# Controlled Symmetry with Woods-Saxon Stochastic Resonance Enabled Weak Fault Detection

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## Abstract

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## 1. Introduction

## 2. The Proposed CSwWSSR

#### 2.1. Introduction of CSwWSSR

#### 2.2. Optimization of CSwWSSR Parameters

## 3. Simulation

#### 3.1. Output Analysis of Simulation Signal

#### 3.2. Capability of Detecting Different Simulation Signals

## 4. Experimental Demonstration of Bearing

## 5. Conclusions

- We analyze the potential functions of WSSR, CSSR, and CSwWSSR to investigate the effects of parameters on the potential structure and dynamical properties. The intermediate potential well of the CSwWSSR is discovered to be controlled by the parameters (H, W, a), which have the same impact on the potential structure as the WSSR, and the potential wells on both sides depend on the parameter k. The CSwWSSR parameters are transparent to the control of the potential structure, which means that each parameter controls a different aspect of the potential structure.
- The output SNR curves of WSSR, CSSR, and CSwWSSR were examined for various noise intensities and various fault characteristic frequencies, demonstrating how well CSwWSSR combines the benefits of WSSR and CSSR both in terms of anti-noise and enhancement of high frequency signals. It has both the features of stable particle motion in WSSR and the controlled adjustment of potential wells on both sides in CSSR.
- We compare the output signal time domain and frequency domain diagrams of the WSSR, CSSR, and CSwWSSR after PSO optimized parameters in the simulation signal and bearing experiment. In terms of output SNR and amplitude of fault characteristic frequency, it is apparent that CSwWSSR outperforms CSSR and WSSR, which indicates that CSwWSSR has robust engineering applicability in the future.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 8.**The output signal time-frequency diagram of each model in simulation signal. (

**Left**column: Time domain diagram;

**Right**column: Frequency domain diagram).

**Figure 11.**The output signal time-frequency diagram of each model in simulation signal. (

**Left**column: Time domain diagram;

**Right**column: Frequency domain diagram).

**Table 1.**Parameter combination and output SNR of each model after optimization of simulation signal.

Model | H | W | a | k | SNR/dB |
---|---|---|---|---|---|

WSSR | 8.5624 | 6.2607 | 3.5634 | - | −10.16 |

CSSR-depth | - | - | - | 0.0417 | −10.85 |

CSSR-width | - | - | - | 1.6886 | −11.63 |

CSwWSSR-depth | 5.5987 | 5.1328 | 0.6898 | 0.0015 | −9.863 |

CSwWSSR-width | 1.6607 | 2.3355 | 0.1399 | 3.3592 | −9.777 |

Inner Diameter/mm | Outer Diameter/mm | D${}_{1}$/mm | n${}_{1}$ | d${}_{1}$/mm | Contact Angle/(°) |
---|---|---|---|---|---|

25.001 | 51.999 | 39.040 | 9.000 | 7.940 | 0 |

Model | H | W | a | k | SNR/dB |
---|---|---|---|---|---|

WSSR | 8.2734 | 3.8796 | 0.7568 | - | −11.21 |

CSSR-depth | - | - | - | 0.2897 | −9.813 |

CSSR-width | - | - | - | 1.1389 | −9.911 |

CSwWSSR-depth | 3.4026 | 3.4026 | 0.0367 | 0.1631 | −9.702 |

CSwWSSR-width | 3.2067 | 2.2868 | 0.0164 | 1.7698 | −8.996 |

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**MDPI and ACS Style**

Liu, J.; Guo, J.; Hu, B.; Zhai, Q.; Tang, C.; Zhang, W.
Controlled Symmetry with Woods-Saxon Stochastic Resonance Enabled Weak Fault Detection. *Sensors* **2023**, *23*, 5062.
https://doi.org/10.3390/s23115062

**AMA Style**

Liu J, Guo J, Hu B, Zhai Q, Tang C, Zhang W.
Controlled Symmetry with Woods-Saxon Stochastic Resonance Enabled Weak Fault Detection. *Sensors*. 2023; 23(11):5062.
https://doi.org/10.3390/s23115062

**Chicago/Turabian Style**

Liu, Jian, Jiaqi Guo, Bing Hu, Qiqing Zhai, Can Tang, and Wanjia Zhang.
2023. "Controlled Symmetry with Woods-Saxon Stochastic Resonance Enabled Weak Fault Detection" *Sensors* 23, no. 11: 5062.
https://doi.org/10.3390/s23115062