Incipient Fault Feature Extraction of Rolling Bearing Based on Signal Reconstruction

Abstract

In the incipient fault vibration signals of rolling bearings, weak fault features are easily submerged in strong background noise and difficult to be extracted. The sparse decomposition method can perform well in the extraction of weak fault features, but the low signal-to-noise ratio (SNR) would cause excessive decomposition. To enhance the fault features and maintain the time–frequency structure of fault impulses, a novel incipient fault feature extraction of rolling bearing based on signal reconstruction is proposed. Firstly, the Teager energy operator (TEO) is used to obtain the envelope of the impulse components in the vibration signal, which is also sensitive to noise and would be seriously affected by strong noise. Secondly, a Savitzky–Golay (S–Golay) filter based on the particle swarm optimization (PSO) algorithm is adopted to suppress the noise in the TEO envelope and generate a smooth envelope signal. Finally, the fault signal is reconstructed by the multiplication of the filtered TEO envelope signal and the original signal. The reconstructed signal can maintain the structural characteristics of the original fault impact signal and can provide reliable feature enhancement signals for further sparse decomposition, multi-source vibration separation, and other operations. Simulation signals and experiments verify the effectiveness of this method in extracting early fault features under low SNRs.

1. Introduction

The early faults of rolling bearings have characteristics such as unclear symptoms, weak feature information, and low SNR [1]. In actual working conditions, the vibration signal of bearing faults is a non-stationary and nonlinear signal, and the fault features contained in it are often submerged in strong background noise, making it difficult to extract early fault features of rolling bearings.

The TEO is a differential operator proposed by Kaiser in 1990, which has been effectively applied in signal demodulation and bearing fault diagnosis. The Teager energy operator provides a method for the recognition of impact features in rolling bearing signals [2]. Zhu used a combination of parameter adaptive modal decomposition and the TEO to introduce the maximum weighted kurtosis index for parameter optimization, extracting fault features from early faults in rotating machinery, and achieving the fault diagnosis of rolling bearings [3]. Zhou proposed a method to enhance the weak fault features of motor bearings by combining the advantages of adaptive morphological filtering and the TEO, achieving the recognition of different fault features of motor bearings [4]. Galezia proposed a multi-band demodulation analysis, which utilizes a combination of bandpass filters and the TEO to extract fault feature frequencies from bearing fault signals [5]. Han applied the method of combining complementary ensemble empirical mode decomposition with the TEO to low-speed rolling bearings, effectively extracting fault features of low-speed bearings [6].

Most of these methods use the TEO as a means of shock signal demodulation, which requires obtaining shock components with high SNRs through other signal decomposition methods [7,8]. Therefore, it is difficult to achieve good results when analyzing low-SNR fault signals. After theoretical derivation and simulation experiment analysis, it was found that multiplying the TEO with the original signal can greatly amplify the impact component, while the structure of the impact component itself remains unchanged. However, the envelope of the TEO is susceptible to high-frequency noise interference, which affects the effectiveness of feature extraction. To obtain a smoother envelope, the least squares method can be used to filter and suppress high-frequency noise.

Least squares filtering is an adaptive filtering method that does not rely on the model and relevant statistical features of the signal, and the computational complexity is very small [9]. S–Golay filtering is a method of fitting signals in the time domain using the least squares method, which can remove noise without changing the signal structure. Zhang used a combination of the S-Golay filter and the local mean decomposition method to remove sharp pulse noise from the fault signal and then used the denoised signal as the input of local mean decomposition to successfully extract the fault feature frequency of the signal [10]. Zhu proposed using the signal-denoising methods of singular value decomposition and the S–Golay filter. First, singular value decomposition was performed on the signal to obtain the reconstructed signal, and then the singular value of the reconstructed signal was optimized using an S–Golay filter, effectively reducing the interference of noise in the signal [11].

Under complex operating conditions such as variable speed rotation or start–brake operation, equipment often lacks periodicity in impact signals. Some devices even have multiple sources of vibration, resulting in the severe coupling of fault signals [12,13]. Traditional time–frequency methods are difficult to diagnose; therefore, it is necessary to analyze the structure of the impact signal itself, using sparse decomposition and its improved methods [14,15]. To improve the accuracy of such methods, this article will enhance the feature of the fault signal without changing the structural features of the impact signal itself.

