An Approach on Fault Detection in Diesel Engine by Using Symmetrical Polar Coordinates and Image Recognition

Abstract

Vibration technique provides useful information in fault detection of diesel engine, bringing significant cost benefits to diesel engine condition monitoring. Usually, time-frequency calculation on vibration signal is so complex that it is difficult to achieve online fault detection. In this paper, a method of fault detection in diesel engine is developed based on symmetrical polar coordinates and image recognition. In this method, time-domain waveform of vibration signal is transformed into snowflake-shaped in mirror symmetry pattern without time-frequency analysis. By the comparison of the geometric features of the snowflake images from different wear conditions of crankshaft bearing in diesel engines, we use centroid position and direction angle of the petal in snowflake image as features to detect the fault. Then, fuzzy c-means (FCM) are used to detect the conditions of the engine according to these features. In order to validate the methods, some experiments have been performed, the experimental results show that the centroid position and direction angle of the petal in snowflake image can reflect the information of different wear conditions in crankshaft bearing, and the fault of crankshaft bearing can be detected accurately. Hence, the method can work as fault detection in diesel engine, which is simple and effective, compared with time-frequency calculation method.

1. Introduction

In a diesel engine fault detection, vibration analysis method is one of the main diagnostic tools, along with the development of modern signal processing technology, a variety of time-frequency analysis methods, such as short-time Fourier analysis, frequency analysis, Wigner-Ville, higher-order spectral analysis, Hilbert-Huang Transform, and other methods are gradually adopted to the analysis of the vibration signal processing [1–4]. In recent years, many researchers utilized image processing technology into the field of diesel engine mechanical fault detection. In articles [5–7], wavelet, EMD-WVD, and other methods were used to get the time-frequency distribution, and then the spectrum images of vibration signals were used in feature extraction, classification, and recognition. The basic idea is vibration signals–time frequency analysis–image generation–image feature extraction–image classification [2, 8, 9]. However, these time-frequency analysis methods are complex and time-consuming; a clear beacon signal was needed in order to ensure comparability of time-frequency image. Hence, these time-frequency analysis methods may not work well for online troubleshooting.

In this paper, we present a novel analysis method for fault detection in diesel engine in which the sampled signals are converted into mirror symmetry image in symmetrical polar coordinate system. Then, the image recognition is used for an analysis. In the method, the sampling signals are directly converted into image from the time domain without time-frequency transform; hence, the method is simple and effective. A type of diesel engine crankshaft bearing wear fault is taken as an example. The vibration signal is studied in polar coordinates and converted to images. Then, the geometrical configuration features of the signal images are extracted. Finally, FCM clustering method is used to recognize different faults in the engine's crankshaft bearing. Some experiments are performed to validate the analysis results.

2. The Basic Principle of Symmetry Polar Coordinate Method and Image Recognition

In polar symmetrical image generation method, the time-domain waveform signal will be collected, and each signal point will be mapped to polar radius in polar coordinates, and then the adjacent points will be mapped to the polar angle. Finally, hexagonal flakes mirror symmetry graphics are produced in polar coordinates [10]. Changes in amplitude and frequency of the sampled signal can be expressed by the difference of the image. In this approach, the time-domain signal can be processed without frequency analysis; hence, it is simple, convenient, and fast.

2.1. Polar Symmetry Method

In discrete signal sample data sequence, we assume that the amplitude of the time i was x _i and the amplitude of the time i + L was x_{i + L}. Substituting symmetrical polar coordinates calculated in formula (1), they can be transformed to points in polar coordinate space P(r(i), Θ(i), ϕ(i)). Figure 1 shows the basic schematic symmetry polar coordinates method.

Figure 1:

Schematic diagram of symmetrical polar coordinates method.

In Figure 1, r(i) is the radius of polar coordinates, Θ(i) is the counterclockwise rotation angle from the start line in polar coordinates, and ϕ(i) is the clockwise rotation angle from the start line in polar coordinates. By changing the rotation angle of the start line, a set of signals (x _i , x_{(i + L)}) can be transformed to mirror-image symmetry in polar coordinates. The specific formula is as follows:

$\begin{array}{c}r\left(i\right)=\frac{x_i-x_{\min }}{x_{\max }-x_{\min }},\\ \text{Θ}\left(i\right)=\theta +\frac{x_{\left(i+L\right)}-x_{\min }}{x_{\max }-x_{\min }}g,\\ \varphi \left(i\right)=\theta -\frac{x_{\left(i+L\right)}-x_{\min }}{x_{\max }-x_{\min }}g,\end{array}$

(1)

where x_max is the maximum of the sample data, x _min is the minimum of the sample data, L is the time interval (typically ranging between 1 and 10), θ is the rotation angle of start line, and g is the amplification factor of angle.

