Incrementally accumulated holographic SDP characteristic fusion method in ship propulsion shaft bearing fault diagnosis

In a ship propulsion shafting system (see figure 1), the intermediate bearing is one of the most important components because it is indispensable in realizing the transmission function [1]. A rolling bearing has the advantages of a simple structure and good load capacity, thereby enabling it to be widely used as an intermediate bearing in a ship propulsion shafting system. Accordingly, the health of bearings should be diagnosed to address ship outages, economic losses, and personal safety threats [2, 3] that may be caused by bearing faults.

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Bearing fault diagnosis research is mainly divided into three categories. (a) The method based on extracting the bearing fault characteristic frequency. This method mainly studies signal filtering [4], signal decomposition [5], and the envelope spectrum [6], among others. In the literature [7], the parameter of variational modal decomposition is optimized to decompose the vibration signal and, combined with the envelope spectrum weighted kurtosis index, to detect the bearing fault degree. The literature [8] proposes an empirical mode decomposition (EMD) filter based on time-varyingadaptive optimization to extract clear and rich bearing early fault features. This method can remove background noise and extract the fault characteristic frequency accurately. However, the adaptiveness should be studied further.

(b) The feature self-extraction diagnosis method based on deep learning models, such as the deep belief network (DBN) [9], convolutional neural network (CNN) [10], and stacked autoencoder [11]. In this method, the characteristics of the input signal are adaptively extracted to achieve accurate fault diagnosis. In the literature [12], a rolling bearing vibration signal is acquired as the input of the DBN model, and fault diagnosis is realized through the feature self-extraction function. The literature [13] proposes an adaptive Nesterov momentum CNN for rolling bearing fault diagnosis. Although these methods can realize the intelligent diagnosis of bearing faults, some disadvantages are observed, such as extensive calculation, high complexity, and the need for numerous training samples.

(c) The fault identification method based on vibration signal time- and frequency-domain characteristic parameters. In this method, the data distribution characteristics of the vibration signal and spectrum are studied and such parameters as kurtosis, peak-to-peak value, and skewness are extracted [14, 15]. Fault diagnosis is realized by setting the threshold of the normal state and combining classification models, such as the support vector machine, radial basis function neural network, and extreme learning machine [16, 17]. In the literature [18], a characteristic extraction method based on EMD energy entropy is combined with artificial neural networks to classify bearing faults. The literature [19] states that fault features are extracted to identify the fault location in combination with the grey correlation method. These methods have achieved good results in early bearing fault diagnosis, but the characteristic parameters of the extraction and fusion method should be studied further because of the observed shortcomings in the visualization of fault diagnosis.

The symmetrical dot pattern (SDP) method can reflect the difference between signals because it can amplify fault characteristics. In the literature [20], SDP images are combined with the squeeze-and-excitation CNN classification model to visually diagnose the bearing fault. In the literature [21], vibration signal components after EMD are represented by SDP images, and image features are calculated using the improved Chebyshev distance. These studies have proven that SDP visualizes signals and amplifies their differences. However, ship propulsion shaft equipment works in water; where the acquired vibration signal is nonlinear, the fluctuation range is large, and the background noise is strong. Although directly transforming the vibration signal to SDP will amplify noise, having a large amount of input data and high computational complexity are problems that can occur.

This study proposes an incrementally accumulated holographic SDP characteristic fusion method for ship propulsion shaft bearing fault diagnosis. Combined with the incremental accumulation method, vibration signal time- and frequency-domain characteristics are extracted simultaneously. On the basis of the SDP method, the characteristics are merged into an image to enlarge the differences between fault types. Simulations are verified in this method. Moreover, the results of engineering experiments also prove that the proposed method can effectively fuse the characteristics and improve the accuracy of diagnosis. The innovations of this method are summarized as follows:

• (a)  
  An incremental accumulation characteristic parameter extraction method is proposed to comprehensively record the characteristic change trend of bearing vibration signals.
• (b)  
  A holographic SDP characteristic fusion method is proposed. By fusing the characteristic parameters into one SDP image, the discrimination between bearing fault vibration signals is increased.

