# A New Second-Order Tristable Stochastic Resonance Method for Fault Diagnosis

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## Abstract

**:**

## 1. Introduction

## 2. STSR System

_{ws}) [28] is a nonlinear symmetric potential, proposed by Woods and Saxon. U

_{ws}can be expressed as follows:

#### 2.1. Tristable Potential

#### 2.2. The STSR

#### 2.3. Influence of Parameters on STSR and Procedure of STSR

_{0}meets the requirements of small parameters, A < 1, f

_{0}< 1. Thus, we can simulate a small parameter signal to explore the effect of noise intensity D on STSR. The simulation signal is defined as:

_{0}= 0.01 Hz and n(t) is the zero mean Gaussian white noise.

_{s}= 5 Hz and the length of the signal N = 5000. The time domain and amplitude spectrum of the signal without noise are shown in Figure 5a. Here, given a set of parameters that can produce SR, a = 0.4, b = 2.7, V = 11, r = 39, c = 20, and k = 0.47. The time domain waveform and amplitude spectrum of the STSR output signal are shown in Figure 5b when D = 0.3, and the frequency 0.01 Hz cannot be seen. Further, adding noise to the input signal, as the noise increases, STSR obtains the output signal, as shown in Figure 5c when D = 0.5. The time domain waveforms can see obviously periodic components, and spectral peaks appear at 0.01 Hz in the amplitude spectrum. When the noise D = 0.8, and the frequency 0.01 Hz cannot be recognized from Figure 5d. It is found that the signal amplitude at the noise intensity D = 0.5 is higher than the amplitude of the signal without noise comparing Figure 5c and Figure 5a. It shows that, as the noise increases, the characteristic components of effective signal increases greatly. On the other hand, the energy of the noise transfers to the signal. However, the characteristic components of signal attenuate, while the noise increase further, which is a typical feature of the SR system.

#### 2.4. Procedure of Proposed Method

_{0}represents the frequency at the highest spectral peak and f

_{s}is the sampling frequency of the signal. SNR can be defined as:

- (1)
- Signal preprocessing. The vibration signal is filtered by band-pass filter to eliminate some noise components. Hilbert transform is used to demodulate the filtered signal, in which the obtained signal is marked as S
_{1}. ${S}_{2}={S}_{1}-mean\left({S}_{1}\right)$, $S=\frac{{S}_{2}}{2\times \mathrm{max}\left(abs\left({S}_{2}\right)\right)}$. S is the input signal of STSR system. - (2)
- Parameter initialization. Setting population size, number of iterations, and range of parameters including a, b, V, R, c, k, and R.
- (3)
- Calculate the objective function value of each location according to Equation (11).
- (4)
- If the number of iterations reaches the set value, the process goes to step (5), otherwise it returns to step (3).
- (5)
- The optimal record of output is the value of a, b, V, r, c, R, and k at the maximum SNR.
- (6)
- Weak signal detection. The optimal parameters are substituted for the STSR system. Additionally, the pre-processed signal is used as the input signal of the STSR to obtain the output signal. Restore the signal amplitude and frequency scale to plot time-domain waveform and amplitude spectrum.

## 3. Simulation Analysis

_{0}= 0.2, f

_{0}= 50 Hz, n(t) is Gaussian white noise, and the intensity is D = 2.4.

_{s}= 5 kHz, and sampling number is 5000. Experimental environments are Intel (R) core (TM) i5-7400 (Beijing, China) CPU 3.00 GHz, 8 G RAM, Windows 10, and MATLAB R2018a. The time-domain waveform and amplitude spectrum of signal are shown in Figure 8a. The periodicity of the signal cannot be seen from the time domain waveform and 50 Hz cannot be discovered from the amplitude spectrum. According to Equation (11), the SNR of the signal with noise is −29.82 dB. The optimal output result of CSR [13] is shown in Figure 8b where the parameters are a = 19.72, b = 7217.256, R = 464.03, and SNR = −20.53 dB. While the SNR is improved, the noise energy is also enhanced. Additionally, the low-frequency components are amplified. Figure 8c shows the optimal output of the frequency-shifted and re-scaling SR [11]. The cutoff frequency of the high-pass filter is set to 35 Hz, the modulation frequency is 40 Hz, and the target frequency is 10 Hz in this method with optimization results are a = 1.86, b = 4794.73, R = 140.97, and SNR = −13.38 dB. It can be seen that the SNR of frequency-shifted and re-scaling SR is further improved than CSR, but the interference frequency f