From these formulas and symmetry principle, we can find that the midpoint location of polar coordinate space P is the key method of symmetrical polar coordinates, and the position of each point is determined by two time intervals for the time-domain signal amplitude of L. If a given signal is predominant by middle and low frequency component, there will be a smaller radius and larger deflection angle in the polar coordinate space, and vice versa.

To illustrate the characteristics of symmetrical polar coordinate method, where L is 5, g is 40°, and θ is 60°, it shows that the three different frequencies, sinusoidal signals, and a random noise signal simulation are transformed to mirror symmetry snowflakes images in symmetrical polar coordinates according to this method in Figure 2. As apparent sinusoidal signal frequency increases, gradually petals will be full of snowflakes images and increase the overlap portion. These features reflect the changes in visual image of the original signal. Comparing the snowflakes images of sinusoidal signals and random noise signal, the difference between the periodic signal and nonlinear signal can be distinguished by the characteristics of the image.

Figure 2:

Mirror symmetry snowflakes images of simulated signal ((a) f = 200 Hz, (b) f = 400 Hz, (c) f = 600 Hz, and (d) random noise).

2.2. Parameters Selection of Symmetrical Polar Coordinates Method

In the polar coordinate space P, the choice of θ, g, and L is very important. The choice of parameters decides the mirror symmetry snowflake polar coordinates map feature, where θ is the same as the mirror symmetry plane rotation angle. Its mirror symmetry planes are 0°, 60°, 120°, 180°, 240°, and 300°. In polar coordinates, the six-mirror plane form a hexagonal of snowflake image. Large θ will result in the number of polar coordinates mirror symmetry decrease, and the amount of information about graphical visualization will be reduced. Small θ will result in mirror symmetry overlap increase; the characteristics of the image are hard to be found. When θ is 60°, the signal characterization is clear and it is easy to observe the feature. Therefore, we choose θ = 60°.

When θ is determined, the nuances of the different signals generated images depend mainly on g and L. The optimum values of g and L can improve the image resolution and amplify the difference of signals [11]. Figure 3 shows the effects on generating images of different g and L. With L ​​ ranging from 1 to 10, the petals of symmetrical snowflakes image is more plump. With the increasing value of g, the angle of centroid to the horizontal axis becomes larger in the petals of symmetrical snowflakes image. Because the resulting mirror symmetry snowflakes image is created by rotating θ and overlapping of each petal, too large or small values of g and L will affect the image of each petal overlapping portion symmetrically and produce a nonsignificant difference of the original signal feature information. Therefore, the optimum range of g is 20°∼60° and the optimum range of L is 3∼10.

Figure 3:

Different image with different g and L.

2.3. Mirror Symmetry Image Feature Extraction and Noise Reduction Processing

2.3.1. The Mirror Symmetry Image Feature Extraction

Currently, the characteristic parameters for image pattern recognition can be summarized as follows: (1) statistical characteristics of color or grayscale, (2) texture and edge features, (3) geometric characteristics of the shape, and (4) characteristics of transform coefficients or filter coefficient characteristics [12]. Typically, the use of the feature extraction method is determined by the characteristics of the image. Due to the characteristic mirror symmetry snowflakes graph generation method, after generating a set of petals, six-rotating-mirror plane hexagonal shape is formed by superimposing images of snowflakes. Therefore, different shapes in the snowflakes image will be generated from different signal. So, we choose the geometry parameters extraction method based on image characteristics.

There are a lot of features of image geometry parameters, such as regional area, the boundary region, the long axis and short axis of the ellipse with the same second-order central moments to the region, area centroid, and direction angle. In this paper, the area centroid and direction angle are used as the characteristic parameter.

(1) Area Centroid. It is a feature of shape, and it can be calculated from the coordinates of all points belonging to the region. It is calculated as follows:

$\stackrel{-}{x}=\frac{\mathrm{1}}{A}{{\displaystyle \sum }}_{\left(x,y\right)\in R}x,\hspace{1em}\hspace{1em}\stackrel{-}{y}=\frac{\mathrm{1}}{A}{{\displaystyle \sum }}_{\left(x,y\right)\in R}y,$

(2)

where x, y are the abscissa and ordinate of the point within the region and A is the area of ​​the region, the same as the number of pixels in the region.

