# Ensemble-Based Semi-Supervised Learning for Milling Chatter Detection

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modellings in Milling Chatter Detection

#### 2.1. Motivation

#### 2.2. Proposed Method

#### 2.2.1. Multifractal Detrended Fluctuation Analysis

#### 2.2.2. Ensemble-Based Semi-Supervised Learning

## 3. Framework of the Proposed Method for Chatter Detection

#### 3.1. Feature Extraction and Selection

#### 3.2. Model Parameters Training

## 4. Experimental Study

#### 4.1. Experiment Setup and Data Acquisition

#### 4.2. Signal Processing

#### 4.3. Results and Analysis

#### 4.4. Comparison and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Experimental setup for chatter detection. (

**a**) Configuration of machine; (

**b**) Data-acquisition and analysis system.

**Figure 9.**Chatter identification results with proposed method (spindle speed 4500 rpm, feed rate 400 mm/min, and radial cutting depth 2.5 mm).

**Figure 10.**Chatter identification results with proposed method (spindle speed 6800 rpm, feed rate 450 mm/min, and radial cutting depth 2 mm).

No. | Feature Name | Expression | No. | Feature Name | Expression |
---|---|---|---|---|---|

1 | Mean | $\mathrm{M}=\frac{1}{T}{\displaystyle \sum _{t=1}^{T}f(t)}$ | 10 | Energy ratio | $PC={\displaystyle \sum _{i=1}^{n}{p}_{i}{}^{2}}$ |

2 | Standard deviation | $\mathrm{STD}=\sqrt{\frac{1}{T}{\displaystyle \sum _{t=1}^{T}{(f(t)-M)}^{2}}}$ | 11 | Centre frequency | $FC=\frac{{\displaystyle {\sum}_{i=1}^{n}{f}_{i}{p}_{i}}}{{\displaystyle {\sum}_{i=1}^{n}{p}_{i}}}$ |

3 | Root mean square | $\mathrm{RMS}=\sqrt{\frac{1}{T}{\displaystyle \sum _{t=1}^{T}{(f(t))}^{2}}}$ | 12 | Variation in frequency | $VF=\frac{{\displaystyle {\sum}_{i=1}^{n}{({f}_{i}-FC)}^{2}{p}_{i}}}{{\displaystyle {\sum}_{i=1}^{n}{p}_{i}}}$ |

4 | Peak-Peak | $\mathrm{PP}=\mathrm{max}\left(\mathrm{f}\right(\mathrm{t}\left)\right)-\mathrm{min}\left(\mathrm{f}\right(\mathrm{t}\left)\right)$ | 13 | Frequency variance | $MSF=\frac{{\displaystyle {\sum}_{i=1}^{n}{f}_{i}{}^{2}{p}_{i}}}{{\displaystyle {\sum}_{i=1}^{n}{p}_{i}}}$ |

5 | Skewness | $\mathrm{S}K=\frac{\frac{1}{T}{\displaystyle {\sum}_{t=1}^{T}{(f(t)-\mathrm{M})}^{3}}}{ST{D}^{3}}$ | 14 | Mean of spectrum | ${M}_{p}=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}S{(f)}_{i}}$ |

6 | Kurtosis | $\mathrm{KU}=\frac{\frac{1}{T}{\displaystyle {\sum}_{t=1}^{T}{(f(t)-M)}^{4}}}{ST{D}^{4}}$ | 15 | Peak of spectrum | $Ma{x}_{p}=\mathrm{max}(S{(f)}_{i})$ |

7 | Crest factor | $CF=\frac{\mathrm{max}(\left|f(t)\right|)}{\mathrm{RMS}}$ | 16 | Variance of spectrum | $VA{R}_{p}=\frac{{\displaystyle {\sum}_{i=1}^{n}{(S{(f)}_{i}-{M}_{p})}^{2}}}{n-1}$ |

8 | Clearance factor | $CLF=\frac{\mathrm{max}(\left|f(t)\right|)}{{\left(\frac{1}{T}\sqrt{{\displaystyle {\sum}_{t=1}^{T}\left|f(t)\right|}}\right)}^{2}}$ | 17 | Kurtosis of spectrum | $K{U}_{p}=\frac{1}{n}\frac{{\displaystyle {\sum}_{i=1}^{n}{(S{(f)}_{i}-{M}_{p})}^{4}}}{VA{R}_{p}{}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$ |

