*Article* **Bearing Fault-Detection Method Based on Improved Grey Wolf Algorithm to Optimize Parameters of Multistable Stochastic Resonance**

**Weichao Huang 1,2,\* and Ganggang Zhang <sup>2</sup>**


**Abstract:** In an effort to overcome the problem that the traditional stochastic resonance system cannot adjust the structural parameters adaptively in bearing fault-signal detection, this article proposes an adaptive-parameter bearing fault-detection method. First of all, the four strategies of Sobol sequence initialization, exponential convergence factor, adaptive position update, and Cauchy–Gaussian hybrid variation are used to improve the basic grey wolf optimization algorithm, which effectively improves the optimization performance of the algorithm. Then, based on the multistable stochastic resonance model, the structure parameters of the multistable stochastic resonance are optimized through improving the grey wolf algorithm, so as to enhance the fault signal and realize the effective detection of the bearing fault signal. Finally, the proposed bearing fault-detection method is used to analyze and diagnose two open-source bearing data sets, and comparative experiments are conducted with the optimization results of other improved algorithms. Meanwhile, the method proposed in this paper is used to diagnose the fault of the bearing in the lifting device of a single-crystal furnace. The experimental results show that the fault frequency of the inner ring of the first bearing data set diagnosed using the proposed method was 158 Hz, and the fault frequency of the outer ring of the second bearing data set diagnosed using the proposed method was 162 Hz. The fault-diagnosis results of the two bearings were equal to the results derived from the theory. Compared with the optimization results of other improved algorithms, the proposed method has a faster convergence speed and a higher output signal-to-noise ratio. At the same time, the fault frequency of the bearing of the lifting device of the single-crystal furnace was effectively diagnosed as 35 Hz, and the bearing fault signal was effectively detected.

**Keywords:** multistable stochastic resonance; adaptive parameter; improved grey wolf algorithm; bearing fault detection

## **1. Introduction**

The failure rate of rolling bearings accounts for about 30% of all rotating machinery failures, which is the main reason affecting the operating efficiency, productivity, and life of mechanical equipment. Almost all rolling bearing fault signals are in a very noisy environment, resulting in early weak faults that are difficult to find. Therefore, how to enhance the signal-to-noise ratio of fault signals under extreme conditions has become a key issue in the direction of fault diagnosis. At the same time, monitoring the status of rolling bearings, promptly identifying faults, and conducting equipment maintenance are of great practical significance for ensuring the smooth working of rotating machinery systems [1]. Nowadays, the main methods used for rolling bearing fault detection are: wavelet decomposition [2], empirical mode decomposition [3], variational mode decomposition [4], principal component analysis [5], stochastic resonance [6], etc. The stochastic resonance

**Citation:** Huang, W.; Zhang, G. Bearing Fault-Detection Method Based on Improved Grey Wolf Algorithm to Optimize Parameters of Multistable Stochastic Resonance. *Sensors* **2023**, *23*, 6529. https:// doi.org/10.3390/s23146529

Academic Editor: Luca De Marchi

Received: 21 June 2023 Revised: 14 July 2023 Accepted: 17 July 2023 Published: 19 July 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

algorithm overturns the view that noise is harmful for a long time. It uses the resonance principle to transfer noise energy to the fault signal, thus improving the detection and diagnosis of the fault signal, and opening up a new method and idea for weak bearing fault-signal detection submerged in strong noise.