The remainder of this paper is organized as follows. Section 2 introduces the SDP principle. Section 3 presents the incremental accumulation characteristic parameter extraction method. Section 4 introduces the bearing fault diagnosis method based on holographic SDP. Section 5 presents the simulation experiment. Section 6 provides the engineering and comparative experiments. Lastly, section 7 summarizes this study.

SDP is a signal conversion method that transforms a 1D signal into a 2D polar coordinate image in which the difference between signals is reflected by the shape distribution. The 1D vibration signal can be expressed as follows:

$$\begin{equation}X = [{x_1},{x_2} \cdots {x_i},{x_{i + 1}} \cdots {x_N}].\end{equation} \tag{ 1 }$$

Hence, X can be converted to a polar coordinate system using equations (2)–(4):

$$\begin{equation}r(i)\, = \,\frac{{{x_i} - {x_{\min }}}}{{{x_{\max }} - {x_{\min }}}}\end{equation} \tag{ 2 }$$

$$\begin{equation}\theta (i)\, = \,\beta \, + \,\frac{{{x_{i + \tau }} - {x_{\min }}}}{{{x_{\max }} - {x_{\min }}}}\xi \end{equation} \tag{ 3 }$$

$$\begin{equation}\phi (i)\, = \,\beta - \frac{{{x_{i + \tau }} - {x_{\min }}}}{{{x_{\max }} - {x_{\min }}}}\xi \end{equation} \tag{ 4 }$$

where $r(i)$ is the polar diameter; $\beta$ is the rotation angle of the mirror symmetry plane, which controls the number of shapes in the SDP image; $\theta (i)$ is a clockwise rotation angle along the initial line $\beta$; $\phi (i)$ is a counter-clockwise rotation angle; $\xi$ is the angle magnification factor, which controls the distribution angle of the shape; and $\tau$ is the delay parameter, which controls the degree of shape distribution.

Figure 2 shows the SDP diagram. Signal X is converted by polar coordinates to form a petal-shaped pattern ($r(i)$, $\theta (i)$). According to equation (4), ($r(i)$, $\phi (i)$) is formed by the mirror symmetry of ($r(i)$, $\theta (i)$). Therefore, an SDP image can be formed according to the rotation angle $\beta$.

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Vibration signal time-domain characteristics are often used as the basis for the bearing fault threshold, which reflects the signal fluctuation, dispersion degree, and impact strength. Frequency-domain characteristics can be used to analyze the frequency change range and reflect the distribution of dominant components in the spectrum.

3.1. Incremental accumulation method

Accurate extraction of a fault characteristic can promptly reflect a signal change, and the incremental accumulation method can reflect the real-time change of the signal. A vibration signal $X = [{x_1},{x_2} \cdots {x_i},{x_{i + 1}} \cdots {x_N}]$ with a data length of N can be divided into several segments, in which the characteristic parameter ${c_d}$ of each segment can be described as follows:

$$\begin{equation}{c_d} = c({x_{(d - 1) * n + 1}}:{x_{d * n}})\end{equation} \tag{ 5 }$$

where $n$ is the segment length incremented each time, which can be calculated according to the sampling frequency ${f_s}$ and RPM (i.e. $n > \frac{{60{f_s}}}{{{\text{RPM}}}}$), and $d$ is the rounding of the total data length N and n. Thus, $d = [\frac{N}{n}]$.

Thereafter, the characteristic parameters of vibration signal X obtained using the incremental accumulation method is as follows:

$$\begin{equation}C = [{c_1},{c_2}, \cdots {c_d}].\end{equation} \tag{ 6 }$$

Figure 3 shows the diagram of the incremental accumulation methods.