9 | Impact factor | $IF=\frac{\mathrm{max}(\left|f(t)\right|)}{\frac{1}{T}{\displaystyle {\sum}_{t=1}^{T}\left|f(t)\right|}}$ | 18 | Skewness of spectrum | $S{K}_{p}=\frac{1}{n}\frac{{\displaystyle {\sum}_{i=1}^{n}{(S{(f)}_{i}-{M}_{p})}^{3}}}{VA{R}_{p}{}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$ |

No. | Shape | Ns (r/min) | fd (mm/min) | ap (mm) | ar (mm) | C | No. | Shape | ns (r/min) | fd (mm/min) | ap (mm) | ar (mm) | C |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | #1 | 4500 | 400 | 5 | 0.1 | +1 | 36 | #2 | 7200 | 550 | 3 | 0.2 | - |

2 | #1 | 4500 | 400 | 5 | 0.2 | +1 | 37 | #2 | 7200 | 550 | 3 | 0.3 | - |

3 | #1 | 4500 | 400 | 5 | 0.3 | +1 | 38 | #2 | 7200 | 700 | 3 | 0.3 | - |

4 | #1 | 4500 | 400 | 5 | 0.5 | +1 | 39 | #2 | 7200 | 850 | 3 | 0.3 | - |

5 | #1 | 4500 | 400 | 5 | 0.7 | +1 | 40 | #2 | 7200 | 950 | 3 | 0.3 | - |

6 | #1 | 4500 | 400 | 5 | 1.0 | - | 41 | #2 | 7200 | 300 | 3 | 0.3 | - |

7 | #1 | 4500 | 400 | 5 | 1.5 | - | 42 | #2 | 7200 | 450 | 3 | 0.3 | - |

8 | #1 | 4500 | 400 | 5 | 2.0 | - | 43 | #3 | 6500 | 420 | 1 | 0.5 | −1 |

9 | #1 | 4500 | 400 | 5 | 3.0 | - | 44 | #3 | 6000 | 420 | 1 | 0.1 | +1 |

10 | #1 | 6000 | 500 | 5 | 3.0 | - | 45 | #3 | 6000 | 420 | 2 | 0.1 | +1 |

11 | #1 | 6000 | 500 | 5 | 4.0 | - | 46 | #3 | 6000 | 420 | 2 | 0.2 | - |

12 | #2 | 6000 | 550 | 3 | 0.3 | +1 | 47 | #3 | 6500 | 420 | 2 | 0.1 | −1 |

13 | #2 | 6000 | 550 | 3 | 0.5 | +1 | 48 | #3 | 6500 | 600 | 2 | 0.1 | −1 |

14 | #2 | 6000 | 550 | 3 | 0.7 | - | 49 | #3 | 6500 | 600 | 1.5 | 0.1 | - |

15 | #2 | 6000 | 550 | 3 | 0.8 | - | 50 | #3 | 6000 | 300 | 2 | 0.1 | - |

16 | #2 | 6000 | 550 | 3 | 0.9 | - | 51 | #3 | 5500 | 400 | 2 | 0.1 | +1 |

17 | #2 | 6000 | 550 | 5 | 0.3 | - | 52 | #3 | 5500 | 400 | 1.5 | 0.1 | +1 |

18 | #2 | 6000 | 550 | 5 | 0.5 | - | 53 | #3 | 5500 | 400 | 1.5 | 0.1 | +1 |

19 | #2 | 6000 | 550 | 5 | 0.7 | - | 54 | #3 | 5000 | 400 | 1.5 | 0.1 | - |

20 | #2 | 6000 | 550 | 5 | 0.9 | −1 | 55 | #3 | 5000 | 400 | 1.5 | 0.2 | −1 |

21 | #2 | 6000 | 550 | 5 | 1.1 | −1 | 56 | #3 | 5000 | 400 | 1.5 | 0.3 | −1 |

22 | #2 | 6700 | 550 | 5 | 0.3 | +1 | 57 | #3 | 5000 | 400 | 0.5 | 0.5 | - |

23 | #2 | 6700 | 550 | 5 | 0.6 | - | 58 | #3 | 6000 | 420 | 2.0 | 0.1 | - |

24 | #2 | 6700 | 550 | 5 | 0.8 | −1 | 59 | #3 | 5000 | 400 | 2.0 | 0.1 | - |

25 | #2 | 4500 | 550 | 5 | 0.8 | - | 60 | #3 | 5000 | 400 | 2.0 | 0.1 | - |

