# Research on an Adaptive Variational Mode Decomposition with Double Thresholds for Feature Extraction

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## Abstract

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## 1. Introduction

## 2. Basic Methods

#### 2.1. Variational Mode Decomposition (VMD) Method

**Step 1.**Initialize the parameters of VMD, including $\{{\widehat{u}}_{k}^{1}\},$ $\{{w}_{k}^{1}\}$, and $n$.

**Step 2.**Update the ${u}_{k}$ and ${w}_{k}$ according to the expression (9) and expression (12).

**Step 3.**Update the $\lambda $.

**Step 4.**Set the error to $\epsilon >0$. If $\frac{{\displaystyle \sum _{k}||{\widehat{u}}_{k}^{n+1}-{\widehat{u}}_{k}^{n}|{|}_{2}^{2}}}{||{\widehat{u}}_{k}^{n}|{|}_{2}^{2}}<\epsilon $ is met, the iteration is terminated. Otherwise, return to Step 2.

#### 2.2. Empirical Mode Decomposition and Ensemble Empirical Mode Decomposition

_{n}(t) represents residual error function, and IMF components, ${c}_{1}$, ${c}_{2}$, ${c}_{3}$, $\dots $, ${c}_{n}$, contain different elements, respectively, from a low to high frequency of signals.

## 3. An Adaptive VMD with the Center Frequency Method of Double Thresholds

#### 3.1. The Center Frequency Method

#### 3.1.1. The Basic Principle and Implementation Steps

**Step 1.**Determine the maximum value, ${k}_{\mathrm{max}}$, of the decomposition scale.

**Step 2.**The VMD method is used to decompose the original signal into a series of IMF components under different scales ($IM{F}_{k}(k=1,2,3,\cdots ,{k}_{\mathrm{max}})$). The central frequency values of each IMF component are calculated to obtain the set, $\Delta cf{}_{IM{F}_{k}}(k=1,2,3,\cdots ,{k}_{\mathrm{max}})$.

**Step 3.**Calculate the maximum value of the center frequency of the set, $cf{}_{IM{F}_{k}}$, under different scales, as $\Delta cf\_\mathrm{max}{}_{IM{F}_{k}}(k=1,2,3,\cdots ,{k}_{\mathrm{max}})$.

**Step 4.**Calculate the maximum values of the center frequencies of the set, $\Delta cf\_\mathrm{max}{}_{IM{F}_{k}}$, as $\Delta cf\_\mathrm{max}{}_{IMF}$.

**Step 5.**Calculate the difference between the maximum center frequency value of all modes and the maximum center frequency value of the mode under scale, $k$, as $\Delta cf\_{\mathrm{max}}_{IM{F}_{K}}=\Delta c{f}_{\_}{\mathrm{max}}_{IFM}-\Delta c{f}_{\_}{\mathrm{max}}_{IF{M}_{k}}$. If there is $\Delta cf\_{\mathrm{max}}_{k}<<\Delta cf\_{\mathrm{max}}_{IMF}$, the $k$ is the mode decomposition number of the VMD method.

#### 3.1.2. Experimental Analysis

#### 3.2. The Center Frequency Method of Double Thresholds

#### 3.2.1. The Idea of the Center Frequency Method of Double Thresholds

#### 3.2.2. The Flow and Steps of the Center Frequency Method of Double Thresholds

**Step 1.**Determine the maximum value, ${k}_{\mathrm{max}}$, of the decomposition scale.

**Step 2.**The VMD method is used to decompose the original signal into a series of IMF components under different scales ($IM{F}_{k}(k=1,2,3,\cdots ,{k}_{\mathrm{max}})$). The central frequency values of each IMF component is calculated to obtain the set, $cf{}_{IM{F}_{k}}(k=1,2,3,\cdots ,{k}_{\mathrm{max}})$.

**Step 3.**Calculate the maximum value of the center frequency of the set, $cf{}_{IM{F}_{k}}$, under different scales, as $cf\_\mathrm{max}{}_{IM{F}_{k}}(k=1,2,3,\cdots ,{k}_{\mathrm{max}})$.

**Step 4.**Calculate the maximum value of the center frequency of the set, $cf\_\mathrm{max}{}_{IM{F}_{k}}$, as $cf\_\mathrm{max}{}_{IM{F}_{}}$.

**Step 5.**Set thresholds, $a=8\%$ and $b=15\%$, for the expressions (15) and (16).

**Step 6.**Calculate the difference between the maximum center frequency value of all modes, as $\Delta cf\_{\mathrm{max}}_{IM{F}_{k}}=c{f}_{\_}{\mathrm{max}}_{IF{M}_{}}-c{f}_{\_}{\mathrm{max}}_{IF{M}_{k}}(k=1,2,3,\cdots ,{k}_{\mathrm{max}})$.

**Step 7.**If there is $\Delta cf\_{\mathrm{max}}_{k}\le {T}_{1}$, Step 8 is executed. Otherwise, execute $k=k+1$ and go to Step 6.

**Step 8.**Calculate the difference value of the central frequency of the $L\mathrm{th}$ IMF component and the value of the central frequency of the $(L-1)\mathrm{th}$ IMF component of the set, $cf{}_{IM{F}_{}}$ ($k=L$), and is calculated as $\Delta cf{}_{IM{F}_{}}=c{f}_{IM{F}_{{k}_{L}}}-c{f}_{IM{F}_{{k}_{L-1}}}$. If there is $\Delta cf{}_{IM{F}_{}}\le {T}_{2}$, the corresponding $k$ $(k=1,2,3,\cdots ,{k}_{\mathrm{max}})$ of $\Delta cf{}_{IM{F}_{}}$ is the optimal mode number of the VMD method. Otherwise, execute $k=k+1$ and go to Step 6.

