Mechanical Fault Feature Extraction under Underdamped Conditions Based on Unsaturated Piecewise Tri-Stable Stochastic Resonance

Abstract

In the case of the rapid development of large machinery, the research of mechanical fault signal feature extraction is of great significance, it can not only ensure the development of the economy but also ensure safety. Stochastic resonance (SR) is of widespread use in feature extraction of mechanical fault signals due to its excellent signal extraction capability. Compared with an overdamped state, SR in an underdamped state is equivalent to one more filtering of the signal, so the signal-to-noise ratio (SNR) of the output signal will be further improved. In this article, based on the piecewise tri-stable SR (PTSR) obtained from previous studies, the feature extraction of mechanical fault signals is carried out under underdamped conditions, and it is found that the SNR of the output signal is further improved. The simulation signals and experimental signals are used to verify that PTSR has better output performance under underdamped conditions.

1. Introduction

Italian scholars Benzi et al. first proposed the concept of SR in 1981 [1]. Since then, SR has been increasingly applied to astronomy, mathematics, biology and other scientific fields [2,3,4,5,6,7,8,9]. With the continuous study of SR theory, its advantages in signal feature extraction have been gradually explored and increasingly applied to the feature extraction of mechanical fault signals [10,11,12,13,14,15,16]. Liu et al. studied chaotic phenomena in the process of SR [17]. In Ref. [18], a variable parameter normalized SR method is applied to the extraction of non-stationary signals. V. Sorokin and Blekhman studied stochastic resonance phenomena in parametric excitation systems [19].

From the point of view of signal processing, SR in the underdamped state is equivalent to adding one more filter to the signal than that of the overdamped state. Therefore, in order to further improve the SNR of the output signal, more and more studies on SR are carried out in the underdamped state. Guo et al. used multiplicative and additive signals to drive the bistable SR system under underdamped conditions [20]. Xu et al. studied a triple-well potential SR under an underdamped state and studied the influence of the damping coefficient on SR [21]. In Ref. [22], an underdamped adaptive SR was proposed and successfully avoided the limitations of a characteristic frequency. Li et al. researched the SR of underdamped coupled systems [23]. Hu et al. proposed an underdamped asymmetric SR and applied it to feature extraction of gear faults [24]. In Ref. [25], the underdamped condition is used in the monostable SR model, and the output characteristics are improved. Li et al. combined variational mode decomposition and underdamped SR to detect gearbox faults [26].

The PTSR model was invented in order to improve output saturation. In order to further improve the ability of the PTSR system to extract features, an underdamped PTSR method is proposed. Section 2 introduces the process of the PTSR method in the underdamped state. In Section 3, using the classical SNR curve in SR theory and the simulation signal, it is verified that the underdamped PTSR system has better feature extraction and noise removal abilities. In Section 4, by comparing the performance of the PTSR system under two conditions with two groups of engineering signals, it is concluded that the proposed method has successfully improved the output characteristics and SNR of the system. Section 5 discusses possible future research directions. Finally, Section 6 gives the conclusions.

2. The PTSR in Underdamped Condition

In prior research, we invented a PTSR system that could improve the output saturation problem, the potential function, as shown below [27]:

$$\mathrm{U}\left(x\right)=\{\begin{array}{c}-\frac{3m{p}^2{q}^2\left(p^2-q^2\right)-m\left(p^6-q^6\right)}{6\sqrt{2}\sqrt{3q^2-p^2}-12q}x+\frac{m{q}^4\left(3p^2-q^2\right)}{12}-\frac{3m{p}^2{q}^3\left(p^2-q^2\right)-mq\left(p^6-q^6\right)}{6\sqrt{2}\sqrt{3q^2-p^2}-12q},x\le -q\\ \frac{m}{2}{p}^2{q}^2{x}^2-\frac{m}{4}(p^2+q^2)x^4+\frac{m}{6}{x}^6 , q<x<q\\ \frac{3m{p}^2{q}^2\left(p^2-q^2\right)-m\left(p^6-q^6\right)}{6\sqrt{2}\sqrt{3q^2-p^2}-12q}x+\frac{m{q}^4\left(3p^2-q^2\right)}{12}-\frac{3m{p}^2{q}^3\left(p^2-q^2\right)-mq\left(p^6-q^6\right)}{6\sqrt{2}\sqrt{3q^2-p^2}-12q} , x\ge q\end{array}$$

(1)

where $m$, $p$, $q$ are the system parameter for PTSR, and $m>0$, $p>0$, $q>0$ and $q>p$. Figure 1 displays its potential function.

