Pulses caused by rotating mechanical faults are weak and often submerged in strong background noise, which can affect the accuracy of fault detection. To solve this problem, we study the stochastic resonance phenomenon of a tristable potential system based on strong noise background and also investigate the influence of time-delayed feedback on this stochastic resonance model. The effects of time-delayed feedback strength on potential energy, steady-state probability density function, and signal-to-noise ratio (SNR) are discussed. The results show that stochastic resonance can be enhanced or suppressed by adjusting the delay time and feedback strength. Combined with bearing fault diagnosis simulation research and experimental verification evaluation, the proposed time-delayed feedback tristable stochastic resonance fault diagnosis method is more effective than the classical stochastic resonance method.
In the early fault diagnosis of rotating machinery, the fault signal is often weak, and the working environment is mostly in the background of strong noise [
To solve the above problems, scholars have done considerable research on noise filtering and improvement of the signal-to-noise ratio (SNR) of useful signals and have proposed various methods. Traditional signal processing methods include wavelet analysis [
Most of the above scholars’ research is directed toward the stochastic resonance of the classical bistable model. Classical bistable stochastic resonance has a single structure and cannot form a richer potential structure. The potential model cannot match the complex vibration signal, which limits the enhancement ability of weak signals. In order to further improve the extraction effect of stochastic resonance, some research scholars proposed the tristable stochastic resonance potential model. For example, Lu et al. [
The rest of the paper is organized as follows: In Section
The process of SR detection of weak signals can be described by particle motion in the potential well. When the external drive is the periodic signal
Relationship between the three steady-state potential functions and
The classic stochastic resonance method does not take into account the influence of historical information on the system, and it focuses on a short-term memory system. The historical information is introduced into the negative feedback of stochastic resonance to form a long-term memory system, and the detection effect of the weak signal can be improved. This long-term memory system is time-delayed feedback stochastic resonance. The classic bistable time-delayed system is as follows [
Equation (
Furthermore, obtaining an effective Langevin equation relative to equation (
Without considering the periodic signal and Gaussian noise, the equivalent time-delayed tristable potential corresponding to equation (
The effective potential function graph is shown in Figures
Potential energy and
The time-delayed feedback stationary probability density function is expressed as
Relationship between the stationary probability density function and
Then, the power spectral density function can be derived as
In equation (
The SNR is given by
Taking
Figure
Figures
Figure
Delay feedback tristable stochastic resonance SNR: effect of (a)
Because stochastic resonance theory is implemented under adiabatic approximation conditions, the above derivation is limited to small-parameter limits. In other words, the required frequency is much less than 1 Hz [
In this paper, the ant colony algorithm is inspired by the behavior of ants searching for food in nature, and it is a group intelligent optimization algorithm. The ant colony algorithm is based on the study of the collective foraging behavior of real ant colonies in nature, simulating the real ant colony collaboration process. As a general stochastic optimization method, the ant colony algorithm [
To verify the effect of the time-delayed feedback tristable stochastic resonance weak fault diagnosis method, the effect of the proposed method is analyzed by simulating the bearing fault signal. The simulated fault signal is as follows:
Bearing simulation: (a) noiseless signal, (b) noisy signal, (c) spectrum, and (d) envelope spectrum.
Figures
Bearing simulation. (a) Time-domain waveform and (b) spectrum of the time-delayed feedback tristable stochastic resonance method. (c) Time-domain waveform and (d) spectrum of the classical stochastic resonance method.
To verify the effectiveness of the time-delayed feedback tristable stochastic resonance method, the proposed method is applied to a fault characteristic frequency extraction experiment of slightly worn rolling bearings. Rolling bearings are an important part of rotating machinery and one of the more easily damaged parts. Therefore, a fault signal extraction experiment with slight wear of the inner ring of the rolling bearing is used to verify the effect of the proposed method. The test bench is a comprehensive experimental bench for mechanical equipment failure, as shown in Figure
Mechanical equipment failure comprehensive test bench.
Rolling bearing signal: (a) time-domain waveform, (b) spectrum, and (c) envelope spectrum.
Rolling bearing signal. (a) Time-domain waveform and (b) spectrum of the time-delayed feedback tristable stochastic resonance method. (c) Time-domain waveform and (d) spectrum of the classical stochastic resonance method.
Based on the proposed method, the robust results are verified in the above experiment. The effect of the time-delayed tristable stochastic resonance in extracting the bearing weak fault signal is better than that of the classical stochastic resonance method, and the extracted fault features are more obvious. The noise interference is smaller. We apply the proposed method to the fault feature extraction experiment of the bearing inner ring of a steel mill to further verify the effectiveness of the proposed method. The experimental bearing is shown in Figure
Fault feature extraction of the bearing inner ring of a steel mill.
Engineering bearing signal. (a) Time-domain waveform, (b) spectrum, and (c) envelope spectrum.
Engineering bearing signal. (a) Time-domain waveform of the time-delayed feedback tristable stochastic resonance method. (b) Spectrum of the time-delayed feedback tristable stochastic resonance method. (c) Time-domain waveform of the classical stochastic resonance method. (d) Spectrum of the classical stochastic resonance method.
The above analysis shows that the time-delayed tristable stochastic resonance system can get better signal output. From physical analysis, the time-delayed tristable stochastic resonance system has three potential wells and adjusts the potential structure with three parameters. Compared with the classical bistable stochastic resonance, the time-delayed tristable potential model can obtain a richer potential structure and match the complex vibration signal to achieve better stochastic resonance effect. Moreover, introducing a delay term in the potential model, the potential model can change the external driving energy to obtain a stable particle transition between the potential wells and finally get the best signal-to-noise ratio, which is the best signal enhancement effect. Therefore, it can be concluded that the signal output is better than the classical bistable stochastic resonance due to the existence of the tristable model and the delay term.
In this paper, we have studied the time-delayed feedback tristable stochastic resonance system. A method for time-delayed feedback tristable stochastic resonance weak faults is proposed. The main conclusions are as follows: The optimal potential energy, steady-state probability density function, and SNR can be obtained by adjusting the time-delayed and feedback strength. In other words, the optimal stochastic resonance effect can be obtained by adjusting the delay term. A new time-delayed feedback tristable stochastic resonance model is established. Compared with the classical bistable stochastic resonance model, the proposed model can obtain a richer structure by adjusting the system parameters, thus achieving matching with complex vibration signals. A new weak fault diagnosis method was proposed. The proposed method can not only extract the simulation and experimental signals of bearing faults, but also effectively extract the fault signals in the project. Compared with the classical stochastic resonance method, the weak fault features extracted by the proposed method have a good recognition.
The experiment is carried out on the mechanical failure comprehensive simulation test platform as shown in Figure
The authors declare that they have no conflicts of interest.
This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grants 51805275, 51865045, and 51565046 and in part by the Inner Mongolia Natural Science Foundation under Grant 2018ZD06.