(2) Direction Angle. It is the region with the same angle of the second central moment of the long axis of the ellipse and the horizontal line. It is a feature parameter of shape.

2.3.2. Image Noise Reduction Based on Mathematical Morphology Open Operation

Mathematical morphology is a new method in the field of image processing and pattern recognition; the basic idea is to use a certain form of structural elements to measure and extract the corresponding shape of image and then achieve the purpose of image analysis and recognition. Due to the high speed operation and ease of implementing morphological opening operation, it is the most widely used method of mathematical morphology noise. The principle is as follows.

A is assumed as a set, B is the structuring element, and the opening operation of A by using B is defined as follows:

$A\circ B=\left(A\text{Θ}B\right)\oplus B,$

(3)

where AΘB is A corrosion by using B as a structural element and A⊕B is A expansion using B as a structural element. Figure 4(a) shows a binary image.

Figure 4:

Noise reducing of opening operation ((a) original image, (b) structural element, and (c) image after opening operation).

Figure 4(a) shows a binary image, Figure 4(b) shows a structural element, and Figure 4(c) shows the result of the binary image's opening operation using (b) as a structural element. The results show that the opening operation can eliminate a small area of ​​the object, separate the slim point of the object, and smooth large object boundaries with not significant change of its area.

3. Feature Extraction of Crankshaft Bearing Wear in Diesel Engine

3.1. Vibration Signal Acquisition in Diesel Engine

In this paper, Cummins 6BT5.9 diesel engine was used. The fourth crankshaft bearing was selected as the trial bearing, and four kinds of fit clearance were set to simulate the wear conditions. According to the wear patterns and troubleshooting experience, the wear of adjacent crankshaft bearing will occur when local crankshaft bearing wear occurring. Hence, when a wear fault was set in the fourth channel of the crankshaft bearing, generally a less fault should be set in the neighbor channel. Table 1 shows the fit clearance in the fourth channel and adjacent bearing crankshaft bearings, respectively.

Table 1:

Different fit clearance of diesel engine crankshaft bearing.

Fault	Position	Failure degree (mm)
		Normal	Slight	Moderate	Severe
Crankshaft bearing abnormal sound	The third channel	0.08–0.1	0.16–0.18	0.24–0.26	0.38–0.4
	The fourth channel	0.08–0.1	0.24–0.26	0.38–0.4	0.53–0.55
	The fifth channel	0.08–0.1	0.16–0.18	0.24–0.26	0.24–0.26

Vibration signals of four kinds of engine states were captured in point A of Figure 5, the measurement speed was 1800 r/min, the sampling frequency was 20000 Hz, and the sampling data length was 80000 points every time. The body temperature was maintained at 60–70°C with the test signal acquisition and monitoring engine oil pressure. Figure 6 shows, respectively, the vibration time-domain waveform of the fourth channel in normal, slight, moderate, and severe state from point A.

Figure 5:

Testing point location map.

Figure 6:

Vibration signals of 4 states ((a) normal, (b) slight, (c) moderate, and (d) severe).

3.2. Symmetrical Mirror Image Generation of the Vibration Signal

3.2.1. Vibration Signal Preprocessing

Diesel engine is a complex mechanical system; its vibration signals are typical of nonstationary signals with noise. Therefore, it is necessary to denoise the vibration signal before transformed to image.

Currently, there are a lot of noise reduction methods for vibration signal. In this paper, EMD is used to reduce noise, and the main idea is as follows [13].

• The signal S(t) is decomposed using EMD, and then N IMF components (P _i , i = 0, 1, 2,…, N) are obtained.

• The correlation coefficients (γ(i), i = 0, 1, 2,…, N) are calculated between S(t) and IMF components P _i .

• The correlation coefficients (λ(i), i = 0, 1, 2,…, N) are calculated between the signal under normal conditions S_nor(t) and IMF components P _i .

• The joint correlation coefficients (ϕ(i), i = 0, 1, 2,…, N) are calculated according to γ(i) and λ(i). (ϕ(i) = γ(i) − λ(i)).

• The correlation coefficient factors (β(i), i = 0, 1, 2,…, N) are calculated. Consider β(i) = (ϕ(i) − min⁡(ϕ))/(max⁡(ϕ) − min⁡(ϕ)), where ϕ = {ϕ _i }, i = 0, 1, 2,…, N.

• The correlation coefficient factors β(i) are sorted by descending, the IMF components corresponding to the first 5 β(i) are added up to as the signal, and other IMF components will be deleted as noise.