The SR in the system of Equation (1) in the underdamped state can be expressed as:

$$\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}=\frac{\mathrm{dU}\left(x\right)}{\mathrm{d}x}-\mathsf{\gamma }\frac{\mathrm{d}x}{\mathrm{d}t}+A\cos \left(2\pi f_{d}t\right)+\xi \left(t\right)$$

(2)

where $x$ is system output, $A$ and $f_d$ are the amplitude and frequency of sine signal, $\mathsf{\gamma }$ denotes the damping ratio, $\mathrm{U}\left(x\right)$ is the $\mathrm{U}\left(x\right)$ mentioned in Equation (1).$\xi \left(t\right)$ represents additive white gaussian noise (AWGN) and it fits the following equation:

$$\{\begin{array}{c}\langle \xi \left(t\right)\rangle =0\\ \langle \xi \left(t\right)\xi \left(t-\tau \right)\rangle =2D\delta \left(t\right)\end{array}$$

(3)

where $\tau$ represents the time interval, $D$ represents noise intensity.

First, turning Equation (2) into a system of first order differential equations for the convenience of solving:

$$\{\begin{array}{c}\frac{\mathrm{d}x}{\mathrm{d}t}=y\\ \frac{\mathrm{d}y}{\mathrm{d}t}=-\frac{\mathrm{dU}\left(x\right)}{\mathrm{d}x}-\mathsf{\gamma }y+Acos\left(2\pi f_{d}t\right)+\xi \left(t\right)\end{array}$$

(4)

The fourth-order Runge–Kutta method commonly used in stochastic resonance theory is used to solve the above equation [28]:

$$\{\begin{array}{c}{k}_1=y_i\\ l_1=S_i-\mathsf{\gamma }{k}_1- {\mathrm{U}}^{\prime }\left(x_i\right)\\ k_2=y_i+\frac{1}{2}h{l}_1\\ l_2=S_i-\mathsf{\gamma }{k}_2- {\mathrm{U}}^{\prime }\left(x_i+\frac{1}{2}h{k}_1\right)\\ k_3=y_i+\frac{1}{2}h{l}_2\\ l_3=S_{i+1}-\mathsf{\gamma }{k}_3- {\mathrm{U}}^{\prime }\left(x_i+\frac{1}{2}h{k}_2\right)\\ k_4=y_i+h{l}_3\\ l_4=S_{i+1}-\mathsf{\gamma }{k}_4- {\mathrm{U}}^{\prime }\left(x_i+h{k}_3\right)\\ x_{i+1}=x_i+\frac{1}{6}h\left(k_1+2k_2+2k_3+k_4\right)\\ y_{i+1}=y_i+\frac{1}{6}h\left(l_1+2l_2+2l_3+l_4\right)\end{array}$$

(5)

where $S_i$ represents system input, $x_i$ represents system output, $h$ is step size, $k_i$ and $l_i$ are the slopes of $x_i$ and $y_i$, respectively. We note that $i$ in the formula is the length of the data, which is the number of sampling points.

The above calculation is equivalent to changing the overdamped state to the underdamped state on the basis of the PTSR method in the previous study.

3. Performance Analysis of the PTSR in the Underdamped State

In signal processing theory, integral operation is equivalent to filtering, so solving the differential equation of underdamped SR is equivalent to adding an additional filter on the basis of the traditional overdamped SR. It is necessary to verify whether a better system output can be obtained under the condition of multiple filtering once.

3.1. The Output SNR Comparison

SNR is an important standard with which to evaluate the performance of the system. In the article, an SNR invented in Ref. [29] is used as a measure to compare the PTSR method under the underdamped state with the previous studies:

$$SNR=10{\log }_{10}\frac{A_d}{A_n}$$

(6)

where $A_d$ represents amplitude at $f_d$ of Equation (2), $A_n$ represents the sum of other amplitudes.

We adjust the parameters of the STSR method [30], the PTSR and the PTSR in an underdamped state to be the same ($m=4, p=0.5,q=1$, $\gamma =1$), and input the same periodic signal with noise into all three systems ($\mathrm{A}=0.1$, $f_d=0.1$ Hz, D goes from 0.1 to 0.8). Due to the randomness of noise, the experiment is repeated ten times and de-averaged, then the SNR curves of the three systems are plotted in Figure 2. It shows that all three curves are going up and then down, which proves that the phenomenon of SR is generated in all the three systems. At the same time, the SNR of the PTSR system under the underdamped condition is significantly higher than that under the traditional overdamped condition, indicating that the PTSR method under the underdamped condition can obtain better output characteristics.

3.2. The Simulation Results

In order to show more intuitively that the PTSR in the underdamped state has better output, a periodic signal with noise is input into the PTSR system under underdamped and overdamped conditions, respectively ($A=0.5$, $f_d=5$). Sampling frequency $f_s$ and noise intensity D are set to 2000 Hz and 30, respectively, Figure 3 is the original signal.

It can be seen from Figure 3 that both the time domain and frequency domain diagrams are disorderly, the periodicity and the characteristic frequency bar cannot be recognized. Meanwhile, the original SNR is calculated to be −32.4038 dB.

