Abstract
Bearing is widely used in the rotating machinery and prone to failure due to the harsh working environment. The bearing faultinduced impulses are weak because of poor background noise, long vibration transmission path, and slight fault degree. Therefore, the bearing fault detection is difficult. A novel adaptive stochastic resonance method based on multiagent cuckoo search algorithm for bearing fault detection is proposed. Stochastic resonance (SR) is like a nonlinear filter, which can enhance the weak faultinduced impulses while suppressing the noise. However, the parameters of the nonlinear system exert an influence on the SR effect, and the optimal parameters are difficult to be found. Multiagent cuckoo search (MACS) algorithm is an excellent heuristic optimization algorithm and can be used to search the parameters of nonlinear system adaptively. Two bearing fault signals are used to validate the effectiveness of our proposed method. Three other adaptive SR methods based on cuckoo search algorithm, particle swarm optimization or genetic algorithm are also used for comparison. The results show that MACS can find the optimal parameters more quickly and more accurately, and our proposed method can enhance the faultinduced impulses efficiently.
Highlights
 Adaptive stochastic resonance (SR) based on multiagent cuckoo search (MACS) is effective for weak bearing fault detection.
 MACS can find the better SR parameters in same max iteration or same calculation time.
 Both of SR parameters H and K affect the SR effect.
1. Introduction
Bearing, which can support the rotating parts, is among the most widely used components in rotating machinery. Its failure is a common cause of machine breakdown as the harsh working environment, which can lead to tremendous economic losses and even cause disasters [1]. Earlier the bearing failure is discovered, more time people can take to avoid the unnecessary losses. However, the faultinduced impulses are weak due to the poor background noise, the long vibration transmission path, and the slight fault degree. Some methods have been proposed to detect the bearing failure [24]. For example, Sawalhi et al. [2] combined minimum entropy deconvolution with spectral kurtosis for rolling bearing failure. Smith et al. [3] proposed a simple spectral kurtosisbased approach for selecting the best demodulation band to extract bearing faultrelated impulses from vibration. Mishra et al. [4] diagnosed bearing fault under slow speed operation by wavelet denoising. However, the bearing faultinduced impulses are too weak and submerged in the strong background noise. Although much work has been done, there are still many shortcomings in bearing fault detection.
Stochastic resonance (SR) is a special physical phenomenon that the weak lowfrequency signal is enhanced when the mixture of the weak signal and noise are inputted into a proper nonlinear system. The phenomenon means that noise can be utilized for the weak signal enhancement under certain circumstances. Since SR was proposed by Benzi et al. in 1981 [5], it has been studied and applied in many fields such as bioengineering [6], medical equipment [7], energy engineering [8] and so on. According to adiabatic approximation theory, the traditional SR only can deal with the lowfrequency signal (the frequency is smaller than one) [9]. However, the bearing fault signal is a highfrequency signal (fault characteristic frequency is usually larger than one) and cannot be processed by the traditional SR. To solve the problem, researchers have proposed some methods such as rescaling frequency SR [10], adaptive stepchanged SR [11] and parameter normalized SR [12]. These methods can transform the highfrequency signal into a lowfrequency signal that can be processed by the traditional SR. Because parameter normalized SR is simple and comprehensible, it will be used in this paper.
The parameters of the nonlinear system have a big influence on the SR effect. For example, the two parameters of the bistable system determine the barrier height. If the barrier is too high, it will be difficult for the particle to transfer from one potential well to another, and the bistable SR will not be triggered. If the barrier is too low, the effect of SR is not obvious. Heuristic optimization algorithms can search the optimal parameters of SR easily and have been applied in adaptive SR. For example, Chi et al. [13] used an adaptive SR based on cuckoo search algorithm for bearing fault diagnosis. López et al. [14] applied the underdamped SR based on particle swarm in weak signal detection. Qiao et al. [15] proposed an adaptive unsaturated bistable SR based on the genetic algorithm for mechanical fault diagnosis. However, these algorithms have relatively long computing time and their local optimization accuracies are insufficient. Multiagent cuckoo search (MACS) algorithm applies the multiagent strategy to cuckoo search (CS) algorithm and is a more efficient algorithm. In this paper, an adaptive SR based on MACS is proposed for bearing fault detection.