3.2.2. The Parameter Selection of Symmetrical Polar Coordinate Method

In order to characterize accurately engine's vibration signal by using mirror symmetry snowflakes image, it is necessary to determine L and g in the symmetry polar coordinate method. In this paper, the optimum seeking method is based on image correlation; this means that L changes as a step of 1 from 1 to 10 and g as a step of 10° from 20° to 60°, then the values of L and g are fixed, respectively, and the correlations of normal and abnormal wear condition are analyzed in snowflake image. Because we focus on the distinction of normal and abnormal conditions, the summary of correlation coefficients of the images of normal and abnormal conditions is used as optimum seeking parameter. The L and g corresponding to the minimum correlation coefficient between the normal and abnormal conditions are selected. Quantitative analysis using the correlation coefficient between images is calculated as follows:

$R\left(A,B\right)=\frac{{\sum }_m{\sum }_n\left(A_{mn}-\stackrel{-}{A}\right)\left(B_{mn}-\stackrel{-}{B}\right)}{\sqrt{\left[{{\sum }_m{\sum }_n\left(A_{mn}-\stackrel{-}{A}\right)}^2\right]\left[{{\sum }_m{\sum }_n\left(B_{mn}-\stackrel{-}{B}\right)}^{\mathrm{2}}\right]}}.$

(4)

In formula (4), A and B are two-dimensional matrixes in images and $\overline{A}$ and $\overline{B}$ are the averages of two-dimensional matrixes in images. The correlation coefficient R can be obtained by calculating from this formula, and its absolute value is between 0∼1. R = 1 means that the two images are identical, and R = 0 means that they are two completely different images. When L and g were given different values​​, respectively, the summaries of correlation coefficients of images between normal and different abnormal wear conditions were calculated according to the above method; then we found that the summary of correlation coefficients was minimum when g = 30° and L = 3. It indicates that the images have the maximum difference between normal and abnormal conditions, and it is conducive to the next image feature extraction. Thus, according to the formulas (1), the vibration signals with different engine conditions are transformed to mirror symmetry snowflake images after noise reduction. They are shown in Figure 7 after the opening operation of mathematical morphology.

Figure 7:

Mirror symmetry snowflake images under different fit clearance ((a) normal, (b) slight, (c) moderate, and (d) severe).

3.3. Feature Extraction Based on the Shape of the Image

According to the characteristics of a mirror symmetry snowflake image, the two former petals from 0° to 360° in the first quadrant can represent the characteristics of the whole piece snowflake image, so we can extract the features from these petals.

Firstly, the two petals are segmented from the first quadrant of the images with different conditions; they are shown in Figure 8. Secondly, the centroid position and direction angle are calculated from the upper and lower petals, respectively. In Figure 8(a), A is the centroid of the upper petal, B is the centroid of the lower petal, and u and v are the direction angles of the petals. The calculated features from different wear conditions are shown in Table 2.

Table 2:

The features from different wear fit clearance.

Wear degree of crankshaft bearing	Group of signals	x A /pixel	y A /pixel	u/°	x B /pixel	y B /pixel	v/°
Normal	1	35.266	65.012	50.153	51.1	88.764	10.018
	2	34.704	65.432	50.142	50.264	89.741	9.2399
	3	33.333	66.182	50.334	48.987	90.404	8.1477
Slight	1	41.12	62.783	40.925	53.984	85.758	18.55
	2	40.342	63.1	41.185	53.902	85.688	19.745
	3	38.171	64.204	42.438	52.332	86.682	18.923
Moderate	1	39.221	62.933	45.986	52.828	87.105	13.321
	2	42.619	61.614	45.582	56.561	84.726	13.618
	3	40.192	62.522	46.709	54.143	86.158	13.97
Severe	1	37.549	65.173	39.329	50.266	87.286	20.563
	2	41.513	63.331	38.113	53.423	84.921	22.377
	3	38.844	64.258	39.49	51.598	86.743	20.823

Figure 8:

The petals from different wear fit clearance ((a) normal, (b) slight, (c) moderate, and (d) severe).

It can be observed from Table 2 that there are some obvious differences in the features of each petal image. In the upper petal image, there are some characteristics as follows: the position of centroid from normal wear petal image is in the bottom left and that from slight and severe wear petal image is in the right. The direction angle from normal wear petal image is about 50° and that from moderate wear petal image is about 45–46°, and it is about 40° in slight and severe conditions. In the lower petal image, there are some characteristics as follows: the position of centroid from normal wear petal image is downward, and there is not much difference in that of slight, moderate and severe. The direction angle from normal wear petal image is about 10° and that from moderate wear petal image is about 13°, and it is about 20° in slight and severe conditions. As the process of crankshaft bearing wear is gradual, the variation of features from different conditions is fuzzy and it is difficult to distinguish directly the conditions from the features; hence, we use fuzzy clustering method to solve this problem.

