Feature Extraction Method for Weak Faults Based on Time-Delayed Feedback Mixed Potential Stochastic Resonance

To extract weak faults under strong noise, a method for feature extraction of weak faults with time-delayed feedback mixed potential stochastic resonance (TFMSR) is proposed. -is method not only overcomes the saturation characteristics of classical bistable stochastic resonance (CBSR), but also verifies a new potential function model. Based on this model, considering the short memory characteristics of the CBSRmethod, a method is proposed that can add historical information to the negative feedback process of the stochastic resonance (SR). -rough the combination of the above two methods, the weak fault extraction under strong background noise is realized. -e article analyzes the effects of the delay term, feedback term, and system parameter on the effect of SR and uses the ant colony algorithm (ACA) to optimize the above parameters. Finally, through simulated and engineering experimental results, it is proved that the proposed method has more advantages than the CBSR method in weak fault feature extraction.


Introduction
With the development of technology, equipment becomes large scale and intelligent with high speed [1,2]. Failure of any component in the system can jeopardize the operational safety of the entire device. erefore, early prediction of weak faults in mechanical systems is particularly important for the extraction of the weak fault feature. It is an emerging science to study how to extract weak signals from strong noise. It requires multiple disciplines to work together, so it is also an interdisciplinary [3,4].
Benzi et al. [5] first proposed SR when explaining the problem of ancient glaciers. It transfers some parts of the noise energy to the weak signal. While enhancing the faint fault characteristics, some noise becomes weakened, thereby improving the signal-to-noise ratio (SNR). Existing noise reduction methods, such as wavelet analysis [6][7][8], empirical mode decomposition (EMD) [9], variational mode decomposition [10], and dictionary learning [11,12], can weaken the useful weak signal when filtering the noise. As a signal processing tool, SR has an interesting feature; that is, the noise added into the nonlinear system can excite and amplify the early weak fault features. Utilizing this principle, Yang et al. [13] obtained an amazing result. ey recover unknown signals completely submerged in strong noise through a new SR method. Due to this unique characteristic of SR, it has attracted wide attention in the fields of physics [14], medicine [15], and machinery.
Initially, SR theory suffers from the small frequency limitation under adiabatic approximate conditions [16]. erefore, it cannot be applied in actual engineering. In order to make SR universal [17,18], many researchers have made their own contributions. Leng et al. [19] studied a recalibration of the frequency SR method to achieve a large parameter signal SR. Tan et al. [20] used frequency shift variable scale processing SR to alleviate the contradiction between the number of sample points and sampling frequency. Later, the CBSR model became the most widely used SR model. Li et al. [21] proposed a Morlet wavelet transform CBSR method that can obtain a finer frequency, which is used to extract wind turbine fault characteristics. Lei et al. [22] proposed an adaptive SR method for mechanical fault diagnosis. He et al. [23] proposed a new multiscale noise adjustment method to improve SR performance. Lu et al. [24] proposed a multiscale noise adjustment SR method, which was proved in the detection of actual sound signals from train bearings. Shi et al. [25] found that feedback parameters, time-delay parameters, and asymmetric parameters all have an effect on SR, and the time-delay parameters are more sensitive. Li and Shi [26] proposed an SR with a bistable constrained potential and applied to weak fault feature extraction of bearings. Nishiguchi and Fujiwara [27] suggested boosting an effect of CBSR and silhouette noise from noise via a field-effect transistor. e above method mainly uses the CBSR method to lay a foundation for the rapid development of SR. Later, Zhao et al. [28] proved the existence of the output saturation characteristics of the CBSR system. Gosak et al. [29] studied the negative effects of output saturation on the system output in an ultradamped oscillator. Zhang et al. [30] studied SR output saturation based on the additive and multiplicative noises. ese studies have recognized the effect of output saturation characteristics on SR, but have not proposed an effective solution. When saturation occurs, the amplitude of the output signal will not increase as the amplitude of the input signal increases. is makes it difficult to extract weak fault characteristics. erefore, this paper proposes a mixed potential SR model to solve the saturation problem of the CBSR model. Further, the CBSR system is also a short-memory system, without considering the effects of time delay and feedback.
is makes that the output of the system depend on the output of the previous one. Lv et al. [31,32] studied the special role of delay in nonlinear systems. Zhuo et al. [33,34] studied the application of memory and feedback models in fault diagnosis. In summary, adding historical information to the system's feedback process in a certain form will greatly improve the output signal-to-noise ratio (SNR). is paper proposes a method for extracting weak fault features based on TFMSR. is method not only overcomes the saturation characteristics of the CBSR, but also adds historical information to the system's feedback process in a certain form, effectively solving the two major shortcomings of the CBSR method. rough simulated and experimental results, it proves that the proposed method has more advantageous than the CBSR method in weak fault feature extraction.