The rests of this paper are arranged as follows. Section 2 will analyze the bistable SR theory, the SR implements for the discrete signal and the evaluation of SR effect. Section 3 will introduce MACS and propose the adaptive SR based on MACS for bearing fault detection in detail. Section 4 will validate our proposed method by two bearing fault signals. Finally, Section 5 will provide conclusions.
2. Stochastic resonance theory
2.1. Bistable stochastic resonance
Bistable SR phenomenon is described as follows: a particle is driven by a weak driving signal and a noise in the bistable system, and then the weak oscillation is enhanced with the assistance of the noise. Such a phenomenon can be illustrated by Langevin Equation (LE) as:
where $A$ and ${f}_{d}$ are the amplitude and frequency of the driving signal. $N\left(t\right)=\sqrt{2D}\epsilon \left(t\right)$ is Gaussian white noise (GWN) with noise intensity $D$. $\epsilon \left(t\right)$ is the standard GWN with zero mean and unit variance.
Fig. 1Sketch map of the bistable potential function Ux
$U\left(x\right)=a{x}^{2}/2+b{x}^{4}/\text{4}$ is the bistable potential function as shown in Fig. 1. $a$ and $b$ are the potential parameters, and $a\in {R}^{+}$, $b\in {R}^{+}$. It is obvious that the bistable potential function has two minimums which are located at $\pm {x}_{m}=\pm \sqrt{a/b}$ and a potential barrier which is located at ${x}_{b}=$ 0 with the height $\mathrm{\Delta}U={a}^{2}/\text{4}b$.
Thus, Eq. (1) can be rewritten as:
Supposing that $z=\sqrt{a/b}x$ and $\tau =at$, Eq. (2) can be rewritten as:
Because both parameters of the bistable system are transformed into one, it is called parameter normalized transformation of SR. After the transformation, the driving frequency reduces $a$ times. Thus, the highfrequency signal can be transformed into a lowfrequency signal by the transformation with the appropriate parameters ($a$, $b$). The transformed signal will satisfy the limitation of the traditional SR.
2.2. SR implement for discrete signal
The collected vibration signal is discrete and contains weak fault characteristic signal and strong noise. Let $S\left(n\right)$ denote the vibration signal. For a discrete signal $S\left(n\right)$, Eq. (3) can be solved by the fiveorder RungeKutta algorithm as:
${k}_{1}={z}_{n}{z}_{n}^{3}+KS\left(n\right),$
${k}_{2}=\left({z}_{n}+\frac{H}{2}{k}_{1}\right){\left({z}_{n}+\frac{H}{2}{k}_{1}\right)}^{3}+KS\left(n\right),$
${k}_{3}=\left({z}_{n}+\frac{H}{2}{k}_{2}\right){\left({z}_{n}+\frac{H}{2}{k}_{2}\right)}^{3}+KS\left(n+1\right),$
${k}_{4}=\left({z}_{n}+\frac{H}{2}{k}_{3}\right){\left({z}_{n}+\frac{H}{2}{k}_{3}\right)}^{3}+KS\left(n+1\right),$
${k}_{5}=\left({z}_{n}+H{k}_{4}\right){\left({z}_{n}+H{k}_{4}\right)}^{3}+KS\left(n+1\right),$
where $z$ is the SR output. $K=\sqrt{b/{a}^{3}}$ is the amplitude factor. $H=a/{f}_{\mathrm{s}}$ is the integral step.
2.3. Evaluation of SR effect
A criterion is needed to evaluate the SR effect. Some indexes have been proposed such as approximate entropy [11], signaltonoise ratio (SNR) [16], local signal to noise ratio (LSNR) [13] and weighted power spectrum kurtosis (WPSK) [17]. Because of the precise definition and easy application, LSNR will be used as the evaluation criterion:
where ${S}_{R}\left({f}_{d}\right)$ is the power of the frequency ${f}_{\mathrm{d}}$. ${S}_{R}\left({f}_{d},{\u2206f}_{N}\right)$ is the power in frequency domain $\left[\left.{f}_{d}{\u2206f}_{N},{f}_{d}\right)\cup \left(\left.{f}_{d},{f}_{d}+{\u2206f}_{N}\right]\right.\right.$. Larger the LSNR is, better the signal is.
