# Extraction of Frictional Vibration Features with Multifractal Detrended Fluctuation Analysis and Friction State Recognition

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experiments

#### 2.1. Tribological Pair

#### 2.2. Apparatus

#### 2.3. Methods

^{3}and viscosity of 80.247 mm

^{2}/s at 40 °C. The UMT-TriboLab friction and wear testing machine displayed and recorded the friction coefficient data, and the friction states of the friction pair were determined by controlling the amounts of lubricant and the friction coefficient. Reference [30] gives the typical friction coefficient values of different friction states, e.g., the friction coefficient of mixed lubrication is about 0.01~0.12, the friction coefficient of boundary lubrication is about 0.12~0.3 and the friction coefficient of dry friction is more than 0.3. During the mixed lubrication test, the block sample was fixed in the oil tank and immersed in the lubricating oil completely. The mixed lubrication test was carried out for 60 min. After the completion of the mixed lubrication test, the lubricating oil was completely removed, and the friction surface of the block sample and the pin sample were cleaned by non-woven paper with very thin oil film left. The friction state of the boundary lubrication was determined by the friction coefficient and the boundary lubrication test was carried out for 4 min. After completing boundary lubrication test, the surface of the friction pair was thoroughly cleaned by gasoline and the subsequent dry friction test was carried out for 20 min. The friction coefficient data of the three friction state tests of the mixed lubrication, boundary lubrication and dry friction tested in this paper are shown in Figure 2.

## 3. MFDFA Algorithm

_{k}of length N is the signal to be analyzed. The MFDFA algorithm includes the following six steps [33]:

_{s}= int(N/s) nonoverlapping segments of equal length s.

_{s}segments is calculated by a least-square fit of the series. Then determine the variance:

_{s}and

_{s}+ 1, N

_{s}+ 2, …, 2N

_{s}. In this expression, y

_{v}(i) is the fitting polynomial in segment v.

_{q}(s) versus s for each value of q:

## 4. Applying MFDFA to Signals

#### 4.1. Applying MFDFA to Frictional Vibration Time-Domain Signals

_{min}) − f(α

_{max}), describes how much the number of subset elements formed by a fractal object changes as the segmentation length decreasing. The maximum f

_{max}represents the dominant value in quantity of the signal distributions. The multifractal spectrum critical value α

_{f}

_{max}shows the signal distributions dominant in quantity. The spectrum parameter α

_{max}indicates the minimum value of the signal distributions. The spectrum parameter I

_{α}= α

_{f}

_{max}− α

_{min}represents the numerical difference between the signal distribution dominant in quantity and maximum value, manifesting the uniformity of the signal.

_{max}, α

_{f}

_{max}, α

_{max}, and I

_{α}were calculated based on the multifractal spectra of the time-domain signals of the frictional vibrations.

_{max}also shows decreasing trends with ranges between 1.01 and 0.93. The multifractal spectrum critical value of frictional vibrations decreases, and the α

_{f}

_{max}illustrates decreasing trends with ranges between 0.85 and 0.53. The multifractal spectrum parameter α

_{max}indicates decreasing trends with ranges between 1.92 and 1.31. Frictional vibrations tend to be uniform, and I

_{α}demonstrates decreasing trends with ranges between 0.20 and 0.17.

_{max}also shows increasing trends with ranges between 0.93 and 0.95. The α

_{f}

_{max}illustrates increasing trends with ranges between 0.68 and 0.70. The α

_{max}indicates increasing trends with ranges between 1.61 and 1.94. The multifractal spectrum parameter I

_{α}demonstrates increasing trends with ranges between 0.23 and 0.28.

_{max}also shows increasing trends with ranges between 0.94 and 0.98. The α

_{f}

_{max}illustrates increasing trends with ranges between 0.59 and 0.73. The α

_{max}indicates increasing trends with ranges between 1.56 and 1.84. The multifractal spectrum parameter I

_{α}demonstrates increasing trends with ranges between 0.29 and 0.43.

#### 4.2. Applying MFDFA to Friction Coefficient Data

_{max}, α

_{f}

_{max}, α

_{max}, and I

_{α}calculated based on the multifractal spectra.

_{max}also shows decreasing trends with ranges between 1.0231 and 1.0228. The multifractal spectrum critical value of the friction coefficient data decrease, and the α

_{f}

_{max}illustrates decreasing trends with ranges between 1.027 and 1.023. The multifractal spectrum parameter α

_{max}indicates decreasing trends with ranges between 1.186 and 1.154. The multifractal spectrum parameter I

_{α}demonstrates decreasing trends with ranges between 0.053 and 0.028.