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3.2. Time-domain characteristic parameters

For vibration signal $X = [{x_1},{x_2} \cdots {x_i},{x_{i + 1}} \cdots {x_N}]$, the mean value ($\bar x$) and standard deviation ($\sigma$) are given as follows:

$$\begin{equation}\bar x = \frac{1}{N}\sum\limits_{i = 1}^N {{x_i}} \end{equation} \tag{ 7 }$$

$$\begin{equation}\sigma = \sqrt {\frac{1}{{N - 1}}\sum\limits_{i = 1}^N {{({x_i} - \bar x)}^2}} .\end{equation} \tag{ 8 }$$

The following time-domain characteristic parameters are selected according to the bearing fault vibration signal distribution changes:

• (a)  
  Kurtosis: characteristic index for detecting the breadth of vibration signals, highly sensitive to changes in amplitude, and suitable for surface damage fault detection

  $$\begin{equation}{P_1} = \frac{{\sum\limits_{i = 1}^N {{({x_i} - \bar x)}^4}}}{{N{\sigma ^4}}}.\end{equation} \tag{ 9 }$$
• (b)  
  Skewness: asymmetry of positive and negative amplitudes, which is equivalent to zero. Thus, the vibration signal is symmetrically distributed. The larger the skewness value, the stronger the asymmetry

  $$\begin{equation}{P_2} = \frac{{\sum\limits_{i = 1}^N {{({x_i} - \bar x)}^3}}}{{N{\sigma ^3}}}.\end{equation} \tag{ 10 }$$
• (c)  
  Root mean square (RMS): degree of data dispersion. The greater the RMS value, the higher the degree of dispersion

  $$\begin{equation}{P_3} = \sqrt {\frac{1}{N}\sum\limits_{i = 1}^N {x_i}^2} .\end{equation} \tag{ 11 }$$
• (d)  
  Peak-to-peak value: fluctuation range of the vibration signal. The larger the peak-to-peak value, the larger the signal fluctuation range, the more severe the fluctuation, and the more unstable the signal

  $$\begin{equation}{P_4} = \max ({x_i}) - \min ({x_i}).\end{equation} \tag{ 12 }$$

3.3. Frequency-domain characteristic parameters

By using the Fourier transform, signal X is converted to the spectrum, and $F({f_i}),\,i = 1,2 \cdots I$ is the spectral component at frequency ${f_i}$. The following frequency-domain characteristic parameters are selected:

• (e)  
  Average frequency: magnitude of vibration energy in the spectrum. The larger the average frequency value, the greater the energy of the spectrum

  $$\begin{equation}{P_5} = \sqrt {\frac{{\sum\limits_{i = 1}^I F({f_i})}}{I}} .\end{equation} \tag{ 13 }$$
• (f)  
  Center frequency: change of the main energy peak position of the power spectrum:

  $$\begin{equation}{P_6} = \frac{2}{{{f_s}}} \cdot \frac{{\sum\limits_{i = 1}^I {{f_i} \cdot F({f_i})} }}{{\sum\limits_{i = 1}^I {F({f_i})} }}\end{equation} \tag{ 14 }$$

  where ${f_s}$ is the sampling frequency.
• (g)  
  Ricean frequency: distribution of the dominant components in the spectrum

  $$\begin{equation}{P_7} = \frac{2}{{{f_s}}}\sqrt {\frac{{\sum\limits_{i = 1}^I {f_i}^2 \cdot F({f_i})}}{{\sum\limits_{i = 1}^I F({f_i})}}} .\end{equation} \tag{ 15 }$$
• (h)  
  Frequency standard deviation: degree of dispersion of the spectrum

  $$\begin{equation}{P_8} = \sqrt {\frac{{\sum\limits_{i = 1}^I {{({f_i} - {P_6})}^2} \cdot F({f_i})}}{{I - 1}}} .\end{equation} \tag{ 16 }$$

Figure 4 shows the SDP images corresponding to the eight characteristic parameters. These images differ in shape distribution and distinctions.

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To solve the problem of ship propulsion shaft bearing fault visual diagnosis, the difference between the SDP images converted by the vibration signal is not evident. Thus, this study proposes an incrementally accumulated holographic SDP characteristic fusion method. Figure 5 shows the specific flow chart of this method.