Benzi raised the concept of stochastic resonance (SR) in 1981 when studying the changes of the Earth's ice ages [7]. After 40 years of development, SR theory has been widely used in fault diagnosis [8], optics [9], medicine [10], image denoising [11], and other fields, and has achieved many remarkable results. The SR algorithm makes use of the synergy generated by the joint excitation of nonlinear systems, input signals, and noise to make Brownian particles oscillate, improve the output signal-to-noise ratio, and effectively detect the measured signal, which is a typical method to enhance the measured signal. Therefore, it is widely concerned with the domain of signal detection [12]. Classical bistable and monostable SR models have been extensively used in the study of signal detection [13]. However, for the signal to be measured with ultra-low amplitude, due to the potential function structure constraints, particles are often unable to effectively jump between potential wells, and SR-detection methods for bistable and monostable models are also powerless. When studying multistable stochastic resonance systems, Li et al. found that the multistable model can better enhance the output signal-to-noise ratio and improve the noise utilization ratio than the bistable and monostable models [14]. Therefore, more and more scholars have carried out relevant research on multistable SR [15]. For example, Zhang et al. proposed a piecewise unsaturated multistable SR (PUMSR) method which overcomes the weakness of tri-stable SR output saturation and enhances the ability of weak signal detection [16].

However, whether it is a monostable, bistable, or multistable SR algorithm, it is inevitably difficult to select model parameters in practical applications. Mitaim et al. [17] put forward the adaptive SR theory to enhance useful signals by automatically adjusting the structural parameters of nonlinear systems. But, the adaptive SR method, which takes a single parameter of the system as the optimization object, often ignores the interaction between the parameters of the system. With the rise of the swarm intelligence optimization algorithm, finding the global optimal solution through the swarm intelligence algorithm can solve the limitations of traditional adaptive SR systems, and this concept has been extensively used in the domain of bearing fault detection [18]. However, in the existing research results, the adaptive selection of SR model parameters still depends on the performance of intelligent optimization algorithms, so there are generally issues such as a low solving accuracy and being prone to falling into local optima [19]. Therefore, the feasible method to effectively enhance the parameter performance of adaptive selection of SR systems is to improve the defects of the intelligent optimization algorithm, so that it can more quickly and accurately optimize the parameters of the SR system. The grey wolf optimization algorithm can find the optimal solution by simulating the tracking, encircling, pursuit, and attack stages of the group predation behavior of the grey wolf. With few parameters and a simple structure, it is easy to integrate with other algorithms for improvement, but there are also the problems that it is easy to fall into local optimal solutions and low computational efficiency [20]. Therefore, it is of great research value to improve the basic grey wolf algorithm and improve its optimization performance [21]. Vasudha et al. proposed a multi-layer grey wolf optimization algorithm to further achieve an appropriate equivalence between exploration and development, thereby improving the efficiency of the algorithm [22]. Rajput et al. proposed an FH model based on the sparsity grey wolf optimization algorithm, which helps to minimize the computational overhead and improve the computational accuracy of the algorithm [23].

This article takes bearing fault-signal detection as the research object. Aiming at the problem of difficult parameter selection of multistable SR systems, a bearing fault-detection method based on an improved grey wolf algorithm to optimize multistable SR parameters is raised. This method improves the basic grey wolf optimization algorithm. Firstly, considering the quality of the initial solution, a Sobol-sequence initialization population

strategy is proposed to make the distribution of the initial grey wolf population more uniform. Secondly, a convergence-factor adjustment strategy based on exponential rules is proposed to coordinate the global exploration and local development stages of the algorithm. Meanwhile, an adaptive position-update strategy is proposed to improve the accuracy of the algorithm, and Cauchy–Gaussian mixture mutation is used to enhance the algorithm's ability to escape from local optima. Experimental verification is conducted on the performance of the improved grey wolf algorithm using fifteen benchmark test functions from the CEC23 group of commonly used test functions. The verification results display that the multi-strategy improved grey wolf optimization algorithm (MSGWO) has a faster convergence speed and a higher convergence accuracy. Then, on the basis of the model of the multistable SR system, the parameters of the multistable SR system are optimized through the MSGWO, so as to enhance the fault signal and realize the effective detection of the bearing fault signal. Finally, the bearing fault-detection method raised in this article is used to analyze and diagnose a bearing data set from Case Western Reserve University (CWRU) and a bearing data set from the Mechanical Fault Prevention Technology Association (MFPT), and is compared with the optimization results of other improved algorithms. Meanwhile, the method raised in this article is used to diagnose the fault of the bearing of the lifting device of a single-crystal furnace. The test results display that the bearing fault-detection method raised in this article has a fast convergence speed and a large output signal-to-noise ratio, and can detect bearing fault signals accurately and efficiently.