_{max}also shows increasing trends with ranges between 1.0229 and 1.0231. The α

_{f}

_{max}illustrates increasing trends with ranges between 1.0257 and 1.0264. The α

_{max}indicates increasing trends with ranges between 1.152 and 1.177. The multifractal spectrum parameter I

_{α}demonstrates increasing trends with ranges between 0.025 and 0.027.

_{max}also shows increasing trends with ranges between 1.0230 and 1.0232. The α

_{f}

_{max}illustrates increasing trends with ranges between 1.0265 and 1.0269. The α

_{max}indicates increasing trends with ranges between 1.205 and 1.216. The multifractal spectrum parameter I

_{α}demonstrates increasing trends with ranges between 0.018 and 0.022.

#### 4.3. Applying MFDFA to Frictional Vibration Frequency-Domain Signals

_{max}, α

_{f}

_{max}, α

_{max}, and I

_{α}calculated based on the multifractal spectra.

_{max}also shows decreasing trends with ranges between 1.0172 and 1.0168. The multifractal spectrum critical value α

_{f}

_{max}illustrates decreasing trends with ranges between 1.084 and 1.072. The multifractal parameter α

_{max}indicates decreasing trends with ranges between 1.82 and 1.51. Frictional vibrations tend to be uniform, and I

_{α}demonstrates decreasing trends with ranges between 0.47 and 0.53.

_{max}also shows increasing trends with ranges between 1.0170 and 1.0174. The α

_{f}

_{max}illustrates increasing trends with ranges between 1.07 and 1.11. The α

_{max}indicates increasing trends with ranges between 1.50 and 1.62. The multifractal spectrum parameter I

_{α}demonstrates increasing trends with ranges between 0.56 and 0.61.

_{max}also shows increasing trends with ranges between 1.0171 and 1.0174. The α

_{f}

_{max}illustrates increasing trends with ranges between 1.08 and 1.11. The α

_{max}indicates increasing trends with ranges between 1.54 and 1.68. The I

_{α}demonstrates increasing trends with ranges between 0.48 and 0.67.

#### 4.4. Analysis and Discussions

_{a}is the mean roughness, and S

_{a}can be used to indicate significant deviations in the texture. The S

_{sk}is the skewness of the surface texture representing the degree of symmetry of the surface heights about the mean plane, the S

_{sk}shows the preponderance of peaks (that is, S

_{sk}> 0) or valley structures (S

_{sk}< 0) comprising the surface. The S

_{z}is the ten points height over the surface, and the S

_{z}characterizes the maximum peak to valley magnitude for the entire surface [36].

_{max}, α

_{f}

_{max}, α

_{max}, and I

_{α}of the frictional vibrations are large. As the mixed lubrication test proceeds, many micro-convex peaks of different shapes on the surface of the friction pair are continuously flattened. Figure 13b shows the surface topography of the block specimen at the end of mixed lubrication test, and the surface roughness reduces obviously. Besides, the parameter S

_{sk}is negative with shrinking value, indicating the preponderance of valley structures decreases. The S

_{z}value decreases, indicating the maximum peak to valley magnitude decreases. Thus, the frictional vibrations excited tend to be stabilized with the roughness reduced and the relative contact area increased. The spectrum parameters Δα, Δf, f

_{max}, α

_{f}

_{max}, α

_{max}, and I

_{α}of the frictional vibrations show decreasing trends. During the boundary lubrication test, there is very thin oil film existing in the surface of the friction pair, and there are more micro-convex peaks contact. As shown in Figure 13c, the surface roughness reduces and the surface topography consists of a number of small contact plateaus. The frictional vibration is violent and its amplitude is high. Thus, the spectrum parameters Δα, Δf, f

_{max}, α

_{f}

_{max}, α

_{max}, and I

_{α}of the frictional vibrations show increasing trends. During the dry fraction test, the lubrication condition is so bad that the micro-convex peaks are contact directly. Besides, the surface roughness of friction pair increases, the S

_{sk}is negative with growing value, indicating the preponderance of valley structures increases. The S

_{z}value increases, indicating the maximum peak to valley magnitude increases. Figure 13d shows the surface topography of the block specimen at the end of dry friction test. The surface is severely damaged due to the generation of many deep furrows, and the surface heights fluctuate violently in a large range. The frictional vibrations excited are severe. Thus, the spectrum parameters Δα, Δf, f

_{max}, α

_{f}

_{max}, α

_{max}, and I

_{α}of the frictional vibrations show increasing trends.

