Shock and Vibration

Volume 2019, Article ID 2097164, 13 pages

https://doi.org/10.1155/2019/2097164

## Time-Delayed Feedback Tristable Stochastic Resonance Weak Fault Diagnosis Method and Its Application

School of Mechanical Engineering, University of Science & Technology Inner Mongolia, Baotou, China

Correspondence should be addressed to Jianguo Wang; nc.tsumi@cykgjw

Received 18 January 2019; Revised 7 April 2019; Accepted 17 April 2019; Published 9 June 2019

Academic Editor: Stefano Marchesiello

Copyright © 2019 Zhixing Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1–3]. For example, in the early fault diagnosis of rolling bearings, the fault vibration signal itself is very weak, and it can easily be submerged in the surrounding noise, which causes interference in the identification of fault features [4]. Therefore, how to detect and extract weak fault signals from a background of strong noise is especially important for fault diagnosis of rotating machinery.

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 [5, 6], empirical mode decomposition [7, 8], Hilbert transform [9, 10], and singular value decomposition [11, 12]. Because noise plays a negative role in most cases, traditional methods mostly focus on filtering out noise. However, early fault signals tend to be weak, so filtering of useful signals can easily occur in the process of filtering out noise, which is very unfavorable to the detection results [13, 14]. In fact, noise is not harmful in all cases, and sometimes, it can enhance the strength of the useful signal, which is conducive to signal extraction and detection. For example, the ensemble empirical mode decomposition method adds a moderate amount of noise to each empirical mode decomposition to obtain multiple intrinsic mode functions and performs a lumped average for all intrinsic mode functions [15, 16]. Finally, the chaotic phenomenon of traditional empirical mode decomposition is solved, and the participation noise is reduced to a very low level [17]. The stochastic resonance used in this paper is another way to detect weak fault signals using noise. Stochastic resonance is the introduction of noise and periodic signals in nonlinear systems. With the aid of noise, the amplitude and power of the weak signal increase, and the particles transition in the potential well, enabling the detection of weak signals. Stochastic resonance was first proposed by Benzi et al. [18] and used in ancient meteorological problems and then used by scholars in various fields of research. In the field of fault diagnosis, early stochastic resonance research is mainly aimed at overcoming the small-parameter limitation problem. In other words, according to approximate adiabatic theory, the classical stochastic resonance theory requires the input signal frequency to be much less than 1. However, the characteristic frequencies in mechanical systems in practical applications are usually in the range of tens, hundreds of hertz, to several kilohertz, far exceeding the frequency range required by classical stochastic resonance theory [19]. To solve this problem, scholars use the frequency-shifted and scaling transform method to solve the small-parameter problem to achieve large-parameter stochastic resonance [20, 21]. Subsequently, in order to further study stochastic resonance, He et al. [22] proposed a new multiscale noise-tuning method that overcomes the limitations of the small parameters of classical stochastic resonance and improves the performance of classical stochastic resonance by using multiscale noise. Lei et al. [23] proposed an adaptive stochastic resonance method that utilizes the optimization ability of the ant colony algorithm to make more accurate diagnosis of faults. Lu et al. [24] proposed a sequence algorithm based on a multiscale noise-tuned stochastic resonance method to achieve signal demodulation, multiscale noise tuning, and bistable stochastic resonance sequences. Li and Shi [25] proposed a fault diagnosis method for rolling bearings based on strong background noise. This method not only overcomes the difficulty of selecting sensitive intrinsic mode function but also enhances the weak fault feature by combining it with adaptive stochastic resonance. Shi et al. [26] proposed a weak signal detection method based on adaptive stochastic resonance and analytical mode decomposition-ensemble empirical mode decomposition. This method can not only enhance the signal amplitude but also effectively detect the submerged weak multifrequency signal. Qiao et al. [27] proposed an adaptive unsaturated bistable stochastic resonance method that solved the problem of inherent output saturation in classical stochastic resonance.

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. [28] proposed a new method for enhancing the periodic fault signal of a rotating mechanical vibration using a tristable mechanical vibration amplifier. Lu et al. [29] proposed a new stochastic resonance method based on the Woods–Saxon potential model and used it for fault diagnosis of bearings. In summary, in most of the studies, although numerous tristable stochastic resonances and other models were studied, the effect of delay feedback on stochastic resonance was not considered. However, some scholars have studied the delay stochastic resonance. For example, Li et al. [30] proposed a weak signal detection method based on a time-delayed feedback monostable stochastic resonance system and adaptive minimum entropy deconvolution; this method achieves resonance detection of weak signals by selecting an appropriate time delay, feedback strength, and rescale ratio combined with a genetic algorithm. Lu et al. [31] proposed a nonstationary weak signal detection method based on a time-delayed feedback stochastic resonance model. This method is more suitable for detecting signals with strong nonlinear and nonstationary properties, as well as signals subjected to severe multiscale noise interference. Aiming at the problems existing in the above research, in this paper, we propose a new time-delayed feedback tristable stochastic resonance method and apply it to the diagnosis of weak bearing faults. The periodic signal and Gaussian white noise are introduced into the nonlinear tristable potential well, and the delay term (including feedback strength and delay time) is introduced. In the new potential system, through the adjustment and optimization of the potential function parameters and the delay term parameters, the potential function, the periodic signal, and the noise are optimally coupled to obtain the optimal SNR. Compared with the classical stochastic resonance method, the SNR is higher, the fault characteristics are more apparent, and the output effect is better than that of the classical stochastic resonance method.

The rest of the paper is organized as follows: In Section 2, the characteristics of the potential model are analyzed and the influence of each parameter on the potential model is discussed. In addition, a delay feedback term is introduced, the Fokker–Planck equation is given, and the SNR is derived. In Section 3, the process of fault detection and the simulation signal of fault extraction are studied. In Section 4, the experimental verification of the bearing inner ring is conducted to verify that the proposed method has higher SNR for an actual measurement, and the reliability of the proposed method is verified by using actual engineering data. Finally, the conclusions are drawn.

#### 2. Potential Model and Time-Delayed Feedback Tristable Stochastic Resonance System

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 , the particles oscillate slightly on the side of the potential well. When the noise is added to the system, the input signal becomes , and the noise energy will be partially transferred to the particle to overcome the barrier height of the system. When the external drive is too large, the particles in the system are too fast, called the overresonance. When the external drive is too small, the particles cannot break through the barrier and can only move in a potential well, which is called the underresonance. Since the periodic signal and noise are often fixed, in order to ensure the best stochastic resonance effect, it is necessary to adjust the potential model to make the particles move stably between the potential wells. Since the classical stochastic resonance is a bistable potential, the potential model has a single structure and cannot form a rich potential structure to match the complex vibration signal. Therefore, in this paper, the classical potential function is introduced [32]:where , , and are potential model parameters and are nonzero real numbers, used to adjust the potential structure. In a nonlinear system, noise and periodic signals work together to produce a synergistic effect, and a stochastic resonance phenomenon occurs. As shown in Figure 1, the particles transition between the potential wells under the action of noise and periodic signals. The higher the position of the particles in the ordinate, the larger the particle energy. In Figure 1(a), it can be seen that as increases, the potential barrier of the intermediate potential well increases, and the positions of the potential wells on both sides are raised. In the synergy of the periodic signal and the noise, it is easy to jump from the potential wells on both sides to the intermediate potential well. In Figure 1(b), it can be seen that as increases, the depth of the wells on both sides decreases, and the transition between the wells is easier and less energy is required; in Figure 1(c), it can be seen that as increases, the barrier of the intermediate potential well increases, and it is difficult for the particles to transition between the three potential wells, and the required energy will increase.