The rest of this article is arranged as below: The Section 2 introduces the specific cases of bearing failure in rotating machinery in different industries. The Section 3 introduces the basic principle of multistable SR. The Section 4 introduces the principle of the basic grey wolf optimization algorithm and the MSGWO, and compares it with some basic optimization algorithms and improved optimization algorithms, respectively. At the same time, the population diversity and the exploration and development stage of the MSGWO are analyzed. The Section 5 introduces the bearing fault-diagnosis method based on the MSGWO to optimize the multistable SR parameters, and uses the proposed method to analyze and diagnose the bearing data sets from CWRU and the MFPT. Meanwhile, the raised method is used to diagnose the bearing fault of the monocrystal furnace lifting device. The Section 6 is the summary.

#### **2. Specific Cases of Bearing Failure**

Due to the diverse working environments of bearings during the operation of rotating machinery, they are easily affected by wear, corrosion, and other factors, making it easy for various faults to occur. For example, in June 1992, during the overspeed test of a 600 MW supercritical active generator set at the Kansai Electric Power Company Hainan Power Plant in Japan, the bearing failure of the unit and the critical speed drop caused strong vibration of the unit, resulting in a crash accident and economic losses of up to JPY 5 billion. From September 2003 to October 2004, the China Railway Beijing–Shanghai Line, Shitai Line, and Hang-gan Line had a total of five traffic incidents. According to relevant statistics, four of these accidents were caused by train bearing-fatigue fracture, with a total economic loss of up to CNY 2 billion. In April 2015, China Dalian West Pacific Petrochemical Co., LTD., due to the serious distortion and fracture of the inner ring of the driving end bearing and the serious wear and deformation of the bearing ball, the seal of the bottom pump of the stripping tower of a hydrocracking unit quickly failed, and the medium leaked, which caused a fire. The accident caused three pumps, the frame above the pump, and a small number of meters and power cables to set fire; a local pipeline to crack; and direct economic losses of CNY 166,000. In 2018, the US Navy's "Ford" aircraft carrier had to return to the shipyard for maintenance due to a thrust bearing failure during a mission. In August 2019, when a drone was spraying pesticides at a farm in Hebei, China, its motor rolling bearing failed, causing the drone to lose control, and a large amount of pesticides were spilled into the river, causing serious pollution. In December 2021, there were two recessive

cracks in the bearing of unit #33 of a wind farm in Liaoning, China. Due to the limited installation position, the appearance inspection could not find them. As a result, the shaft cracks were promoted by the wind wheel's alternating load during operation, resulting in a spindle fracture and the impeller's overall fall. Therefore, the research on fault-diagnosis technology of rolling bearings is very necessary and has great practical significance. resulting in a spindle fracture and the impeller's overall fall. Therefore, the research on fault-diagnosis technology of rolling bearings is very necessary and has great practical significance. **3. Basic Principles of Multistable SR** *3.1. The Basic Theory of Multistable SR*

In August 2019, when a drone was spraying pesticides at a farm in Hebei, China, its motor rolling bearing failed, causing the drone to lose control, and a large amount of pesticides were spilled into the river, causing serious pollution. In December 2021, there were two recessive cracks in the bearing of unit #33 of a wind farm in Liaoning, China. Due to the limited installation position, the appearance inspection could not find them. As a result, the shaft cracks were promoted by the wind wheel's alternating load during operation,

#### **3. Basic Principles of Multistable SR** The principle of SR is that weak characteristic signals can be enhanced and detected