_{1}of the filter can be seen from Figure 8c. Additionally, the artificially set cutoff frequency of the filter may also affect the stochastic resonance system performance.

_{0}can be clearly seen from the amplitude spectrum, while other frequencies are almost nonexistent. The optimization parameters are a = 10.23, b = 17186.47, k = 0.179, V = 31.56, r = 7.254, c = 6.5, R = 60.508, and SNR = −6.53 dB at this time. Therefore, the proposed method can obtain larger output SNR and better filtering performance than traditional SR methods for detecting weak signals.

## 4. Experimental Verification and Analysis

_{s}is 12 kHz and sampling points are 4096. The fault characteristic frequencies for outer ring and inner ring are expressed by f

_{BPFI}and f

_{BFO}, respectively, that can be theoretically calculated as follows:

_{r}is rotating frequency of shaft, d is ball diameter, D is pitch diameter, and $\alpha $ is the contact angle, respectively.

#### 4.1. Outer Race Fault Signal Detection

_{BFO}= 105.9 Hz with an approximate rotating speed of 1772 r/min. Figure 10a shows the original vibration signal wave and its power spectrum. It is found that the periodic component cannot be seen from time domain and the low frequency signal is modulated to the high frequency band. Firstly, the fault signal is filtered using the band-pass filter at 2000–4000 Hz. Then, Hilbert transform demodulated the filtered signal. Additionally, the enveloped signal and its amplitude spectrum are displayed in Figure 10b. While the characteristic frequency f

_{BFO}can be seen from the amplitude spectrum, and the noise is still obvious in spectrum with output SNR = −15.02 dB.

_{BFO}is amplified and the energy of the rotating frequency is greatly amplified. Then, the enveloped signal is analyzed by the USSSR method. Additionally, the results are exhibited in Figure 10d where a = 0.46, b = 11471, R = 221.5, k = 0.434, and output SNR = −8.357 dB. Finally, the proposed STSR method deals with the same signal and Figure 10e display the time domain waveform and amplitude spectrum of the output signal. The fault characteristic frequency f

_{BFO}can be clearly seen in the whole spectrum diagram where a = 8.282, b = 14244.13, k = 0.434, V = 35.725, r = 15.451, c = 14.328, R = 143.929, and the output SNR = −5.09 dB.

#### 4.2. Inner Race Fault Signal Detection

_{BPFI}= 156.1 Hz with approximate rotating speed of 1730 r/min. Figure 11a shows the time waveform of inner race fault and its power spectrum. It is hard to see the fault characteristic frequency from Figure 11a, due to the strong noise. The origin vibration signal was filtered by a band-pass filter in a bandwidth of 2000–4000 Hz and demodulated by Hilbert transform, then we got the envelop signal which is displayed in Figure 11b with output SNR = −17.53 dB. The CSR method was employed to test the envelop signal and the results are shown in Figure 11c. It can be observed that most of the high frequency noises are suppressed, but low frequency noises are enhanced where a = 0.0015, b = 5235.68, R = 672.536, and output SNR = −15.511 dB. Output results of the USSSR system are shown in Figure 11d with parameters of a = 0.908, b = 12476.8, k = 0.222, and R = 304.395. It can be seen the f

_{BPFI}clearly in the amplitude spectrum and the output SNR is improved to −10.42 dB. The proposed STSR method is utilized to analyze the same signal. The optimal diagnosis result is displayed in Figure 11e where a = 7.534, b = 9801.33, c = 10.97, V = 34.07, R = 206.5, r = 12.71, k = 0.222, and the output SNR = −6.23 dB. It can be seen that fault characteristic frequency f

_{BPFI}is prominent in the whole spectrum, which also demonstrate the effectiveness of the proposed STSR method than the traditional SR methods in bearing fault diagnosis.