Theoretical Foundation
where s(t) � A cos(Ωt) is a weak periodic signal, A is the amplitude, and ξ(t) is a zero mean Gaussian white noise. e following condition is satisfied: where D is the noise intensity. e potential function of the CBSR system is expressed as where a and b represent the system parameters. In Figure 1, is the potential function curve of the CBSR model.
Without loss of generality, we set a � 1 and b � 1. Random boost is related to the specific nonlinear form of the potential function. By selecting different nonlinear potential functions and adjusting related parameters, the random boost can be effectively enhanced. erefore, in order to avoid the output saturation characteristic of the CBSR potential function, we propose a mixed potential stochastic resonance model. It is expressed as where m is the system parameter. e mixed potential function model curve U(x) and the CBSR potential function curve U 1 (x) are shown in Figure 1. e potential function of the mixed potential model is wider than that of CBSR, which reflects that the proposed method has more advantages in antisaturation characteristics than CBSR [30].
Considering the effects of the delay term and the feedback term, the Langevin equation is obtained as follows: e probability density of Fokker-Planck with approximate delay feedback can be expressed as [35] zP( where h n is the conditional average drift. h n can be obtained [36] as Without considering the effects of periodic signals, equation (7) can be deduced as Figure 1: e mixed potential function model curve U(x) and the CBSR potential function curve U 1 (x) (a � 1, b � 1).

Shock and Vibration
Compared with the CBSR, the potential well of TFMSR is affected by the system parameter m, the feedback term K, and the delay term σ. As shown in Figure 2, by adjusting the values of m, K, and σ, we can observe that the shape of the potential well is constantly changing.
In Figures 2(a) and 2(b), as the feedback intensity K decreases from 1.1 to 0.7, the height of the potential barrier remains unchanged, but the bottom of the potential well moves slowly upward. In Figures 2(c) and 2(d), by adjusting the delay term σ, the depth of the potential well continuously changes. As σ increases from 0.4 to 1.2, the steepness of the potential wall increases. In Figures 2(e) and 2(f ), as m decreases from 1 to 0.6, the steepness of the potential wall does not change, but the depth of the potential well increases.
According to the theory of the two-state model [37], the probability density of particles moving between two potential wells can be expressed as Under small delay condition, equation (9) can be deduced as Under adiabatic approximation [38], the power spectral density expression is where the output power spectral density of the signal can be expressed as e power spectral density of the noise can be expressed as e output SNR of the system can be expressed as [39] SNR � In Figure 3(a), the system parameter m and delay term σ remain unchanged. As the noise intensity D increases, a single peak appears in the output SNR. As analyzed in Section 2, as K decreases, the depth of the potential well is reduced. e energy required for a particle to transition from one potential well to another decreases, the output signal becomes larger, and the effect of SR is obvious. In Figure 3(b), the system parameter m and the feedback intensity K remain unchanged. When σ � 0.4, because σ is small, the particles cannot break through the barrier of the potential and realize the transition from one potential well to another. e effect of SR is not obvious. Increasing σ to 1.2, the particles have enough energy to cross the potential barrier. At this time, the particles oscillate back and forth between the two potential wells. In Figure 3(b), we can see that the effect of SR is obvious. In Figure 3(c), the feedback intensity K and the delay term σ remain unchanged. When the system parameter m � 0.6, the potential wall is too steep, and the particles cannot obtain sufficient acceleration to break through the barrier constraint. Increasing m to 1, the potential well depth decreases, and the potential function structure can form the best match with the periodic force, so as to achieve the best SR effect.

Proposed Model
rough the above analysis, we can conclude that the effect of SR is most obvious when the delay term, feedback intensity, and system parameters are optimally matched. is article uses ACA to match and optimize the above parameters. e detection strategy of the TFMSR method is as follows: (1) Signal Preprocessing. First, the acquired signal is demodulated by the Hilbert transform, and then the small parameters (with signal amplitude, noise intensity, and signal frequency all being far less than 1) required for SR are obtained by frequency shift variable scale processing [20]. (2) Parameter Initialization and Optimization. First, initialize the parameters and optimize delay term, feedback intensity, and system parameter. e range of parameter optimization is set to [0, 5]. (3) Calculating the SNR. e optimal match of parameters are inputted into the SR system. e maximum output SNR of the TFMSR method is calculated by the Runge-Kutta equation [40]. (4) Postprocessing.
rough the postprocessing of the signal by the TFMSR system, the faint fault features are extracted. Figure 4 shows the detection strategy of the TFMSR method.