The signals processed by both methods are displayed in Figure 4. By contrasting with Figure 4a,b, it can be seen that both signals show a trend of resuming periodicity, but the signal in Figure 4b is obviously clearer. At the same time, by contrasting Figure 4c,d, it can be seen that the amplitude of the characteristic frequency is extracted, but the characteristic frequency in Figure 4d is obviously more prominent, and the SNR of the PTSR and the PTSR in the underdamped state are calculated to be −18.8225 dB and −13.8189 dB, respectively. Thus, it can be concluded that the PTSR method in the underdamped state has a stronger degree of noise filtering and better output characteristics.

4. The Experimental Results

In order to be more convincing, two groups of actual engineering signals are used to further verify PTSR method in the underdamped state, which can further improve the effect of feature extraction on the original method.

4.1. Experimental Data from Case Western Reserve University

Figure 5 shows the experimental equipment, the drive bearing is SKF 6205-2RSJEM,

Table 1. Dimensions of the drive roller bearings.

Inner Diameter /mm	Outer Diameter /mm	Pitch Diameter /mm	Contact Angle /(°)	Number of Balls	Balls Diameter /mm
25.001	51.999	39.040	0	9.000	7.940

shows its parameters, the sampling frequency $f_{s1}$ is 12,000 Hz, the fault characteristic frequency $f_{d1}$ is 156 Hz; Figure 6 shows the original signal.

It can be seen from Figure 6 that the periodicity and characteristic frequency of the original signal are all covered by noise, so the original signal is, respectively, compared with the PTSR model in the overdamped and underdamped states to compare the effect of feature extraction. Due to SR needing to satisfy the adiabatic approximation, the original signal must be scaled [31]. First, setting the variable scale parameter $k_1$ to 2000, the secondary sampling frequency is $f_{sr1}=f_{s1}/k_1=6 \mathrm{Hz}$ and the step size is $h_1=1/f_{sr1}=1/6$. Since the system parameters have a great influence on the output signal, a particle swarm optimization algorithm (PSO) is used to search for the best parameter [32]. In the PSO algorithm, both learning factors are set to 2, population size is set to 60, inertia weight is set to 0.5 and the adaptive function is the SNR mentioned in Equation (6). Meanwhile, the search range of parameters $m$, $p$ and $q$ are set to $m\in \left(0.0001,4\right), p\in \left(0.0001,1.5\right) \mathrm{and} q\in \left(\mathrm{0.0001.2}\right),$ respectively. Finally, the search results are $m=3.0340, p=0.4869 \mathrm{and} q=1.0326,$ and the output signal is shown in Figure 7.

Figure 7 shows that both methods restore the periodicity of time-domain signals to a certain extent, but the waveform in Figure 7b has a higher clarity and larger amplitude. In Figure 7c,d, the amplitude of the characteristic frequency is obviously prominent from other frequencies, but the value is higher in Figure 7d. In addition, the SNR is increased from −28.0015 dB to −19.1526 dB and −17.8624 dB by the two methods, respectively. Through these, it can be concluded that the PTSR method in the underdamped state has stronger noise filtering effect and can obtain better output effects.

4.2. Experimental Data of a Company

Figure 8 is ER-16K which a medium-speed outer ring bearing failure platform of a company. The sampling frequency $f_{s2}=12,000\mathrm{Hz}$, the fault characteristic frequency $f_{d2}=41.02\mathrm{Hz}$ and Figure 9 display the original signal.

Because of the signal clutter, we cannot conclude from Figure 9 whether there is a fault or not. The signal is processed by two methods to further compare the advantages and disadvantages of the output characteristics. Then, the scale changes as in the previous experiment ($k_2=1500$) after searching for the best parameters ($m=3.3014, p=0.6147,q=0.8790$). Signals processed by both methods are shown in Figure 10. Compared with Figure 10a,b, both output signals recover obvious periodicity in the time domain, but the signal in Figure 10b is significantly cleaner, indicating that PTSR has better denoising effect in the underdamped state. By comparison with Figure 10c,d, both methods are successful in feature lifting in the frequency domain, but the value in Figure 10d is higher, and further calculation results show that the SNR of the two methods increases from −27.4205 to −14.5002 and −12.8098, respectively. All the above results indicate that PTSR has better output characteristics in the underdamped state.

5. Discussion

Through the above research, it can be proved that the PTSR system under underdamping has better output characteristics, which can further improve the SNR of the original system. At the same time, in the theory of signal processing, the integral is equivalent to the filter, which is also consistent with the above conclusion. In subsequent research, we can try to continue to increase the order of differential equations of the SR system, so that more times of integration are needed in the process of solving this differential equation, which is equivalent to using more filter times for the signal and generating a higher output signal-to-noise ratio.

6. Conclusions

Through the study of the PTSR method in the underdamped state, the following conclusions are obtained:

• Compared with the overdamped state, the PTSR method in the underdamped state has better denoising ability, which can make the time domain signal smoother.
• Compared with the overdamped state, the PTSR method in the underdamped state has better feature extraction ability, and the amplitude at the characteristic frequency is higher.
• The PTSR method can improve the output SNR under both overdamped and underdamped conditions, but the SNR is higher under underdamped conditions.