An incipient fault feature extraction of rolling bearings based on signal reconstruction is proposed. Firstly, TEO processing is performed on early bearing fault signals. Then, the particle swarm optimization algorithm is used to find the optimal window length and fitting order of the S–Golay filter, and then the S–Golay filter is used for filtering to obtain a smooth TEO envelope of the shock signal. Finally, by multiplying the filtered envelope signal with the original signal and enhancing the impact component, the feature-enhanced fault signal is obtained, achieving the early fault feature extraction of rolling bearings.

The proposed method focuses on fault diagnosis specifically in rotating mechanical equipment. It aims to identify various types of faults that would occur in rolling bearings, such as peeling, corrosion, fracture, fatigue, and others. By applying this method, operators can analyze these specific faults to ensure the reliable and efficient operation of the equipment.

2. Basic Principles

2.1. TEO Principle

TEO is a nonlinear operator with the characteristics of no complex calculation and low computational complexity, which can effectively extract the impact components in early rolling bearing faults. For continuous signal, TEO is defined as:

$$\psi (x(t))=\dot{x}{(t)}^2-x(t)\ddot{x}(t)$$

(1)

wherein, $x(t)$ represents continuous signal.

2.2. The S–Golay Filter

The S–Golay filter is a filtering method based on the least squares method to fit curves. It consists in the following steps: take the left and right points at a certain point in the original data, that is, continuous points, as the window array; select the fitting order to perform the least squares curve fitting; take the value of the fitted curve data center point as the filtered value; repeatedly move the window; and finally get the value of all data. The fitting polynomial is

$$q(n)={\displaystyle \sum _{k=0}^p{a}_k{n}^k}$$

(2)

wherein $-M\le n\le M$, $p\le 2M+1$, $k$ is the degree of polynomial fitting, and $M$ is the number of points.

The minimum mean square error function is

$${\epsilon }_D={\displaystyle \sum _{n=-M}^M(q(n)-x(n))=}{\displaystyle \sum _{n=-M}^M({\displaystyle \sum _{k=0}^p{a}_k{n}^k}-x(n))}$$

(3)

wherein ${\epsilon }_D$ is the mean squared error, $q(n)$ is a fitting polynomial, and $x(n)$ is the original signal.

When the ${\epsilon }_D$ is the smallest, the S–Golay filter has the best fitting effect, and then all the values of the original data can be obtained by moving the window. Therefore, the two most important parameters that determine the effectiveness of the S–Golay filter are the window size and fitting order. If the window is small and the fitting order is large, many noise signals will be retained. A large window with a small fitting order can distort the signal.

2.3. Principles of Particle Swarm Optimization

The optimization process of particle swarm optimization comes from people’s imitation of the aggregation and dispersion of bird flocks during bird foraging, using the individual or food location of the bird flock as the solution to the optimization problem [16]. The main principle is as follows: assuming there are dimensional particles in the population, the population size is, and the spatial position $x_i$ and velocity of each particle $v_i$ in the population are

$$\begin{array}{c}{x}_i=(x_{i1},x_{i2},x_{i3},\cdots x_{iN})\\ v_i=(v_{i1},v_{i2},v_{i3},\cdots v_{iN})\end{array}$$

(4)

Update the position $x_i$ and velocity $v_i$ of particles as shown in Formula (5)

$$\begin{gathered} v_i(t+1)=w(t)v_i(t)+c_1{r}_{{}_1}[p_{best}(t)-x_i(t)]+c_2{r}_2[g_{best}(t)-x_i(t)] \\ {x}_i(t+1)=x_i(t)+v_i(t+1) \end{gathered}$$

(5)

where $w$ is the weight coefficient, the acceleration factor $c_1$, $c_2$ are constants in [1.5, 2], the random factor $r_1$, $r_2$ are constants in [0, 1], and the number of iterations is set to $t_{\max }$. $v_i(t)$ is the velocity of the particle, $x_i(t)$ is the position of the particle, $p_{best}(t)$ is the local optimal value of the particle, $g_{best}(t)$ is the global optimal value.