Simulation
To verify the effort of the TFMSR method, we decided to conduct a simulation experiment. First, a periodic signal with Shock and Vibration  Figure 6(b) shows that the spectral peak value A max obtained by using the TFMSR method is 0.07659. e value of the second highest point is 0.03546. e difference value ΔA is 0.04113. We propose a new evaluation index to evaluate the effectiveness of the two methods. We use ΔA/A max to get a dimensionless constant. e value of the dimensionless constant is 0.537.
For comparison, we will use the CBSR method to process the simulated signal. e initial parameters are settled consistent with the TFMSR method. e obtained parameters are a � 0.0369 and b � 0.117. e parameters are substituted into the CBSR system. Figures 6(c) and 6(d) show the time domain and spectrum of the simulated signal. In Figure 6(d), the spectral peak value A 1max is 0.04329. e value of the second highest point is 0.03364. e difference value ΔA 1 is 0.00965. ΔA 1 /A 1max � 0.2229. In summary, the proposed TFMSR method can enhance weak signals effectively.

Engineering Experiment Verification
In the field of metallurgy, rolling mills are widely used. Due to the harsh working environment, the fault signals of the components in the equipment are easily overwhelmed by strong noise. Some minor failures, if cannot be found in   Shock and Vibration time, may create serious consequences over time. e hot rolling production site of a thin plate factory is shown in Figure 7. e working environment of the equipment is very bad. rough routine inspections, we found that the gearbox of the rolling mill failed, but it was not possible to determine which component was faulty. e on-site signal acquisition device is the ZonicBook/618E, and the sampling position is shown in Figure 8. Table 1 lists the basic parameters of the gearbox. e acquired signal is processed to obtain the original signal, spectrum, and envelope spectrum as shown in Figures 9(a), 9(b), and 9(c), respectively. Due to the strong background noise, we were unable to clearly distinguish the fault characteristics and the specific location of the damaged components. In order to find the specific location of the fault from the strong background noise, the proposed TFMSR method is used to process the signal. e obtained parameters are σ � 1.0691, K � 0.1048, and m � 0.8175. Figures 10(a) and 10(b) show the time domain waveform and the spectrum obtained by the acquired signal. Figure 10(b) shows that the fault characteristic frequency is 25 Hz. By comparing the characteristic frequency calculated in Table 1, we can see the characteristic frequency of the bearing 1. e characteristic frequency of the inner ring is 24.3 Hz. erefore, we boldly conclude that the inner ring of the bearing 1 has failed. Further, we use CBSR method to process the signal. e obtained parameters are a � 0.0708 and b � 0.195.  Figure 10(d) shows that the fault characteristic frequency is 27.6 Hz. By comparing with the data in Table 1, we cannot find the characteristic frequency which is close to 27.6 Hz. is further proves the advancement of the proposed TFMSR method. To finally confirm the fault location, we use the endoscope to check the inside of the gearbox. It was finally confirmed that cracks occurred in the inner ring of the bearing 1, and the crack is shown in Figure 11. is is in complete agreement with our speculation, further validating the usability of the TFMSR method.
From the above analysis, we can conclude that the TFMSR method has a significant detection effect at l-speed single faults. However, the availability of this method has yet to be verified in the face of weak fault detection under highspeed and multifault conditions. is will also be our next research work.    Shock and Vibration 9

Conclusion
Considering the saturation characteristics of the CBSR model, this paper explores a new potential function model. Because of the short memory characteristics of the CBSR method, we add historical information to the feedback process of the SR. rough the combination of the above two methods, the weak fault extraction under strong background noise is realized. Based on the theoretical analysis and experimental results, the main conclusions are as follows: (1) A potential function model with antisaturation characteristics is proposed, and the characteristics of the potential function model are analyzed. (2) Considering the short memory characteristics of the CBSR method, a delay term and a feedback term are introduced on the basis of the proposed potential function model, and the output SNR is theoretically derived. e analysis of the effect of parameters on the output SNR is completed. (3) e proposed method is used to simulate bearing fault extraction. Simulated results show that the proposed method effectively extracts fault features and is superior to the CBSR method. (4) e proposed TFMSR method was used to verify the failure of the rolling mill gearbox. e experimental results show that the TFMSR method has better antisaturation characteristics and feedback characteristics than the CBSR method in extracting weak fault characteristics.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.
Acknowledgments is work was supported in part by the National Natural Science Foundation of China (51805275) and in part by the Beijing University of Science and Technology Innovation Talents Fund.  10 Shock and Vibration