_{max}, α

_{f}

_{max}, α

_{max}, and I

_{α}of the friction coefficient show decreasing trends. During the boundary lubrication test, there is very thin oil film existing in the surface of the friction pair, and the friction forces increase with more micro-convex peaks contact. The friction coefficients display upward trends. Thus, the spectrum parameters Δα, Δf, f

_{max}, α

_{f}

_{max}, α

_{max}, and I

_{α}of the friction coefficient data show increasing trends. During the dry fraction test, the lubrication condition is so bad that the micro-convex peaks are contact directly. The friction coefficients show upward trends with the friction forces increasing. Thus, the spectrum parameters Δα, Δf, f

_{max}, α

_{f}

_{max}, α

_{max}, and I

_{α}of the friction coefficient data show increasing trends.

## 5. Friction State Recognition

#### 5.1. Principal Component Analysis Algorithm

_{0}is normalized first by the following equation to eliminate the influences of variable dimension and sample quantity in the original data:

_{i}in T are orthonormal. These orthonormal vectors are the linear combination of X that show how samples are related to each other. P is the loading matrix which is derived from the covariance matrix of X. Vectors p

_{i}in P are orthonormal, and they are the dimensions in the new orthonormal coordinate system. P can be derived from the following eigenvalue problem:

_{i}in the new space.

#### 5.2. Analysis and Discussions

_{max}, ${\alpha}_{{f}_{\mathrm{max}}}$, α

_{max}, I

_{α}of the frictional vibration time-domain signals and the frequency-domain signals are shown in Figure 8 and Figure 12. The 12 variables consist of six time-domain multifractal spectrum parameters and six frequency-domain multifractal spectrum parameters. Each of the eight frictional vibration signals selected from the states of mixed lubrication, boundary lubrication and dry frication, respectively, constitute the data matrix of 24 samples and 12 variables. By analyzing results, the first three principal component eigenvalues and percent variance are shown in Table 2. The CPVs of the first three principal components are shown in Table 3. The CPV of the first three principal components is 91.9811%, indicating that the first three principal components contain most of the characteristic information.

## 6. Conclusions

- (1)
- The multifractal detrended fluctuation analysis algorithm can extract the fractal characteristics of the frictional vibration signals effectively. In different friction states, the multifractal spectrum parameters of the frictional vibrations have different parameter ranges and present different trends. The analysis shows that it is symmetric in the variation trends of the multifractal spectrum parameters of the frictional vibrations and the friction coefficients within the same lubrications. The multifractal spectra and their parameters can characterize the nonlinear characteristics of the frictional vibrations.
- (2)
- The principal component analysis based on the multifractal spectrum parameters of the frictional vibrations can realize the friction state recognition. The first three components of the frictional vibration multifractal spectrum parameters of the mixed lubrication, boundary lubrication and dry friction have their respective positions in a three-dimensional space and close to each other with different states in different spatial zones. The multifractal spectrum parameters of the frictional vibrations can be used to identify the friction states of the friction pair.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The friction coefficient data of the three friction state tests. (

**a**) mixed lubrication; (

**b**) boundary lubrication; (

**c**) dry friction.

**Figure 3.**The time-domain waveforms and frequency spectra of frictional vibrations with mixed lubrication. (

**a**) 1st min; (

**b**) 5th min; (

**c**) 10th min; (

**d**) 20th min; (

**e**) 30th min; (

**f**) 40th min; (

**g**) 50th min; (

**h**) 60th min.

**Figure 4.**Typical time-domain waveform and frequency spectrum of frictional vibrations with boundary lubrication.

**Figure 5.**Typical time-domain waveform and frequency spectrum of frictional vibrations with dry friction.

**Figure 6.**Multifractal detrended fluctuation analysis of frictional vibration signal. (

**a**) the qth order fluctuation function; (

**b**) the q-order Hurst exponent; (

**c**) the q-order singularity exponent and singularity dimension; (

**d**) multifractal spectrum.

**Figure 7.**Multifractal spectra of time-domain signals with frictional vibrations in different friction states. (

**a**) mixed lubrication; (

**b**) boundary lubrication; (

**c**) dry friction.

**Figure 8.**Variation law of multifractal spectrum parameters of frictional vibration time-domain signals.

**Figure 9.**Multifractal spectra of friction coefficient data in different friction states. (

**a**) mixed lubrication; (

**b**) boundary lubrication; (

**c**) dry friction

**Figure 11.**Multifractal spectra of frequency-domain signals with frictional vibrations in different friction states. (

