# A Sparsity-Promoted Method Based on Majorization-Minimization for Weak Fault Feature Enhancement

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## Abstract

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## 1. Introduction

## 2. Basic Theory

#### 2.1. Basis Pursuit Algorithm

_{0}norm is the best method to measure the sparsity. Nevertheless, it doesn’t belong to a continuous function and, in this case, the continuous derivative cannot be achieved. To solve this problem, Chen S., Donoho D. L. et al. [23] proposed that utilizing the l

_{1}norm to measure the sparsity, which can be converted in the following equation.

_{1}norm minimization, sparse representation is defined as a series of constrained extremum problems that can be solved by a linear programming approach. Therefore, the aforementioned optimization problems can be expressed as seen in Equation (5).

#### 2.2. Majorzation-Minimization Algorithm

## 3. Weak Fault Feature Enhancing Strategy Using Sparsity-Promoted Method

- (1)
- Collect the vibration signals y using acceleration sensors,
- (2)
- Construct the Majorization iterative function and initialize k = 1, sparse matrix D, penalty factor $\lambda $, and ${\mathsf{\Lambda}}_{k}=diag\left(\left|{\alpha}_{k}\left(n\right)\right|\right)$,
- (3)
- According to Equation (18), carry out the Majorization iterative steps and update the iterative number k←k + 1.
- (4)
- Obtain sparse signal x
_{k}_{+1}, which only contains the transient components. - (5)
- Perform envelope analysis and extract the fault characteristic frequency.

## 4. Application Cases

#### 4.1. Simulation Analysis

_{s}= 10 kHz, the system resonance frequency is f

_{n}= 3 kHz, the initial phase angle is ${\varphi}_{0}=5$ rad, and the relative damping ration is $\xi $ = 0.1. T

_{k}represents the trigger time of the kth impulse and ∆T

_{k}= 0.01 is the time between the (k − 1)th and kth impulse in which fault characteristic frequency is 100 Hz. In order to simulate the background noise, Gaussian noise v(t) is added to the simulated signal and the SNR is −5 dB [28].

#### 4.2. Experimental Verification and Discussion

#### 4.2.1. Test Rig

#### 4.2.2. Detection of the Bearing Fault in the Outer Race

#### 4.2.3. Detection of the Bearing Fault in the Inner Race

#### 4.2.4. Detection of the Gearbox Fault with Broken-Tooth

#### 4.3. Comparison with the Traditional MM Algorithm

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 4.**Simulated signals: (

**a**) time-domain waveform of simulated signal, (

**b**) envelope spectrum of simulated signal, (

**c**) simulated signal with noise, and (

**d**) envelope spectrum of signal with noise.

**Figure 5.**Detection results of simulated signals using the proposed method: (

**a**) waveform of sparse signal and (

**b**) envelope spectrum of the sparse signal.

**Figure 6.**Experimental system, (

**a**) experimental table, (

**b**) gear broken-tooth fault, (

**c**) bearing outer-race fault, and (

**d**) bearing inner-race fault.

**Figure 7.**Bearing signals: (

**a**) time-domain signal with outer-race fault, (

**b**) spectrum of the time-domain signal, and (

**c**) envelop spectrum of the time-domain signal.

**Figure 8.**Detection results of outer-race fault using the proposed method: (

**a)**waveform of sparse signal and (

**b**) envelope spectrum of the sparse signal.

**Figure 9.**Bearing signals: (

**a**) time-domain signal with inner-race fault, (

**b**) spectrum of the time-domain signal, and (

**c**) envelope spectrum of the time-domain signal.

**Figure 10.**Detection results of inner-race fault using the proposed method: (

**a**) waveform of sparse signal and (

**b**) envelope spectrum of the sparse signal.

**Figure 11.**Gearbox signals: (

**a**) time-domain signal with tooth-broken fault and (

**b**) envelope spectrum of the time-domain signal.

**Figure 12.**Detection results of tooth-broken fault using the proposed method: (

**a**) waveform of sparse signal and (

**b**) envelope spectrum of the sparse signal.

**Figure 13.**Detection results of outer-race fault signals using the traditional method: (

**a**) Optimal wavelet basis, (

**b**) sparse coefficients, (

**c**) reconstructed signal, and (

**d**) envelope spectrum of the reconstructed signal.

**Figure 14.**Detection results using the traditional method: (

**a**) envelope spectrum of simulated sparse signal, (

**b**) envelope spectrum of the inner-race sparse signal, and (

**c**) envelope spectrum of the gearbox sparse signal.

Number of Rollers | External Diameter (mm) | Inner Diameter (mm) | Width (mm) |
---|---|---|---|

11 | 47 | 20 | 14 |

Research Object | Number of Teeth | Rotating Period (s) | Rotating Frequency (Hz) | Meshing Frequency (Hz) |
---|---|---|---|---|

Driving Gear I | 80 | 3.448 | 0.29 | 23.15 |

Driven Gear II | 19 | 0.824 | 1.214 | |

Driving Gear II | 80 | 0.824 | 1.214 | 97.13 |

Driven Gear III | 17 | 0.175 | 5.714 |

Fault Category | Outer Race | Inner Race | Broken-Tooth |
---|---|---|---|

Fault characteristic frequency (Hz) | 86.32 | 145.84 | 5.714 |

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## Share and Cite

**MDPI and ACS Style**

Ren, B.; Hao, Y.; Wang, H.; Song, L.; Tang, G.; Yuan, H. A Sparsity-Promoted Method Based on Majorization-Minimization for Weak Fault Feature Enhancement. *Sensors* **2018**, *18*, 1003.
https://doi.org/10.3390/s18041003

**AMA Style**

Ren B, Hao Y, Wang H, Song L, Tang G, Yuan H. A Sparsity-Promoted Method Based on Majorization-Minimization for Weak Fault Feature Enhancement. *Sensors*. 2018; 18(4):1003.
https://doi.org/10.3390/s18041003

**Chicago/Turabian Style**

Ren, Bangyue, Yansong Hao, Huaqing Wang, Liuyang Song, Gang Tang, and Hongfang Yuan. 2018. "A Sparsity-Promoted Method Based on Majorization-Minimization for Weak Fault Feature Enhancement" *Sensors* 18, no. 4: 1003.
https://doi.org/10.3390/s18041003