Fig. 2 shows the output LSNR with the increase of noise intensity $D$ under different low driving frequencies without transformation. With the increase of $D$, the input LSNR decreases while the output LSNR increases at first and then decreases. Thus, the noise improves the SR output before the output LSNR reaches the maximum. Also, with the increase of the driving frequency, LSNR becomes smaller and smaller. It means that SR without transformation is effective for the lowfrequency signal but bad for the highfrequency signal.
Fig. 2Output LSNR vs. D under different driving frequencies without transformation (fs= 10 Hz, signal length n= 104, A= 0.2, K= 1, H=1/fs=0.1, and ∆fN= 20 Hz)
Fig. 3 shows the output LSNR with the variation of amplitude factor $H$ and integral step $K$. Output LSNR varies with the change of parameter $H$ and $K$. Larger the output LSNR is, better the parameter $H$ and $K$ are. However, the 3D surf in Fig. 3 is complicated. It is difficult to find the optimal parameter ${H}_{o}$ and ${K}_{o}$. Therefore, a heuristic optimization algorithm is needed to find the optimal parameters.
Fig. 3Output LSNR vs. joint amplitude factor H and integral step K (fs= 6000 Hz, signal length n= 104, A= 1, fd= 30 Hz, D= 2 and ∆fN= 20 Hz)
3. Multiagent cuckoo search algorithm
3.1. Cuckoo search algorithm
Cuckoo search (CS) algorithm is a new heuristic optimization algorithm and has been used in many fields [13, 1820]. It simulates the parasitic reproductive strategy of cuckoos: the solutions are seen as cuckoo eggs, the optimal solution which has the optimal fitness as the optimal cuckoo egg, the feasible region as the search zone of the cuckoos. Three hypothetical rules are set as follows:
• A cuckoo only lay an egg at a time and dump it in a nest randomly.
• The current best egg should be kept to the next generation.
• The nest number is fixed. The host birds should find some cuckoo eggs with a probability ${P}_{a}=\left(\left.\text{0, 1}\right]\right.$, and these eggs will be replaced with new eggs.
When generating the new candidate solutions ${Y}_{i}^{t}$, the update by Lévy flights is performed as [19]:
$v~N\left(\mathrm{0,1}\right),u~N\left(0,{\sigma}_{u}^{2}\right),{\sigma}_{u}^{2}={\left[\frac{\mathrm{\Gamma}\left(1+\beta \right)\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{\pi \beta}{2}\right)}{{2}^{\left(1+\beta \right)/2}\beta \mathrm{\Gamma}\left[\frac{\left(1+\beta \right)}{2}\right]}\right]}^{1/\beta},$
where ${X}_{i}^{t}$ is the $i$th solution of the $t$th generation. $R$ is the stepsize scale. ${X}_{best}^{t}$ is the best solution of the $t$th generation. $u$ and $v$ obey the Gaussian distribution. $\mathrm{\Gamma}\left(\bullet \right)$ is the standard gamma function. $n$ is the number of the solutions. $\beta =$ 1.5 is a constant.
3.2. Multiagent cuckoo search algorithm
CS is an excellent algorithm. However, it lacks the communication among the populations and knowledge learning of the cuckoos, which limits the performance of CS. We regard the cuckoos as intelligent agents. These agents can communicate with others and learn the knowledge. Thus, a new improved CS is proposed called multiagent cuckoo search (MACS) algorithm. Two operations are designed to realize these behaviors. They are the competitive cooperation operation for communication and the selflearning operation for knowledge learning.