• (a)  
  An acceleration sensor is used to collect vibration signals of the bearing under normal and faulty states.
• (b)  
  Optimal signal segment length n is calculated according to RPM and sampling frequency. The incremental accumulation method is used to extract the time- and frequency-domain characteristics.
• (c)  
  SDP parameters, the mirror symmetry rotation angle, the angle amplification factor, and time delay parameters, among others, are set. Information fusion is performed on eight characteristic parameters based on SDP.
• (d)  
  Bearing fault diagnosis is realized upon the combination of SDP image similarity.

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To reduce the complexity of fault identification, polar coordinates are divided into eight regions based on rotation angle $\beta$, and the characteristic parameters are fused into an SDP image. Figure 6 shows that the SDP image is the fusion result of eight characteristic parameters. Thus, the differences between each parameter are obvious.

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To verify the effect of this proposed method in the visual display of the characteristic parameters, a bearing outer race fault simulation model was established. The specific equation is as follows:

$$\begin{align}f(t) = \sum\limits_{i = 1}^{N{}_0^{}} \left[ {\begin{array}{*{20}{l}} {\sin 2\pi (t - i/{f_0})\sqrt {1 - \xi _0^2} \cdot } \\[6pt] {({\,f_{01}}{{\text{e}}^{ - {\xi _0}2\pi {f_{01}}(t - i/{f_0})}} + {f_{02}}{{\text{e}}^{ - {\xi _0}2\pi {f_{02}}(t - i/{f_0})}})} \end{array}} \right] + n(t)\end{align} \tag{ 17 }$$

where $n(t)$ is the simulated background noise, which is a random number that follows a normal distribution; N₀ is the RPM, which is set to 42; ${f_0}$ is the bearing fault characteristic frequency, which is set to 50 Hz; ${\xi _0}$ is the damping ratio, which is set to 0.05; and ${f_{01}}$ and ${f_{02}}$ are the carrier center frequencies at 1200 Hz and 5200 Hz, respectively.

Figure 7 shows the simulation signal and SDP images. Figures 7(b)–(d) present the noise-free signal, random noise, and with-noise signal SDP images, respectively. Figure 7(a) shows that the regular impact caused by the bearing outer race fault is relatively strong, but the impact characteristics are submerged after the background noise is added. Figures 7(b) and (c) illustrate that the points are relatively concentrated and scattered, respectively. Figure 7(d) also shows that the bearing fault impact is submerged by noise, and the distribution of $f(t)$ and $n(t)$ in the SDP is similar.

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Figure 8 shows the results from the extracted characteristics of the simulation signal, in which the incremental accumulation extracted parameters can better represent the volatility of data changes. However, the difference between each characteristic parameter is not evident.

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Figure 9 shows the holographic characteristic fusion of the SDP image. Thus, the difference between each parameter significantly increased.

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According to equation (18), the similarity between the noise-free and with-noise signals is calculated:

$$\begin{equation}r(X,Y) = \frac{{{\text{Cov}}(X,Y)}}{{\sqrt {{\text{Var}}[X]{\text{Var}}[Y]} }}\end{equation} \tag{ 18 }$$

where Cov represents the covariance of X (noise-free signal) and Y (with-noise signal) and Var represents the variance.

Table 1 shows that the similarity between the original signals is low and improved between the holographic SDP images. Given that the characteristic parameter tends to be stable and is affected relatively minimally by noise, the holographic SDP fusion can amplify the difference between signals.

6.1. Experiment platform

In a ship propulsion shafting system, rolling bearings are used as intermediate supporting components for mechanical energy transmission. Its outer race, inner race, and rollers often fail owing to shafting misalignment, bending, and poor lubrication. To verify the effectiveness of the proposed bearing fault diagnosis method, experiments were conducted with the bearing fault simulation experiment platform. As shown in figure 10, this platform consists of a servo motor, control system, intermediate shaft, and removable bearing base. The system can realize uniform operation and RPM regulation, and has a maximum speed of 1800 RPM. An acceleration sensor is placed above the bearing base to acquire vibration signals. The bearing used in this platform is a rolling bearing from NTN Bearing Co., Ltd with a type of NU204 ET2X. By using wire cutting technology to cut square notches on the bearing, it can conduct bearing inner race, outer race, and roller fault experiments.