*Sensors* **2023**, *23*, x FOR PEER REVIEW 4 of 25

#### *3.1. The Basic Theory of Multistable SR* by noise transfer mechanism under the action of nonlinear system. In general, when in-

where

*x*

The principle of SR is that weak characteristic signals can be enhanced and detected by noise transfer mechanism under the action of nonlinear system. In general, when interpreting the SR model, we should first consider Langevin's dynamic equation [24], which is as follows: terpreting the SR model, we should first consider Langevin's dynamic equation [24], which is as follows: 2 ' 2( ) ( ) ( ) *d x dx U x s t n t* + = − + + (1)

$$\frac{d^2\mathbf{x}}{dt^2} + \frac{d\mathbf{x}}{dt} = -\mathcal{U}'(\mathbf{x}) + \mathbf{s}(t) + n(t) \tag{1}$$

where *x* is the system response of SR, *U*(*x*) is a class of nonlinear multistable potential function, *s*(*t*) is the external incentive, *n*(*t*) is the noise excitation, *m* is the mass of the particle, and *k* is the drag coefficient. function, *st*() is the external incentive, *nt*() is the noise excitation, *m* is the mass of the particle, and *k* is the drag coefficient. The definition formula of the nonlinear multistable potential function is:

+

 *a*

1

The definition formula of the nonlinear multistable potential function is: *a*

$$\mathcal{U}(\mathbf{x}) = \frac{a}{2}\mathbf{x}^2 - \frac{1+a}{4b}\mathbf{x}^4 + \frac{c}{6}\mathbf{x}^6 \tag{2}$$

 *c*

In the formula, *a*, *b*, and *c* are parameters of the nonlinear multistable model, and they are all greater than 0. The potential function model image of the multistable system is displayed in Figure 1. In the formula, *a* , , and are parameters of the nonlinear multistable model, and they are all greater than 0. The potential function model image of the multistable system is displayed in Figure 1.

*b*

*c*

**Figure 1.** Potential function curve of multistable system. **Figure 1.** Potential function curve of multistable system.

Substitute the potential function of the multistable model into Formula (1), add noise with intensity *D* in the system, and then obtain the Langevin equation of the nonlinear multistable system as follows:Substitute the potential function of the multistable model into Formula (1), add noise with intensity *D* in the system, and then obtain the Langevin equation of the nonlinear multistable system as follows:

$$\frac{d\mathbf{x}}{dt} = -a\mathbf{x} + \frac{\mathbf{1} + a}{b}\mathbf{x}^3 - c\mathbf{x}^5 + s(t) + \sqrt{2D}n(t) \tag{3}$$

When periodic signal and noise signal are used as excitation simultaneously, the inclination of potential well in the multistable system will increase. In addition, the periodic When periodic signal and noise signal are used as excitation simultaneously, the inclination of potential well in the multistable system will increase. In addition, the periodic signal will also make the potential well depth of the three potential wells of the multistable potential function change periodically, and can guide the noise signal to switch synchronously. When the signal, noise, and multistable SR system reach a certain matching relationship, particles can make periodic transitions between potential wells, so that the components of the system output with the same frequency as the input signal are strengthened.

#### *3.2. System Parameters' Range*

The fourth order Runge–Kutta formula was used to solve the multistable SR model. The specific calculation formula is:

$$\begin{cases} \begin{aligned} k\_1 &= h(-ax(n) + \frac{1+a}{b}x^3(n) - cx^5(n) + s(n)) \\ k\_2 &= h(-a(x(n) + \frac{k\_1}{2}) + \frac{1+a}{b}(x(n) + \frac{k\_1}{2})^3 - c(x(n) + \frac{k\_1}{2})^5 + s(n)) \\ k\_3 &= h(-a(x(n) + \frac{k\_2}{2}) + \frac{1+a}{b}(x(n) + \frac{k\_2}{2})^3 - c(x(n) + \frac{k\_2}{2})^5 + s(n)) \\ k\_4 &= h(-a(x(n) + k\_3) + \frac{1+a}{b}(x(n) + k\_3)^3 - c(x(n) + k\_3)^5 + s(n)) \\ x(n+1) &= x(n) + \frac{1}{b}(k\_1 + 2k\_2 + 2k\_3 + k\_4) \end{aligned} \end{cases} \end{cases} \tag{4}$$

where *x*(*n*) is the nth sampling value of the system output, *s*(*n*) is the nth sampling value of the noise-added input signal, *h* is the sampling step, and *ki*(*i* = 1, 2, 3, 4) is the slope of the output response at the relevant integration point.