A multiagent system should be constructed at first. Fig. 4 shows a multiagent system with the Von Neumann structure. It contains $M$×$N$ circles. Each circle represents a cuckoo. For a cuckoo, other cuckoos that connected with it by the lines are regarded as its neighborhoods. Let ${L}_{i,j}$ denote the cuckoo in $i$th row and $j$th column. Its neighborhoods are ${L}_{i1,j}$, ${L}_{i2,j}$, ${L}_{i,j1}$, and ${L}_{i,j2}$:
${i}_{2}=\left\{\begin{array}{l}i+1,i\ne N,\\ 1,i=N.\end{array}\right.$
${j}_{1}=\left\{\begin{array}{ll}j1,j\ne 1,& \\ M,j=1.& \end{array}\right.$
${j}_{2}=\left\{\begin{array}{l}j+1,j\ne M,\\ 1,j=M.\end{array}\right.$
Fig. 4The multiagent system with Von Neumann structure
Cuckoo ${L}_{i,j}$ only communicates with its neighborhoods by competitive cooperation operation. Supposing ${L}_{max}$ is the best neighborhood of ${L}_{i,j}$. Competitive cooperation operation is performed only when the fitness of ${L}_{max}$ is better than ${L}_{i,j}$. The operation can be described as:
where ${L}_{cc}$ is the new cuckoo. $U\left(\mathrm{1,1}\right)$ is a random number that obeys the (–1, 1) uniform distribution. ${p}_{c}\in \left(\mathrm{0,1}\right)$ is the crossover probability. $r$ is a random number that obeys the (0, 1) uniform distribution. $H\left(\bullet \right)$ is the Heaviside function:
Only the current best cuckoo can learn knowledge through selflearning operation. Supposing ${L}_{cb}$ is the current best cuckoo. Number ${N}_{s1}$ of cuckoos ${L}_{s1}$ are produced in a narrow range by Eq. (10). If the fitness of the produced best cuckoo is better than ${L}_{cb}$, the produced best cuckoo will replace ${L}_{cb}$. Otherwise, ${L}_{cb}$ will not be changed:
where ${R}_{s}\in \left(\mathrm{0,1}\right)$ is the local search radius. $U\left(\text{1}{R}_{s},\text{1}+{R}_{s}\right)$ is a random number that obeys the $\left(\text{1}{R}_{\mathrm{s}},\text{1}+{R}_{\mathrm{s}}\right)$ uniform distribution.
The simple pseudocode of the MACS algorithm can be provided as shown in Fig. 5.
Fig. 5Pseudocode of the MACS algorithm
3.3. The proposed adaptive SR for bearing fault detection
For the deterministic input signal, SR output depends on the parameters $H$ and $K$ according to Eq. (4). However, the optimal parameters ${H}_{o}$ and ${K}_{o}$ are different to be found. In the proposed adaptive SR, MACS is used to search the optimal parameters ${H}_{o}$ and ${K}_{o}$. LSNR is used as the evaluation criterion of the SR effect. For the bearing fault detection, the detailed steps are stated as follows:
Step 1: Bearing vibration signal acquisition. Longer the vibration transmission path is, weaker the impact signal is. When acquiring the bearing vibration signals, the sensors should be near the bearings.
Step 2: Bearing vibration signal preprocessing. Some common techniques are used to preprocess the bearing vibration signal. Resonance demodulation is one of the most preprocessing technologies. Resonance band can be chosen by kurtogram, spectral kurtosis or visual identification [3, 21]. Visual identification is selected in this paper. The envelope signal is obtained by the conventional Hilbert transform method. The envelope signal contains not only the faultinduced impulses but also other known signals such as the shaft rotation signal. These known signals, especially whose frequencies are lower than the frequency of the impulses, are harmful to SR effect. Thus, these signals should be removed by spectral editing. The envelope signal is a unipolar signal, which is worse than the bipolar signal for SR effect improvement [22]. Highpass filtering is used to transform the envelope signal into a bipolar signal and filter some low frequencies which are harmful to the SR effect. Through the preprocessing, the preprocessed signal is obtained.
Step 3: Parameter initialization and optimization by MACS. Intervals of parameters ($H$, $K$) and other related parameters are set. MACS is used to search for the optimal parameters (${H}_{o}$, ${K}_{o}$) which may solve Eq. (11):
Step 4: Signal postprocessing. The optimal parameters (${H}_{o}$, ${K}_{o}$), optimal SR output and its corresponding maximum LSNR are collected.
4. Experimental validations
To verify the feasibility of our proposed method, two bearing preset failure tests are carried out through the test rig as shown in Fig. 6. The test rig consists of three parts: the power and control part, the bearing fault simulation part and the data acquisition part (not shown). The power and control parts are composed of a threephase asynchronous motor (supply the power), a motor controller (control the motor speed) and a speed indicator (display the realtime speed of the motor). The bearing fault simulation part consists of two deep groove ball bearings (support the shaft), an optical shaft (supply the radial load for the bearings) and two flywheels (also supply the radial load). In order to realize the power transmission, the coupler should connect the motor to the shaft. The data acquisition part is composed of vibration acceleration sensors, a data acquisition card and a software (display and storage the data).