3. Incipient Fault Feature Extraction Method Based on Signal Reconstruction

3.1. TEO Pre-Treatment

For a fault impulse signal, the expression is

$$S(t)=A{e}^{-a{(t-u)}^2}\cos (\omega t+\phi )$$

(6)

where $A$ is amplitude, $a$ is the scale factor, $u$ is the shift factor, $\omega$ is the frequency factor, and $\phi$ is the phase factor. The TEO of the fault impulse signal is as follows:

$$\psi (t)=\psi [S(t)]\approx A_1^2{\omega }^2$$

(7)

$$A_1=A{e}^{-a{(t-u)}^2}$$

(8)

Multiply the TEO processed signal with the original signal to obtain the reconstructed signal as shown:

$$S_{\psi }(t)={\psi }_s(t)\cdot S(t)\approx A^3{\omega }^2{e}^{-3a{(t-u)}^2}\cos (\omega t+\phi )$$

(9)

Comparing Equations (6) and (9), it can be seen that the modulation signal has not changed; the amplitude of the signal has been greatly enhanced, which is proportional to the square of the natural frequency; and the attenuation coefficient has become larger. The simulation signal and reconstruction signal are shown in Figure 1.

As shown in Figure 1, in the absence of noise, the amplitude of the simulated signal after TEO processing increases, and the reconstructed signal structure remains unchanged. As shown in Equation (9), the amplitude of the reconstructed signal increases by multiplying the cubic power of the original amplitude by the square of the frequency, greatly increasing the impact component in the signal. But in the noised impulse signal, it is obvious that there is noise on the TEO envelope, and the reconstructed signal cannot distinguish the complete impact signal components. Therefore, for signals with noise, filtering processing is required after TEO processing. The S–Golay filter is a principle that uses the least squares method to fit curves for filtering. Using the S–Golay filter, the high frequency noise can be removed, and the low-frequency TEO envelope components can be retained; thus, the smooth TEO envelope can be obtained.

3.2. Performance Index

To evaluate the signal integrity of the reconstructed signal in this method, the correlation coefficient between the reconstructed signal and the original signal is used as the evaluation index. The correlation coefficient $K$ is defined as follows:

$$K=\frac{{\displaystyle \sum _{n=1}^{N}s(n)}\times y(n)}{\sqrt{{\displaystyle \sum _{n=1}^N{(s(n))}^2\times {\displaystyle \sum _{n=1}^N{(y(n))}^2}}}}$$

(10)

where N is the signal length, s(n) is the original signal amplitude, and y(n) is the reconstructed signal amplitude.

3.3. S–Golay Filter Based on PSO

To evaluate the effectiveness of the S–Golay filter on TEO filtering, the amplitude ratio $R$ between the filtered signal and the TEO processed signal is used as the fitness function, with $R$ defined as follows:

$$R=10\lg \frac{{\displaystyle \sum _{n=1}^{N}y(n)}}{{\displaystyle \sum _{n=1}^N(x(n)}-y(n))}$$

(11)

where $y(n)$ is the filtered signal, and $x(n)$ is the processed signal for TEO.

The most important parameters for the S–Golay filter are window length and fitting order. When the fitness function of $R$ reaches the maximum, the window length and fitting order reach the optimal values, and the S–Golay filter produces the best filtering effect.

PSO is an optimization algorithm with global search ability. In the proposed method, PSO is used to search the optimal window length and fitting order of the S–Golay filter. The specific process is as follows:

Step 1: Initialize the particle swarm. Set the number of iterations to $t_{\max }$, the population size to $m$, the weight coefficient to $\omega$, the acceleration factor to $c_1$, $c_2$, and the random factor to $r_1$, $r_2$. Set the initial position and initial speed of each particle. Define the local optimal value of each particle to $p_{best}(t)$, and the optimal value of all particles to $g_{best}(t)$.