Normally, due to noise, particles jump over higher barrier heights by accumulating energy, so *b*, *c*, and *h* take the real numbers of [0, 10]. As the target signal is relatively weak, the interval in [25] is quoted; the range of *a* is set to [0, 0.5].

#### **4. Multi-Strategy Improved Grey Wolf Optimization Algorithm**

#### *4.1. The Primary Theory of Grey Wolf Optimization Algorithm*

Grey Wolf Optimizer (GWO) is a new intelligent swarm optimization algorithm proposed by Mirjalili et al. [26], whose main ideas are the leadership hierarchy and group hunting mode of grey wolf groups. The grey wolf population has a strict hierarchy. The head of the population is *α*, which represents the most coordinated individual in the wolf pack, and is mainly responsible for the decision-making affairs of the group's predation behavior. The *β* wolf is second only to *α* in the population, and its role is to serve the *α* wolf to make decisions and deal with the behavior of the population. The third rank in the population is the *δ* wolf, which obeys the instructions issued by the *α* and *β*, but has command over other bottom individuals. The lowest individual in the group, known as *ω*, is submissive to the instructions of other higher-ranking wolves and is primarily responsible for balancing the relationships within the group. GWO defines the three solutions with the best fitness as *α*, *β*, and *δ*, while the remaining solutions are defined as *ω*. The hunting process (optimization process) is guided by *α*, *β*, and *δ* to track and hunt the prey (position update), and finally complete the hunting process, that is, obtain the optimal solution. Grey wolf groups gradually approach and surround their prey through several formulas:

$$D = \left| \mathbb{C} \cdot X\_p(t) - X(t) \right| \tag{5}$$

$$X(t+1) = X\_p(t) - A \cdot D \tag{6}$$

where *t* represents the number of iterations, *X*(*t*) and *Xp*(*t*) represent the position vector between the wolf and its prey, *A* and *C* represent the cooperation coefficient vector, and *D* is the distance between the individual wolf pack and the target. The formula for calculating coefficient vectors *A* and *C* is:

$$A = 2f \cdot r\_1 - f \tag{7}$$

$$\mathbb{C} = \mathbb{2} \cdot r\_2 \tag{8}$$

where, as the number of iterations increases, *f* decays linearly from 2 to 0. To enable some agents to reach an optimal position, *r*<sup>1</sup> and *r*<sup>2</sup> take values between [0, 1].

When hunting, GWO thinks that *α*, *β*, and *δ* are better at predicting the location of prey. Therefore, individual grey wolves will judge the distance *Dα*, *Dβ*, and *D<sup>δ</sup>* between themselves and *α*, *β*, and *δ*; calculate their moving distances *X*1, *X*2, and *X*<sup>3</sup> toward the three, respectively; and finally move within the circle of the three. The moving formula is shown in Equation (9).

$$\begin{cases} \begin{array}{l} D\_{\alpha} = \left| \mathbf{C}\_{1} \cdot \mathbf{X}\_{\alpha} - \mathbf{X}(t) \right| \\ D\_{\beta} = \left| \mathbf{C}\_{1} \cdot \mathbf{X}\_{\beta} - \mathbf{X}(t) \right| \\ D\_{\delta} = \left| \mathbf{C}\_{1} \cdot \mathbf{X}\_{\delta} - \mathbf{X}(t) \right| \end{array} \tag{9}$$

$$\begin{cases} X\_1 = X\_\alpha - A \cdot D\_\alpha \\\ X\_2 = X\_\beta - A \cdot D\_\beta \\\ X\_3 = X\_\delta - A \cdot D\_\delta \end{cases} \tag{10}$$

$$X(t+1) = (X\_1 + X\_2 + X\_3)/3\tag{11}$$