Fig. 6Bearing fault simulation test rig
The bearing type is ER12K, and its main dimensions are shown in Table 1. The preset faults are the innerrace fault (a groove with 0.5 mm width in the center of the inner race) and the rolling element fault (a deep groove with 0.5 mm width in one of the rolling elements). The fault bearing is located at bearing 1 as shown in Fig. 6. Vibration signals are collected by sensor 1. Sampling frequency ${f}_{s}$ is 12800 Hz, and sampling time $t$ is 0.5 s. Adaptive SR methods based on MACS, CS, particle swarm optimization (PSO) and genetic algorithm (GA) are used to process the bearing vibration signals for comparison. The parameters of these optimization algorithms are set as shown in Table 2. The parameters of bistable SR are set as $H\in \left[{10}^{3},10\right]$ and $K\in \left[{10}^{3},500\right]$. ${\u2206f}_{N}$ is set as 50 Hz.
Table 1Main dimensions of bearing ER12K
Number of rolling elements  Ball diameter (inch)  Pitch diameter (inch)  Contact angle (°) 
8  0.3125  1.318  0 
Table 2Algorithm parameter setting
Algorithms  Parameters 
MACS  Population size $M\times N=$ 6×6, discovery rate ${P}_{a}=$ 0.25, stepsize scale $R=$ 0.35, crossover probability ${p}_{c}=$ 0.5, local search radius ${R}_{s}=$ 0.05, number ${N}_{s1}=$ 25 
CS  Population size = 50, discovery rate ${P}_{a}=$ 0.25, stepsize scale $R=$ 0.35 
PSO  Population size = 100, intertia coefficient = 1, damping ratio of inertia coefficient = 0.99, personal acceleration coefficient = 5, social acceleration coefficient = 10 
GA  Population size = 100, crossover percentage = 0.7, extra range factor for crossover = 0.4, mutation percentage = 0.3, mutation rate = 0.1 
4.1. Bearing innerrace fault detection
The bearing outer race fault signal is analyzed firstly. According to the main dimensions of bearing ER12K in Table 1 and the real motor speed (19.922 r/s), the fault feature frequency ${f}_{BPFI}$ is equal to 98.582 Hz. The bearing signal is preprocessed by the resonance demodulation. The signal is filtered by minimumorder Butterworth bandpass filter with passband [1500, 5000] Hz. The envelope signal is obtained by the Hilbert transform. The envelope signal and its spectrum are shown in Fig. 8(a). Some known frequencies that are lower than ${f}_{BPFI}$ like the frequency 2${f}_{R}$ and 3${f}_{R}$_{}are bad for the SR effect, and they will be removed by the spectral editing. It is obvious that the envelope signal is a unipolar signal, which is worse than the bipolar signal for the SR effect [22]. Thus, the minimumorder Butterworth highpass filter with stopband [0, 5] Hz is used to transform the unipolar envelope signal to a bipolar signal. After preprocessing, the preprocessed signal is obtained as shown in Fig. 8(b).
Then, the preprocessed signal is processed by the SR methods based on MACS, CS, PSO and GA respectively. Max iterations (50 iterations) and limited time (60 seconds) of calculation are set as the termination conditions of the four optimization algorithms respectively. The fitness convergence curves and final optimization results are shown in Fig. 7 and Table 3. We can see that the final LSNR of SR output based MACS is the biggest among the four algorithms under both termination conditions.
Fig. 7The fitness convergence curves of the four algorithms for the innerrace fault signal
a) Max iteration = 50 times
b) Limited time = 60 s
Therefore, the adaptive SR based on MACS is the best among the four methods. The optimal SR output ($H=$ 0.290, $K=$ 94.398) and its spectrum are shown in Fig. 8(c). The fault feature frequency ${f}_{BPFI}$ is enhanced obviously. Thus, our proposed method is effective for bearing fault detection.