4. Pattern Recognition Based on FCM Clustering

Cluster analysis is a mathematical method, which works based on the similarity and classification process as follows [13, 14]. For the sample data set X = {x₁, x₂,…x _k ,…, x _n }, if you want to separate the sample into c (1 < c < n) class, the membership belongs to the ith class of the arbitrary sample x _k which is defined as u _ik :

$\begin{array}{c}\mathrm{0}\le u_{ik}\le \mathrm{1},\\ \forall k,\hspace{1em}{\displaystyle \sum }_{i=\mathrm{1}}^c{u}_{ik}=\mathrm{1},\\ \forall i,\hspace{1em}\mathrm{0}<{\displaystyle \sum }_{k=\mathrm{1}}^n{u}_{ik}<\mathrm{1}.\end{array}$

(5)

In order to find a reasonable classification result in a large number of possible classifications, a reasonable similarity or dissimilarity clustering criteria must be determined; the objective function is shown as the following formula:

$J\left(U,V\right)={\displaystyle \sum }_{k=\mathrm{1}}^n{\displaystyle \sum }_{\hspace{0.17em}i=\mathrm{1}}^c{\left(u_{ik}\right)}^m{\left(d_{ik}\right)}^{\mathrm{2}}.$

(6)

In formula (6), U = [u _ik ] is the fuzzy classification matrix, V = {v₁, v₂,…, v _c } is a set of c cluster centers, m is weighted exponent, d _ik is distance from the kth sample to the ith class, and it is generally calculated by the Euclidean norm, defined as follows:

$d_{ik}=\parallel x_k-v_i\parallel .$

(7)

Obviously, J(U, V) denotes the sum of the weighted distance from the sample to the cluster center in every class, and the clustering criteria are taken as the minimum requirements of J(U, V). When J(U, V) becomes the minimum, the membership matrix and fuzzy cluster center should be calculated as follows.

For membership matrix, considering that d _ik may be 0, two cases are discussed. We define the set I _k = {i ∣ 1 ≤ i ≤ c, d _ik = 0} and ${\stackrel{-}{I}}_k=\{\mathrm{1,\; 2},\dots ,c\}-I_k$ , then J(U, V) becomes the minimum and u _ik should be calculated as follows:

$u_{ik}=\{\begin{array}{cc}\frac{\mathrm{1}}{{\sum }_{j=\mathrm{1}}^c{\left(d_{ik}/d_{jk}\right)}^{\mathrm{2}/\left(m-\mathrm{1}\right)}},\hfill & I_k=\Phi ,\hfill \\ \mathrm{0},\hspace{1em}\forall i\in {\stackrel{-}{I}}_k\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}{\displaystyle \sum }_{j\in I_k}{u}_{ik}=\mathrm{1},\hfill & I_k\ne \Phi .\hfill \end{array}$

(8)

Similarly, J(U, V) is obtained as the minimum and v _i should be calculated as follows:

$v_i=\frac{{\sum }_{k=\mathrm{1}}^n{\left(u_{ik}\right)}^m{x}_k}{{\sum }_{k=\mathrm{1}}^n{\left(u_{ik}\right)}^m},\hspace{1em}\hspace{1em}i=\mathrm{1,2},\dots ,c.$

(9)

In this paper, FCM is performed to classify the features relative with the wear conditions. The calculated procedure is as follows.

Step 1. The cluster center V(0) is initialized; let l = 0; and the maximum permissible error ε (ε > 0) is determined.

Step 2. The membership matrix U^(k) is calculated according to formulas (10) and (11).