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6.2. Applications

The vibration signal is acquired by the acceleration sensor with a sampling frequency of 100 kHz at 1500 RPM, as shown in figure 11. Figures 11(a)–(d) correspond to the bearing normal state (N), outer race fault (O), inner race fault (I), and roller fault (R), respectively. Figure 11 also illustrates that the impact caused by a bearing fault is submerged owing to background noise.

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Figure 12 shows the SDP images corresponding to the vibration signal. Table 2 shows the results from the extraction of the axis length (R), saturation (D), center space (d), and deflection (θ) of the SDP image. Thus, the SDP images possess relatively small differences.

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Table 2. Parameters extracted from the SDP images.

Bearing state	Parameters
	R (cm)	D (cm)	D (cm)	Θ (º)
N	7.1	1.9	1.25	8.9
O	7.1	2.1	1.1	8.1
I	7.1	2.0	1.25	8.5
R	7.1	2.05	1.15	8.5

According to equation (18), calculating the similarity between vibration signals corresponds with the SDP images. Table 3 shows that roller fault (R) was misidentified, and noise in the vibration signal has an impact on bearing diagnosis.

Table 3. Comparison of similarity.

	N_2	O_2	I_2	R_2
N_1	1.0000	0.8836	0.9728	0.8815
O_1	0.8993	1.0000	0.9471	0.9545
I_1	0.9746	0.9345	1.0000	0.9233
R_1	0.8972	1.0000	0.9442	0.9661

Figure 13 shows the result from incrementally and cumulatively extracting the vibration signal time- and frequency-domain characteristic parameters.

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Table 4 shows the parameters extracted by part of the vibration signal in the four states. Under the same parameter, the values of the four states are different, and the differences are evident. Thus, eight parameters are reasonable and can effectively distinguish the types of bearing faults.

Table 4. Characteristic parameters extracted from a section of vibration signal.

Bearing state	Feature parameters
	P1	P2	P3	P4	P5	P6	P7	P8
N	2.139	0.084	2.595	14.348	8.067	0.044	0.115	46 236.899
O	2.340	−0.010	4.534	29.481	12.628	0.061	0.135	85 537.994
I	3.596	0.027	4.617	45.375	12.537	0.060	0.135	84 665.293
R	2.793	0.058	4.601	37.954	13.314	0.054	0.116	77 136.560

Figure 14 shows that the differences in holographic characteristic fusion SDP images are evident.

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On the basis of the holographic SDP, bearing faults are identified based on similarity. Table 5 shows that the four states can achieve one-to-one correspondence and identify bearing faults accurately. The proposed method can effectively enhance the difference between signals and realize bearing fault diagnosis.

Table 5. Comparison of similarity based on holographic SDP.

	N_2	O_2	I_2	R_2
N_1	1.0000	0.9030	0.9132	0.8743
O_1	0.8439	1.0000	0.9401	0.9521
I_1	0.6933	0.8209	1.0000	0.9571
R_1	0.7366	0.8552	0.9327	1.0000

On the basis of the bearing data of Case Western Reserve University, the effectiveness of the proposed method is verified. The similarity recognition results are shown in table 6. The results prove that the incrementally accumulated holographic SDP method in this study can accurately identify bearing faults.

Table 6. Similarity recognition results of the Case Western Reserve signal.

	N_2	O_2	I_2	R_2
N_1	1.0000	0.9108	0.9456	0.9725
O_1	0.9029	1.0000	0.9577	0.9778
I_1	0.9366	0.9340	1.0000	0.9659
R_1	0.9035	0.9344	0.9044	1.0000

6.3. Comparison of image visualization methods

At present, short-time Fourier transform (STFT), continuous wavelet decomposition (CWD), and Wigner-Ville distribution (WVD) are the three common time-frequency feature imaging methods in the field of visual bearing diagnosis research. The definitions, advantages, and disadvantages are shown in table 7.