#### 4.3. Discussion and Anlysis

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The bistable potential function: (

**a**) When a = 1, the potential model for b changes; (

**b**) when b = 1, the potential model for a changes.

**Figure 5.**Effect of noise intensity D on STSR: (

**a**) Input signal without noise and its amplitude spectrum; (

**b**) STSR output signal and its amplitude spectrum when D = 0.3; (

**c**) STSR output signal and its amplitude spectrum when D = 0.5; and (

**d**) STSR output signal and its amplitude spectrum when D = 0.8.

**Figure 6.**Effect of k on STSR when D = 0.3: (

**a**) STSR output signal and its amplitude spectrum when k = 0.534; (

**b**) STSR output signal and its amplitude spectrum when k = 0.885.

**Figure 7.**Effect of k on STSR when D = 0.8: (

**a**) STSR output signal and its amplitude spectrum when k = 0.375; (

**b**) STSR output signal and its amplitude spectrum when k = 0.235.

**Figure 8.**Analysis results of simulation signal: (

**a**) Original signal and its amplitude spectrum; (

**b**) stochastic resonance system parameters (CSR) output signal and its amplitude spectrum; (

**c**) frequency-shifted and re-scaling SR output signal and its amplitude spectrum; (

**d**) USSSR output signal and its amplitude spectrum; (

**e**) STSR output signal and its amplitude spectrum.

**Figure 10.**Analysis results with outer-race fault signal: (

**a**) Original signal and its power spectrum; (

**b**) envelope signal and its amplitude spectrum; (

**c**) CSR output signal and its amplitude spectrum; (

**d**) USSSR output signal and its amplitude spectrum; and (

**e**) STSR output signal and its amplitude spectrum.

**Figure 11.**Analysis results with inner-race fault signal: (

**a**) Original signal and its power spectrum; (

**b**) envelope signal and its amplitude spectrum; (

**c**) CSR output signal and its amplitude spectrum; (

**d**) USSSR output signal and its amplitude spectrum; and (

**e**) STSR output signal and its amplitude spectrum.

Inner Diameter (Inch) | Outsider Diameter (Inch) | Pitch Diameter (Inch) | Ball Diameter (Inch) | Number of the Rollers |
---|---|---|---|---|

0.9843 | 2.0472 | 1.537 | 0.3126 | 9 |

Fault Type | Envelop Signal SNR (dB) | Output SNR of CSR (dB) | Output SNR of USSSR (dB) | Output SNR of STSR (dB) |
---|---|---|---|---|

Outer-race fault | −15.02 | −13.21 | −8.537 | −5.09 |

Inner-race fault | −17.53 | −15.511 | −10.42 | −6.23 |

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## Share and Cite

**MDPI and ACS Style**

Lu, L.; Yuan, Y.; Wang, H.; Zhao, X.; Zheng, J.
A New Second-Order Tristable Stochastic Resonance Method for Fault Diagnosis. *Symmetry* **2019**, *11*, 965.
https://doi.org/10.3390/sym11080965

**AMA Style**

Lu L, Yuan Y, Wang H, Zhao X, Zheng J.
A New Second-Order Tristable Stochastic Resonance Method for Fault Diagnosis. *Symmetry*. 2019; 11(8):965.
https://doi.org/10.3390/sym11080965

**Chicago/Turabian Style**

Lu, Lu, Yu Yuan, Heng Wang, Xing Zhao, and Jianjie Zheng.
2019. "A New Second-Order Tristable Stochastic Resonance Method for Fault Diagnosis" *Symmetry* 11, no. 8: 965.
https://doi.org/10.3390/sym11080965