Table 3Optimization results of the four algorithms and related parameters for the innerrace fault signal
Algorithms  Max iterations = 50 times  Limited time = 60 s  
LSNR (dB)  $\left[H,K\right]$  LSNR (dB)  $\left[H,K\right]$  
MACS  –0.961  [0.290, 94.352]  –0.960  [0.290, 94.398] 
CS  –1.001  [0.284, 93.965]  –0.961  [0.290, 94.652] 
PSO  –1.222  [0.277, 99.096]  –1.200  [0.297, 89.445] 
GA  –1.097  [0.264, 96.203]  –1.243  [0.195, 130.002] 
Fig. 8Analysis results for the innerrace fault signal
a) Envelope signal and its spectrum
b) Preprocessed signal and its spectrum
c) SR output and its spectrum
4.2. Bearing rolling element fault detection
The faultinduced impulses in bearing rolling element fault signal are weaker than that in bearing innerrace fault signal. Then, the bearing rolling element fault signal will be analyzed. According to the main dimensions of bearing ER12K in Table 1 and the real motor speed (19.922 r/s), the fault feature frequency ${f}_{BSF}$ is equal to 39.650 Hz. The bearing signal is preprocessed by the resonance demodulation. The signal is also filtered by minimumorder Butterworth bandpass filter with passband [1500, 5000] Hz, and the envelope signal is also obtained by the Hilbert transform. The envelope signal and its spectrum are shown in Fig. 10(a). It is obvious that the frequency ${f}_{BSF}$ is very weak and submerged in the heavy noise. The envelope signal is a unipolar signal, which is worse than the bipolar signal for the SR effect [22]. Then, the minimumorder Butterworth highpass filter with stopband [0, 5] Hz is used to transform the unipolar envelope signal to a bipolar signal. After preprocessing, the preprocessed signal is obtained as shown in Fig. 10(b).
Then, the preprocessed signal is also processed by the SR methods based on MACS, CS, PSO and GA respectively. Max iterations (50 iterations) and limited time (60 seconds) of calculation are set as the termination conditions of the four optimization algorithms respectively. The fitness convergence curves and final optimization results are shown in Fig. 9 and Table 4. In Fig. 9, the convergence curve of MACS is steep in the early stage of optimization, which means that the MACS converges quickly. The final LSNR of SR output based MACS is the biggest among the four algorithms under both termination conditions. Therefore, the adaptive SR based on MACS is the best among the four methods. The optimal SR output ($H=$ 2.350, $K=$ 58.986) and its spectrum are shown in Fig. 10(c). The fault feature frequency ${f}_{BSF}$ is enhanced obviously. Thus, our proposed method can enhance the weak signal submerged in the strong noise.
Fig. 9The fitness convergence curves of the four algorithms for the ball fault signal
a) Max iteration = 50 times
b) Limited time = 60 s
Table 4Optimization results of the four algorithms for the ball fault signal
Algorithms  Max iterations = 50 times  Limited time= 60 s  
LSNR (dB)  $\left[H,K\right]$  LSNR (dB)  $\left[H,K\right]$  
MACS  –4.855  [1.498, 40.379]  –4.514  [1.350, 58.986] 
CS  –6.401  [1.609, 27.523]  –5.191  [1.380, 70.745] 
PSO  –7.115  [1.650, 27.523]  –8.644  [0.240, 263.966] 
GA  –8.800  [1.612, 11.193]  –8.462  [0.215, 293.586] 
Fig. 10Analysis results for the rolling element fault signal
a) Envelope signal and its spectrum
b) Preprocessed signal and its spectrum
c) SR output and its spectrum
5. Conclusions
In this paper, we propose an adaptive SR based on MACS for bearing fault detection. Two bearing fault signals are used to validate our proposed method, and three adaptive SR methods based on CS, PSO, or GA are compared with our proposed method. The conclusions are drawn as follows: (1) Both of parameters $H$ and $K$ affect the SR effect; (2) MACS can find the best parameters of SR in the same max iteration or the same limited calculation time among the four methods; (3) Our proposed method is effective for bearing fault detection especially when the fault is weak.
However, the proposed adaptive SR based on MACS only can enhance the single effective frequency component. Other effective frequency components may be filtered. Thus, the detection of the multifrequency signal should be researched furtherly.
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