Table 7. Time-frequency feature imaging methods.

Methods	Literature	Formula	Characteristic
STFT	[22, 23]	${\text{STFT}}\,(\tau,\, f\,) = {\displaystyle\int} \left[ {\,f(\tau ) \cdot \omega (\tau - t*)} \right] \cdot {{\text{e}}^{ - 2\pi ik\tau }}{\text{d}}\tau$	High-frequency signals are better reflected in the time-domain, while low-frequency signals are better reflected in the frequency-domain.
CWD	[24, 25]	$W(a,b;\varPsi ) = {\displaystyle\int\limits_{ - \infty }^\infty} x(t){\varPsi _{a,b}}(t){\text{d}}t$	The selection of the wavelet basis is difficult, and different wavelet bases have different results.
WVD	[26, 27]	$W(t,\, f\,) = {\displaystyle\int\limits_{ - \infty }^\infty} x(t + \frac{\tau }{2})x*(t - \frac{\tau }{2}){{\text{e}}^{ - j2\pi f\tau }}{\text{d}}\tau$	The cross term will interfere with it when analyzing multi-component signals.

Figure 15 shows the bearing vibration signal time-frequency images under normal and fault states using three methods. Images generated using STFT and CWT show that the difference between the four states is minimal and difficult to distinguish. To verify the effects of the three methods in bearing fault diagnosis, the results based on similarity are shown in tables 8(1)–(3).

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Table 8(1). Similarity comparison based on STFT.

	N_2	O_2	I_2	R_2
N_1	1.0000	0.9981	0.9984	0.9985
O_1	0.9988	1.0000	0.9983	0.9980
I_1	1.0000	0.9998	0.9989	0.9995
R_1	1.0000	0.9998	0.9998	0.9991

Table 8(2). Similarity comparison based on CWT.

	N_2	O_2	I_2	R_2
N_1	1.0000	0.9997	0.9995	0.9998
O_1	0.9995	1.0000	0.9994	0.9997
I_1	1.0000	0.9996	0.9998	0.9994
R_1	1.0000	0.9998	0.9997	0.9999

Table 8(3). Similarity comparison based on WVD.

	N_2	O_2	I_2	R_2
N_1	1.0000	0.6789	0.5735	0.5821
O_1	0.7636	1.0000	0.6471	0.6363
I_1	0.9168	0.8793	1.0000	0.7620
R_1	1.0000	0.9812	0.9572	0.6764

Tables 8(1) and (2) show that the inner race (I) and roller (R) faults were misdiagnosed, indicating that STFT and CWT cannot effectively distinguish changes in the time-frequency image. Table 8(3) shows that roller fault (R) was misidentified. However, figure 15 shows that the WVD images are relatively different, and the similarity results cannot meet the diagnosis requirements.

The preceding analysis indicates that the incrementally accumulated holographic SDP characteristic fusion method can effectively extract the characteristics of vibration signals, increase the discrimination between signals, and realize accurate diagnosis of ship propulsion shaft bearing faults.

This study proposed an incrementally accumulated holographic SDP characteristic fusion diagnosis method, which is used in the visual fault diagnosis research of ship propulsion shaft bearings. To solve the problem of distinguishing between bearing fault vibration signals, characteristic parameters in time- and frequency-domains were extracted to describe the trend of the change of vibration signals and their spectrum. In addition, the incremental accumulation method was adopted to extract the characteristic parameters, thereby enabling changes to be displayed in time. On the basis of the SDP method, 1D parameters were simultaneously fused to a 2D image, thereby magnifying the difference between signals and laying the foundation for the fault diagnosis of ship propulsion shaft bearings.

This work was supported by the Program for Scientific Research Start-up Funds of Guangdong Ocean University, Key Research and Development Project from Anhui Province of China (Grant No. 202004a05020025), Key Research and Development Project from Anhui Province of China (Grant No. 202104b11020011).

The data generated and/or analyzed during the current study are not publicly available for legal/ethical reasons but are available from the corresponding author on reasonable request.