c}_{1},\phantom{\rule{0.277778em}{0ex}}{c}_{2},\phantom{\rule{0.277778em}{0ex}}{c}_{3}$ in mm | 0.1 | 0.35 |

Cage width ${b}_{\mathrm{c}}$ in mm | 19 | 23 |

Cage height ${h}_{\mathrm{c}}$ in mm | 4 | 8 |

Guidance diameter ${d}_{\mathrm{g}}$ | 122 | 124.2 |

Group | Property | Symbol |
---|---|---|

Mechanical and material properties | Weighted area moment of inertia | $\tilde{I}$ |

Weighted cross-sectional area | $\tilde{A}$ | |

Cage mass | m | |

Cage moment of inertia | ${J}_{\mathrm{x}}$, ${J}_{\mathrm{y}}$ | |

Geometric properties | Pocket shape parameters | ${c}_{0},\phantom{\rule{0.277778em}{0ex}}{c}_{1},\phantom{\rule{0.277778em}{0ex}}{c}_{2},\phantom{\rule{0.277778em}{0ex}}{c}_{3}$ |

Pocket clearance circumferential | ${c}_{\mathrm{c}}$ | |

Pocket clearance axial | ${c}_{\mathrm{a}}$ | |

Guidance clearance | ${c}_{\mathrm{g}}$ | |

Load parameter | Axial Force | ${F}_{\mathrm{x}}/{C}_{0,\mathrm{r}}$ |

Radial Force | ${F}_{\mathrm{y}}/{C}_{0,\mathrm{r}}$ | |

Torque | ${T}_{\mathrm{z}}/({C}_{0,\mathrm{r}}\xb7{d}_{\mathrm{p}})$ | |

Rotational speed inner ring | ${n}_{\mathrm{i}}$ | |

Coefficient of friction | ${\mu}_{\mathrm{c}}$ | |

Predicted cage motion | Cage motion class | C |

**Table 5.**Parameters of the EA for the optimization of the hyperparameters of the regression algorithms.

Parameter | Value | Parameter | Value |
---|---|---|---|

Number of generations | 30 | Population size | 50 |

Elite individuals | 1 | Crossover probability | 0.8 |

Mutation probability | 0.15 |

Model | Parameter | Minimum | Maximum |
---|---|---|---|

XGBoost | Max depth | 20 | 200 |

Number of estimators | 100 | 1500 | |

Learning rate | 0.0001 | 0.01 | |

L1 regularization | 0.0001 | 0.9 | |

L2 regularization | 0.0001 | 0.9 | |

Minimum loss reduction for tree split | 0.00001 | 0.2 | |

Random Forest | Max depth | 20 | 200 |

Number of estimators | 100 | 1500 | |

Minimum samples required for a leaf | 2 | 10 | |

Maximum number of features for a split | 10 | 17 | |

Maximum number of leaf nodes | 10 | 500 | |

Minimal cost-complexity pruning | 0 | 0.5 | |

ANN | Number of neurons in layer 1 | 100 | 600 |

Number of neurons in layer 2 | 100 | 600 | |

Number of neurons in layer 3 | 100 | 600 | |

Number of neurons in layer 4 | 100 | 600 | |

Learning rate | 0.0001 | 0.9 | |

Activation function | ELU, RELU, Leaky_RELU |

Model | Parameter | Minimum | Maximum | Value |
---|---|---|---|---|

XGBoost | Max depth | 20 | 200 | 52 |

Number of estimators | 100 | 1500 | 832 | |

Learning rate | 0.0001 | 0.01 | 0.045 | |

L1 regularization | 0.0001 | 0.9 | 0.865 | |

L2 regularization | 0.0001 | 0.9 | 0.791 | |

Minimum loss reduction for tree split | 0.00001 | 0.2 | 0.021 | |

Random Forest | Max depth | 20 | 200 | 46 |

Number of estimators | 100 | 1500 | 812 | |

Minimum samples required for a leaf | 2 | 10 | 2 | |

Maximum number of features for a split | 10 | 17 | 17 | |

Maximum number of leaf nodes | 10 | 500 | 441 | |

Minimal cost-complexity pruning | 0 | 0.5 | 0.0009 | |

ANN | Number of neurons in layer 1 | 100 | 600 | 447 |

Number of neurons in layer 2 | 100 | 600 | 568 | |

Number of neurons in layer 3 | 100 | 600 | 577 | |

Number of neurons in layer 4 | 100 | 600 | 455 | |

Learning rate | 0.0001 | 0.9 | 0.003 | |

Activation function | ELU, RELU, Leaky_RELU | Leaky_RELU |

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## Share and Cite

**MDPI and ACS Style**

Schwarz, S.; Grillenberger, H.; Graf-Goller, O.; Bartz, M.; Tremmel, S.; Wartzack, S.
Using Machine Learning Methods for Predicting Cage Performance Criteria in an Angular Contact Ball Bearing. *Lubricants* **2022**, *10*, 25.
https://doi.org/10.3390/lubricants10020025

**AMA Style**

Schwarz S, Grillenberger H, Graf-Goller O, Bartz M, Tremmel S, Wartzack S.
Using Machine Learning Methods for Predicting Cage Performance Criteria in an Angular Contact Ball Bearing. *Lubricants*. 2022; 10(2):25.
https://doi.org/10.3390/lubricants10020025

**Chicago/Turabian Style**

Schwarz, Sebastian, Hannes Grillenberger, Oliver Graf-Goller, Marcel Bartz, Stephan Tremmel, and Sandro Wartzack.
2022. "Using Machine Learning Methods for Predicting Cage Performance Criteria in an Angular Contact Ball Bearing" *Lubricants* 10, no. 2: 25.
https://doi.org/10.3390/lubricants10020025