26 | #2 | 5500 | 550 | 5 | 0.8 | - | 61 | #4 | 4500 | 400 | 0–8 | 2.5 | - |

27 | #2 | 4000 | 550 | 5 | 0.6 | - | 62 | #4 | 6000 | 400 | 0–8 | 2.5 | - |

28 | #2 | 4000 | 550 | 5 | 0.5 | - | 63 | #4 | 6500 | 450 | 0–8 | 2.5 | - |

29 | #2 | 4000 | 550 | 5 | 0.3 | - | 64 | #4 | 6700 | 450 | 0–8 | 2.5 | - |

30 | #2 | 4000 | 550 | 5 | 0.1 | +1 | 65 | #4 | 6900 | 450 | 0–8 | 2.0 | - |

31 | #2 | 5000 | 550 | 5 | 0.1 | - | 66 | #4 | 6900 | 450 | 0–8 | 3.0 | - |

32 | #2 | 5000 | 550 | 3 | 0.1 | - | 67 | #4 | 6800 | 450 | 0–8 | 2.0 | - |

33 | #2 | 5200 | 550 | 3 | 0.1 | - | 68 | #4 | 6750 | 450 | 0–8 | 2.0 | - |

34 | #2 | 6750 | 550 | 3 | 0.1 | - | 69 | #4 | 6750 | 450 | 0–8 | 1.0 | - |

35 | #2 | 7200 | 550 | 3 | 0.1 | - | 70 | #4 | 6750 | 450 | 0–8 | 3.5 | - |

Chatter State | ${\mathit{a}}_{\mathit{max}}$ | $\mathit{f}({\mathit{a}}_{\mathit{max}})$ | ${\mathit{a}}_{\mathit{min}}$ | $\mathit{f}({\mathit{a}}_{\mathit{min}})$ | ${\mathit{a}}_{\mathit{0}}$ | $\mathbf{\Delta}\mathit{a}$ |
---|---|---|---|---|---|---|

Stable | 0.2514 | 0.5346 | −0.0386 | 0.7899 | 0.0644 | 0.2900 |

Unlabeled-1 | 0.1364 | 0.7446 | −0.0025 | 0.9103 | 0.0413 | 0.1389 |

Chatter | 0.1093 | 0.8321 | −0.0042 | 0.8794 | 0.0457 | 0.1135 |

Unlabeled-2 | 0.1446 | 0.7742 | −0.0497 | 0.7886 | 0.0461 | 0.1943 |

Machining Condition | Labeled | Unlabeled | Mixed |
---|---|---|---|

No. 61 | 97.27% | 90.91% | 95.45% |

No. 63 | 85.15% | 88.12% | 94.06% |

No. 65 | 96.25% | 88.75% | 93.75% |

No. 67 | 87.78% | 87.78% | 94.44% |

No. 69 | 84.29% | 87.14% | 92.88% |

Machining Condition | EB-SSL | UDEED |
---|---|---|

No. 61 | 95.45% | 92.73% |

No. 63 | 94.06% | 91.09% |

No. 65 | 93.75% | 91.25% |

No. 67 | 94.44% | 92.22% |

No. 69 | 92.88% | 90.00% |

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**MDPI and ACS Style**

Liu, W.; Wang, P.; You, Y.
Ensemble-Based Semi-Supervised Learning for Milling Chatter Detection. *Machines* **2022**, *10*, 1013.
https://doi.org/10.3390/machines10111013

**AMA Style**

Liu W, Wang P, You Y.
Ensemble-Based Semi-Supervised Learning for Milling Chatter Detection. *Machines*. 2022; 10(11):1013.
https://doi.org/10.3390/machines10111013

**Chicago/Turabian Style**

Liu, Weichao, Pengyu Wang, and Youpeng You.
2022. "Ensemble-Based Semi-Supervised Learning for Milling Chatter Detection" *Machines* 10, no. 11: 1013.
https://doi.org/10.3390/machines10111013