Step 2: Using SNR as the fitness value of particles $p_i(t)$, determine the optimal value of the current individual based on the particle parameters $p_{best}(t)$ with the highest amplitude $g_{best}(t)$ ratio according to different particle amplitude ratios.

Step 3: Update the position $x_i(t)$ and velocity $v_i(t)$ of particles $p_i(t)$, as shown in Formula (5).

Step 4: Determine whether the convergence criteria of the iterative algorithm have been met. If they are met, end the iteration and obtain the window length and fitting order for the optimal filtering effect. Otherwise, $t=t+1$, and proceed to step 2.

3.4. Incipient Fault Feature Extraction Method Based on Signal Reconstruction

Based on the above analysis, the flowchart of the incipient fault feature extraction method based on signal reconstruction is shown in Figure 2.

The diagnostic method steps are as follows:

(1)

Perform TEO processing on the original vibration signal.

(2)

Use the particle swarm optimization algorithm to initialize the individual fitness function value, iterate continuously to obtain the maximum fitness function value, and find the optimal window length and fitting order.

(3)

Filter the TEO processed signal using the optimized filter in step 2.

(4)

Multiply the filtered signal by the original signal, completing the reconstruction of the signal, enhancing the impact components in the original signal, and achieving the feature extraction of the fault signal.

4. Simulate Analysis

To verify the effectiveness of the proposed incipient fault feature extraction method based on signal reconstruction, the improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN) method was introduced to compare and set a threshold for the IMF component. The iteration ends when the decomposed IMF component coefficient is less than this value. Its flowchart is shown in Figure 3 [17].

The simulation signal $x(t)$ is

$$x(t)=S(t)+N(t)$$

(12)

$$N(t)=\sqrt{2D}\xi (t)$$

(13)

where $S(t)$ is the periodic attenuation shock signal with a frequency of 5 Hz, $D$ is the noise intensity, $\xi (t)$ is Gaussian white noise with a mean of 0 and a variance of 1. The sampling frequency $f_s$ is 2000 Hz, and the sampling point $N$ is 2000. Two sets of simulated signals with different D are shown in Figure 4.

As shown in Figure 4, the impact components are clear when D = 0.1, and the impact components are submerged in the noise when D = 0.7.

In the process of the PSO algorithm, the initialization parameters are set to $m=100$, $t_{\max }=100$, $c_1$, $c_2$ set to 1.5, 2, $w_{\max }=1$, $w_{\min }=0.1$, $r_1=r_2=0.1$, the initial position $x(t)$ is 0 and the initial speed $v(t)$ is 0; $p_{best}=g_{best}=0$.

Using the proposed method and the ICEEMDAN method to extract feature impacts, the results of the proposed method for simulated signal with D = 0.1 are shown in Figure 5. The results of the ICEEMDAN method for simulated signal with D = 0.1 are shown in Figure 6. The results of the proposed method for simulated signal with D = 0.7 are shown in Figure 7. The results of the ICEEMDAN method for simulated signal with D = 0.7 are shown in Figure 8.

Figure 5 shows that when the noise intensity is not great, the TEO signal is close to the ideal signal, and the S–Golay filter based on PSO reaches the optimal value at the 7th iteration. The filtered TEO and reconstructed signal both express the good performance of the proposed method. Compared to Figure 6, the reconstructed signal in Figure 5 performs better in similarity and accuracy.

As shown in Figure 5e, the comparison between the reconstructed signal and the simulated signal shows that only the attenuation coefficients of the two signals are different. As shown in Equation (6), only $a$ is different between the two signals, while the other three factors $u$, $\omega$, $\phi$ are exactly the same. As shown in Equation (6), the reconstructed signal attenuation coefficient increases three times compared to the original signal. As shown in Figure 7e, only the attenuation coefficient of the impact in the signal has changed, but the overall impact has not changed, and the signal still has noise components after reconstruction.

Therefore, using the sparse decomposition method, only the scale factor of the reconstructed signal needs to be found, and then the impact component in the simulation signal can be calculated through Equation (9), which can quickly search for several factors in the simulation signal and better extract the simulation signal from noise.