#### 3.3. Effectiveness Analysis of the Center Frequency Method of Double Thresholds

## 4. Feature Extraction Method Based on the DTCFVMD and Hilbert Transform

#### 4.1. Feature Extraction Method

#### 4.2. Steps of Feature Extraction

**Step 1**. Preprocess the collected vibration signals of the motor rolling bearing, including de-noising, filtering, and so on.

**Step 2**. Propose a center frequency method of double thresholds to improve the VMD method for obtaining the adaptive VMD(DTCFVMD) method.

**Step 3**. The preprocessed vibration signal is decomposed by using the proposed DTCFVMD method to obtain a series of IMFs.

**Step 4**. Analyze the envelope of each mode component by using the Hilbert transform.

**Step 5**. Calculate the power spectrum of each mode component to extract the fault feature frequency.

**Step 6**. The fault feature frequency is compared with theoretical inherent fault frequency to accurately determine the fault type of the motor rolling bearing.

## 5. Verification and Results Analysis

#### 5.1. The Effectiveness Verification

#### 5.2. Comparison and Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Inside Diameter | Outside Diameter | Thickness | Ball Diameter | Pitch Diameter | Roller Number | Rotating Speed |
---|---|---|---|---|---|---|

25 mm | 52 mm | 15 mm | 8.182 mm | 44.2 mm | 9 | 1797 r/min |

Inner Race | Outer Race | Rolling Element | Switching Frequency |
---|---|---|---|

162.2 (Hz) | 107.3 (Hz) | 141.1 (Hz) | 29.2 (Hz) |

IMF1 (Hz) | IMF2 (Hz) | IMF3 (Hz) | IMF4 (Hz) | IMF5 (Hz) | IMF6 (Hz) | |
---|---|---|---|---|---|---|

$k=4$ | 58.6 | 164.1 | 164.1 | 164.1 | ||

$k=5$ | 58.6 | 164.1 | 164.1 | 29.3 | 29.3 | |

$k=6$ | 58.6 | 164.1 | 164.1 | 164.1 | 29.3 | 29.3 |

IMF1 (Hz) | IMF2 (Hz) | IMF3 (Hz) | IMF4 (Hz) | IMF5 (Hz) | |
---|---|---|---|---|---|

$k=3$ | 87.9 | 105.5 | 46.9 | ||

$k=4$ | 87.9 | 105.5 | 46.9 | 29.3 | |

$k=5$ | 87.9 | 105.5 | 105.5 | 46.9 | 46.9 |

**Table 5.**Frequency values extracted by the power spectrum of each IMF component of the inner race vibration signal.

M1 (Hz) | M2 (Hz) | M3 (Hz) | M4 (Hz) | M5 (Hz) | M6 (Hz) | M7 (Hz) | M8 (Hz) | M9 (Hz) | M10 (Hz) | M11 (Hz) | M12 (Hz) | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

EMD | 164.1 | 164.1 | 164.1 | 29.3 | 23.4 | 41.0 | 5.9 | 5.9 | 5.9 | 5.9 | 5.9 | |

EEMD | 164.1 | 164.1 | 164.1 | 58.6 | 35.2 | 29.3 | 5.9 | 11.7 | 11.7 | 5.9 | 5.9 | 5.9 |

DTCFVMD | 58.6 | 164.1 | 164.1 | 29.3 | 29.3 |

**Table 6.**Frequency values extracted by the power spectrum of each IMF component of the outer race vibration signal.

M1 (Hz) | M2 (Hz) | M3 (Hz) | M4 (Hz) | M5 (Hz) | M6 (Hz) | M7 (Hz) | M8 (Hz) | M9 (Hz) | M10 (Hz) | M11 (Hz) | M12 (Hz) | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

EMD | 105.5 | 105.5 | 111.3 | 87.9 | 17.6 | 23.4 | 11.7 | 11.7 | 11.7 | 5.9 | ||

EEMD | 105.5 | 105.5 | 105.5 | 105.5 | 87.9 | 29.3 | 52.7 | 5.9 | 5.9 | 5.9 | 5.9 | 5.9 |

DTCFVMD | 87.9 | 105.5 | 46.9 | 29.3 |

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## Share and Cite

**MDPI and ACS Style**

Deng, W.; Liu, H.; Zhang, S.; Liu, H.; Zhao, H.; Wu, J.
Research on an Adaptive Variational Mode Decomposition with Double Thresholds for Feature Extraction. *Symmetry* **2018**, *10*, 684.
https://doi.org/10.3390/sym10120684

**AMA Style**

Deng W, Liu H, Zhang S, Liu H, Zhao H, Wu J.
Research on an Adaptive Variational Mode Decomposition with Double Thresholds for Feature Extraction. *Symmetry*. 2018; 10(12):684.
https://doi.org/10.3390/sym10120684

**Chicago/Turabian Style**

Deng, Wu, Hailong Liu, Shengjie Zhang, Haodong Liu, Huimin Zhao, and Jinzhao Wu.
2018. "Research on an Adaptive Variational Mode Decomposition with Double Thresholds for Feature Extraction" *Symmetry* 10, no. 12: 684.
https://doi.org/10.3390/sym10120684