**a**) mixed lubrication; (

**b**) boundary lubrication; (

**c**) dry friction.

**Figure 12.**Variation law of multifractal spectrum parameters of frictional vibration frequency-domain signals.

**Figure 13.**The surface topographies of the block specimen. (

**a**) Block specimen initial; (

**b**) End of mixed lubrication; (

**c**) End of boundary lubrication; (

**d**) End of dry friction

**Figure 14.**Spatial distribution of the first three principal components of frictional vibrations in different lubrication states. (

**a**) Spatial distribution of the first three principal components; (

**b**) 1st to 2nd axis plane; (

**c**) 1st to 3rd axis plane; (

**d**) 2nd to 3rd axis plane.

**Figure 15.**Spatial distribution of the first three principal components of frictional vibration time-domain signals in different lubrication states.

Parameters | Block Specimen Initial | End of Mixed Lubrication | End of Boundary Lubrication | End of Dry Friction |
---|---|---|---|---|

S_{a} (μm) | 4.749 | 3.185 | 2.524 | 2.787 |

S_{sk} | −0.693 | −0.51 | −1.376 | −1.833 |

S_{z} (μm) | 63.785 | 54.635 | 36.367 | 37.756 |

Principal Component | Eigenvalues | Percent Variance% |
---|---|---|

The first | 5.8786 | 48.9889 |

The second | 3.4176 | 28.4801 |

The third | 1.7415 | 14.5121 |

Principal Component | Cumulative Percent Variance (CPV)% |
---|---|

The first | 48.9889 |

The first + the second | 77.4690 |

The first + the second + the third | 91.9811 |

**Table 4.**The first three principal components of frictional vibrations in different lubrication states.

Mixed Lubrication | Boundary Lubrication | Dry Friction | ||||||
---|---|---|---|---|---|---|---|---|

The First Principal Component | The Second Principal Component | The Third Principal Component | The First Principal Component | The Second Principal Component | The Third Principal Component | The First Principal Component | The Second Principal Component | The Third Principal Component |

−0.7896 | 2.0636 | 1.9273 | −0.9732 | −0.3196 | −3.6385 | 0.9897 | −1.3864 | 1.9019 |

−1.4366 | 2.2550 | 0.3800 | −0.7220 | −0.0861 | −3.1720 | 1.9564 | −0.8116 | 0.1820 |

−2.1462 | 2.9779 | 1.6202 | −0.9069 | −1.5494 | −1.6859 | 2.2432 | 0.8836 | −0.3994 |

−2.0151 | 3.2316 | −0.4073 | −0.5071 | −2.2932 | −0.0743 | 1.5996 | −0.3241 | 0.5384 |

−2.0641 | 0.9125 | −0.2715 | −0.0323 | −1.4850 | −0.6901 | 3.3515 | 0.7103 | −0.0342 |

−2.7968 | 0.3663 | 0.5866 | −0.1894 | −3.2170 | −1.1709 | 3.3065 | 2.3175 | −1.8440 |

−2.8594 | 0.3683 | 0.7395 | 0.2357 | −2.9844 | −0.7245 | 3.3298 | 1.5562 | 0.5675 |

−2.7226 | 0.1705 | 0.0153 | −0.5426 | −3.2542 | 0.1757 | 2.8816 | 1.5018 | 0.7907 |

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## Share and Cite

**MDPI and ACS Style**

Li, J.-M.; Wei, H.-J.; Wei, L.-D.; Zhou, D.-P.; Qiu, Y.
Extraction of Frictional Vibration Features with Multifractal Detrended Fluctuation Analysis and Friction State Recognition. *Symmetry* **2020**, *12*, 272.
https://doi.org/10.3390/sym12020272

**AMA Style**

Li J-M, Wei H-J, Wei L-D, Zhou D-P, Qiu Y.
Extraction of Frictional Vibration Features with Multifractal Detrended Fluctuation Analysis and Friction State Recognition. *Symmetry*. 2020; 12(2):272.
https://doi.org/10.3390/sym12020272

**Chicago/Turabian Style**

Li, Jing-Ming, Hai-Jun Wei, Li-Dui Wei, Da-Ping Zhou, and Ye Qiu.
2020. "Extraction of Frictional Vibration Features with Multifractal Detrended Fluctuation Analysis and Friction State Recognition" *Symmetry* 12, no. 2: 272.
https://doi.org/10.3390/sym12020272