Figure 7 shows that when the noise intensity is great, the TEO signal cannot present the feature of impacts; through the S–Golay filter based on PSO, the filtered TEO can suppress most of the noise and present the feature impacts. The reconstructed signal can also extract all the impacts. Compared to Figure 8, the reconstructed signal in Figure 5 performs better in similarity and accuracy.

To further improve the advantage of the proposed method, wavelet reconstruction and ICEEMDAN are introduced to process the signals with different SNRs. A correlation coefficient is used to evaluate the performance of different methods, and the higher the correlation coefficient is, the better the effect. The correlation coefficients of different methods are shown in

Table 1. The correlation coefficients of different methods.

Noise Intensity D	0.1	0.4	0.7
Original signal	0.9140	0.4652	0.3157
Wavelet reconstruction	0.9436	0.5633	0.3702
ICEEMDAN	0.9538	0.6078	0.4461
Proposed method	0.9667	0.6329	0.5031

.

As shown in Table 1, when the noise intensity is 0.1, all three methods have good reconstruction results, and the correlation coefficients are greater than 0.9. When the noise intensity is 0.4, the wavelet reconstruction effect and ICEEMDAN reconstruction effect are both poorer than the proposed method. When the noise intensity is 0.7, the wavelet reconstruction and ICEEMDAN reconstruction have little effect. The method proposed in this paper, on the other hand, has a prominent effect, with a correlation coefficient of 0.5031.

5. Experiment Analysis

The experimental data were obtained from the BTS100 experimental platform. The experiment used electric discharge technology to process a single-point groove on the bearing to simulate early faults. To simulate early fault signals, this article set the signal sampling frequency to 10 kHz, and the motor speed to 600 r/min. The BTS100 experimental platform is shown in Figure 9. It can be seen that the experimental platform consists of the electrical machinery, frequency converter, motor shaft, bearing seat, sensor, etc.

The bearing used in this experiment is the SFK6006 deep groove ball bearing that is suitable for the BTS100 test bench. The selected deep groove ball bearing is SKF6006, and the structural parameters are shown in

Table 2. Structural parameters of SKF 6006 bearing.

Outer Diameter	Inner Diameter	Number of Rolling Elements	Contact Angle
55 mm	30 mm	11	0°

.

5.1. Fault Diagnosis of Outer Ring

The vibration data and waveform are obtained when the outer ring of the bearing is damaged. The sampling frequency is 10 kHz, and the sampling point is 2000. A fault with a depth of 1.2 mm and a width of 0.6 mm was artificially machined on the outer ring of the rolling bearing using a wire cutting method. As shown in Figure 10f, when the store’s speed is 300 r/min, the fault signal characteristics are not obvious and are completely submerged in the noise. Therefore, a signal with a motor speed of 600 r/min was selected as the experimental object. The theoretical outer ring fault characteristic frequency is 47 Hz. Figure 10 shows the results of the outer ring fault.

From Figure 10, it can be seen that there is a lot of noise in the time-domain waveform in Figure 10a, which cannot effectively extract the impact components. Using the proposed method to analyze the signal, the optimal value of the S–Golay filter based on PSO reaches 203 at the 33rd iteration, with a window length of 191 and a fitting order of 10. The impacts are extracted clearly in the reconstructed signal in Figure 10c. Figure 10d shows that the fault characteristic frequency and doubling components of the original signal cannot be seen in the envelope diagram. As shown in Figure 10g, using the ICEEMDAN method, it is difficult to find a fault feature frequency of 47 Hz as the fault feature frequency of the signal is submerged in other noises. But through the proposed method, Figure 10e shows that the frequency of 45 Hz and its doubling components are obvious, which are consistent with theoretical values.

By comparing Figure 10e,g, it becomes evident that the method proposed in this paper outperforms ICEEMDAN in terms of extracting and displaying the first and second harmonic components of the fault characteristic frequency of the outer ring of rolling bearings. Figure 10e clearly demonstrates the effectiveness of the proposed method by accurately capturing and highlighting the fault characteristic frequency components, specifically the first and second harmonics. This showcases the method’s ability to effectively isolate and distinguish these components from the overall signal, allowing for precise fault diagnosis. In contrast, Figure 10g, representing the results obtained using the ICEEMDAN method, fails to exhibit the same level of clarity and accuracy in extracting the fault characteristic frequency components.

