# Experimental Evidence of the Speed Variation Effect on SVM Accuracy for Diagnostics of Ball Bearings

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Setup

#### 2.2. Experimental Campaign

- Constant speed motion. Tested speeds: 10, 20, 30, 40, 50, 55, 60, 65 Hz. All the tests lasted 15 s.
- Linear increasing speed motion. From 10 to 65 Hz in 15 s, corresponding to a constant acceleration of 1320 deg/s
^{2}(3.667 Hz/s^{2}); - Fifth-grade polynomial motion profile. From 10 to 65 Hz in 15 s. The polynomial equation is$$q(t)={a}_{0}+{a}_{1}t+{a}_{2}{t}^{2}+{a}_{3}{t}^{3}+{a}_{4}{t}^{4}+{a}_{5}{t}^{5},$$$${a}_{0}={q}_{i},$$$${a}_{1}=\dot{{q}_{i}},$$$${a}_{2}=\frac{1}{2}\ddot{{q}_{i}},$$$${a}_{3}=\frac{1}{2{T}^{3}}[20({q}_{f}-{q}_{i})-(8\dot{{q}_{f}}+12\dot{{q}_{i}})T-(3\ddot{{q}_{f}}-\ddot{{q}_{i}}){T}^{2}],$$$${a}_{4}=\frac{1}{2{T}^{4}}[30({q}_{f}-{q}_{i})+(14\dot{{q}_{f}}+16\dot{{q}_{i}})T+(3\ddot{{q}_{f}}-2\ddot{{q}_{i}}){T}^{2}],$$$${a}_{5}=\frac{1}{2{T}^{5}}[12({q}_{f}-{q}_{i})-6(\dot{{q}_{f}}+\dot{{q}_{i}})T-(\ddot{{q}_{f}}-\ddot{{q}_{i}}){T}^{2}]$$
^{2}.

#### 2.3. Machine Learning

^{©}Statistics and Classification Learner Toolbox. This toolbox provides several machine learning techniques. It comprises an automatic training and testing of the input data and tries different combinations of settings (e.g., different kernels available) in order to find the optimal machine learning solution to the classification problem.

## 3. Results

- Influence of motion profile on the SVM output;
- Influence of discretization of the speed range in the training step;
- Influence of the length of the signal in the training step;
- Influence of feature arrays in the training step;
- Influence of feature domain in the training step.

#### 3.1. Influence of Motion Profile

#### 3.2. Influence of Discretization of the Speed Range

#### 3.3. Influence of the Length of the Signal

#### 3.4. Influence of Feature Array in Training Step

#### 3.5. Influence of Features Domain

- Time domain: RMS, skewness, and kurtosis.
- Frequency domain: frequency RMS, spectral skewness, and spectral kurtosis.
- Time and frequency domains: RMS, skewness, kurtosis, spectral skewness, and spectral kurtosis.

## 4. Conclusions

- The training data must span the speed range in detail, at least 5 Hz steps.
- Despite the speed range discretization step, there is a limit to the accuracy that depends on the motion profile and cannot be exceeded.
- The accuracy is not sensible to the length of the signal on which the feature array is computed (this is valid for the specific feature array discussed in this paper).
- A proper choice of the feature array can decrease the effect of the variation of the motion profile.
- In non-stationary conditions, time-domain features are preferable to frequency-domain features in the diagnostics of ball-bearings.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Test rig. From left to the right: the electric motor, the elastic coupling, the main shaft, its support, and the custom case with the bearing under test. The loading system is in the lower part of the custom case, the mono-axial accelerometer is in the higher part.

**Figure 2.**Raw data (

**left**) and corresponding spectrum (

**right**) for faulted and healthy bearings. The text arrows point to fault frequencies and rotation frequency.

**Figure 3.**Speed profiles used in the tests. Solid lines show the three main types: constant, linear, and polynomial. Dotted lines refer to the different values of the constant speeds used.

**Figure 4.**Example of a support vector machine (SVM) for the classification of two classes [29].

**Figure 5.**(

**a**) The mean speed of the dataset used considering consecutive 1-s windowing; (

**b**) The mean speed of the dataset used shifting 1-s windowing to get the same mean speed.

**Figure 6.**Classification accuracy of the SVMs as a function of the discretization step of the speed range.

**Figure 7.**Variations of feature values for different duration of data input for linear speed tests. RMS: root mean square.

**Figure 8.**Variations of feature values for different duration of data input for polynomial speed tests.

Deep Groove Ball Bearing (6204) | |
---|---|

Pitch diameter (mm) | 33.5 |

Ball diameter (mm) | 7.94 |

Rotational frequency (Hz) | 55 |

Outer race fault frequency (Hz) | 167.9 |

Inner race fault frequency (Hz) | 272.1 |

Speed Range (Hz) | Samples | Motion Profile | Accuracy | |
---|---|---|---|---|

Training | [10, 20, 30, 40, 50, 60] | 432 | Constant | — |

Test | [10, 20, 30, 40, 50, 60] | 108 | Constant | 99.5% |

[10–60] (Figure 5a) | 96 | Linear | 57.3% | |

90 | Polynomial | 51.1% | ||

[20, 30, 40, 50, 60] (Figure 5b) | 36 | Linear | 75% | |

36 | Polynomial | 69.4% |

Δ(Hz) | Speed Range (Hz) | Constant Speed | Linear Speed | Polynomial Speed |
---|---|---|---|---|

5 | [50–65] | 99.3% | 79.2% | 80.6% |

10 | [10–60] | 99.5% | 57.3% | 51.1% |

20 | [10–50] | 100% | 47.9% | 38.9% |

T(s) | Constant Speed | Linear Speed | Polynomial Speed |
---|---|---|---|

0.5 | 99.1% | 64.1% | 68.3% |

1 | 100% | 66.7% | 66.7% |

1.5 | 99% | 61.7% | 71.7% |

Feature Array | Constant Speed | Linear Speed | Polynomial Speed |
---|---|---|---|

RMS, kurtosis, skewness | 100% | 66.7% | 66.7% |

Kurtosis, skewness | 92.7% | 64.6% | 58.9% |

RMS, kurtosis | 96.2% | 60.42% | 65.56% |

RMS, skewness | 95.8% | 63.54% | 62.22% |

Features Domain | Constant Speed | Linear Speed | Polynomial Speed |
---|---|---|---|

Time domain | 99.5% | 57.3% | 51.1% |

Frequency domain | 99.5% | 53.13% | 37.78% |

Time and frequency domains | 100% | 41.67% | 35.56% |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cavalaglio Camargo Molano, J.; Rubini, R.; Cocconcelli, M. Experimental Evidence of the Speed Variation Effect on SVM Accuracy for Diagnostics of Ball Bearings. *Machines* **2018**, *6*, 48.
https://doi.org/10.3390/machines6040048

**AMA Style**

Cavalaglio Camargo Molano J, Rubini R, Cocconcelli M. Experimental Evidence of the Speed Variation Effect on SVM Accuracy for Diagnostics of Ball Bearings. *Machines*. 2018; 6(4):48.
https://doi.org/10.3390/machines6040048

**Chicago/Turabian Style**

Cavalaglio Camargo Molano, Jacopo, Riccardo Rubini, and Marco Cocconcelli. 2018. "Experimental Evidence of the Speed Variation Effect on SVM Accuracy for Diagnostics of Ball Bearings" *Machines* 6, no. 4: 48.
https://doi.org/10.3390/machines6040048