5.2. Fault Diagnosis of Inner Ring

When the inner ring of the bearing is damaged, vibration data and waveforms are acquired. The data is sampled at a frequency of 10 kHz, with a total of 2000 sampling points. To simulate the fault, an artificial defect is created on the outer ring of the rolling bearing using a wire cutting method. The defect has a depth of 1.2 mm and a width of 0.6 mm. According to theoretical calculations, the characteristic frequency associated with the inner ring fault is expected to be at 64 Hz. The obtained results, specifically related to the inner ring fault, are shown in Figure 11.

From the analysis of Figure 11, it is evident that the time-domain waveform depicted in Figure 11a contains a significant amount of noise, preventing effective extraction of the impact components. However, by employing the proposed method for signal analysis, significant improvements are observed. The optimal value of the S–Golay filter, determined using particle swarm optimization (PSO), reaches a high value of 178 during the 39th iteration. The filter is applied with a window length of 73 and a fitting order of 9. As a result, the impact components are clearly extracted, as demonstrated in the reconstructed signal provided in Figure 11c. Furthermore, upon examining the envelope diagram shown in Figure 11d, it becomes evident that the fault characteristic frequency and doubling components of the original signal are not discernible. As shown in Figure 11f, using the ICEEMDAN method, it is difficult to find a fault feature frequency of 65 Hz as the fault feature frequency of the signal is submerged in other noises. But through the proposed method, Figure 11e shows that the frequency of 65 Hz and its doubling components are obvious, which is consistent with theoretical values.

By comparing Figure 11e,f, it is evident that the fault impact components of 65 Hz and 130 Hz are clearly visible in Figure 11e. However, in the results obtained using ICEEMDAN, it is difficult to identify the fault characteristic components of 65 Hz. This demonstrates the superior performance of the proposed method in accurately capturing and presenting the fault characteristic components associated with the analyzed rolling bearings.

6. Conclusions

This paper proposes an incipient fault feature extraction method of rolling bearing based on signal reconstruction. Through simulation experiments and actual signal analysis, the effectiveness of the proposed method is verified, and the conclusions are as follows:

(1)

The reconstructed signal is obtained by multiplying the filtered Teager–Kaiser energy operator (TEO) with the original signal. This approach proves effective in extracting weak impact components present amidst strong noise. When compared to the ICEEMDAN method, the proposed method demonstrates higher relevance to the original signal. It exhibits greater efficacy in accurately extracting and identifying fault features from the incipient fault signal.

(2)

The S–Golay filter based on PSO is proposed to filter the noise in TEO, which can search the optimal window length and fitting order of the S–Golay filter, and can achieve the optimal filtering effect.

Limitations of this article include its primary suitability for early feature extraction of rolling bearing faults, specifically focused on outer ring and inner ring faults. However, its effectiveness may be limited when dealing with more complex faults such as those related to rolling elements. Additionally, the method may not be applicable to composite faults, where multiple fault types are present simultaneously. Extracting comprehensive fault feature information and isolating fault information for each specific fault type may prove challenging when confronted with multiple faults in rolling bearings.

In the future, we plan to conduct research on the following issues:

(1)

The filter utilized in this method may still retain some residual noise even after processing. To enhance the effectiveness of the filter, further improvements can be made to reduce or completely eliminate the remaining noise. This could involve exploring different filter designs or enhancing the parameters used in the filtering process to achieve better noise reduction.

(2)

This article uses a single signal reconstruction method for processing, and restores signal integrity through multiple reconstruction methods.

(3)

After reconstructing the fault signal using the method proposed in this article, it can undergo additional analysis such as sparse decomposition and blind source separation. These further processing techniques can allow for the determination of several unknown parameters within the signal, leading to improved computational efficiency and accuracy in the sparse decomposition process. By incorporating sparse decomposition and blind source separation, a deeper understanding and characterization of the fault signal can be obtained.