If $\forall j,k,\begin{array}{c}\end{array}{d}_{jk}(l)>\mathrm{0}$ , then

$u_{ik}=\frac{\mathrm{1}}{{\sum }_{j=\mathrm{1}}^c{\left(\left(d_{ik}\left(l\right)\right)/\left(d_{jk}\left(l\right)\right)\right)}^{\mathrm{2}/\left(m-\mathrm{1}\right)}}.$

(10)

If $\forall j,k,\begin{array}{c}\end{array}{d}_{jk}(l)=\mathrm{0}$ , then

$u_{jk}\left(l\right)=\mathrm{1},\hspace{1em}{u}_{ik}\left(l\right)=\mathrm{0},\hspace{1em}i\ne j.$

(11)

Step 3. The cluster center V^{(l + 1)} is calculated according to the following formula:

$v_i\left(l+\mathrm{1}\right)=\frac{{\sum }_{k=\mathrm{1}}^n{\left(u_{ik}\left(l\right)\right)}^m{x}_k}{{\sum }_{k=\mathrm{1}}^n{\left(u_{ik}\left(l\right)\right)}^m}.$

(12)

Step 4. While ∥V(l + 1) − V(l)∥ < ε, classification will be finished, and the iteration will be aborted. Otherwise, let l = l + 1, and repeat Step 2.

5. Experimental Validations

The features (x _A , y _A , u,x _B , y _B , v) of the mirror snowflake Image are used as the input parameters of FCM. Because the set of sample classification is {normal, slight, moderate, severe}, the number of clusters is set as 4. The stop condition of iteration is ε = 10⁻⁵. The feature parameter matrix is calculated using FCM. After 11 iterations, the value of the objective function is 0.5270, and the known fault samples membership matrix U is

$U=\left[\begin{array}{cccccccccccc}\mathrm{0.8797}& \mathrm{0.9968}& \mathrm{0.9208}& \mathrm{0.0043}& \mathrm{0.0026}& \mathrm{0.0315}& \mathrm{0.0585}& \mathrm{0.0474}& \mathrm{0.0015}& \mathrm{0.0281}& \mathrm{0.0208}& \mathrm{0.0061}\\ \mathrm{0.0283}& \mathrm{0}.\mathrm{0008}& \mathrm{0.0200}& \mathrm{0.9329}& \mathrm{0.9591}& \mathrm{0.1807}& \mathrm{0.1264}& \mathrm{0.2445}& \mathrm{0.0057}& \mathrm{0.0712}& \mathrm{0.7288}& \mathrm{0.0467}\\ \mathrm{0.0501}& \mathrm{0.0012}& \mathrm{0.0280}& \mathrm{0.0356}& \mathrm{0.0175}& \mathrm{0.0929}& \mathrm{0.7118}& \mathrm{0.6155}& \mathrm{0.9899}& \mathrm{0.0372}& \mathrm{0.0807}& \mathrm{0.0145}\\ \mathrm{0.0417}& \mathrm{0.0012}& \mathrm{0.0301}& \mathrm{0.0270}& \mathrm{0.0208}& \mathrm{0.6949}& \mathrm{0.1034}& \mathrm{0.0926}& \mathrm{0.0029}& \mathrm{0.8634}& \mathrm{0.1697}& \mathrm{0.9327}\end{array}\right].$

(13)

The cluster center V is

$V=\left[\begin{array}{cccccc}\mathrm{0.1195}& \mathrm{0.8601}& \mathrm{0.9882}& \mathrm{0.1480}& \mathrm{0.8683}& \mathrm{0.0693}\\ \mathrm{0.8132}& \mathrm{0.3079}& \mathrm{0.2018}& \mathrm{0.6425}& \mathrm{0.1452}& \mathrm{0.8108}\\ \mathrm{0.7609}& \mathrm{0.1872}& \mathrm{0.6646}& \mathrm{0.6950}& \mathrm{0.2474}& \mathrm{0.3957}\\ \mathrm{0.5341}& \mathrm{0.6392}& \mathrm{0.1666}& \mathrm{0.3121}& \mathrm{0.3810}& \mathrm{0.8502}\end{array}\right].$

(14)

The cluster centers can be available as standard crankshaft bearing fault modes. In V, the first, second, third, and fourth line are the normal, slight, moderate, and severe wear cluster center of sample data, respectively. In order to validate this method, we collected vibration signals of four sets of diesel engine with the same type, and the six features were calculated in the images transformed from signals. After standardization, the approach degrees of every line in the standard model V were calculated, and the classes of diagnosed objects were determined according to the maximum value of the approach degrees. Parts of the results are shown in Table 3. Bold part of Table 3 is the maximum approach degree of sample data.

Table 3:

Parts of classification results of sample data.

Index	Features of the sample data	Approach degree				Classification results
		Normal	Slight	Moderate	Severe
1	33.371 66.227 51.143 48.011 90.458 9.5365	0.90349	0.26902	0.41221	0.45267	Yes
2	34.077 65.274 50.222 51.025 89.515 9.9216	0.93431	0.39622	0.53942	0.57971	Yes
3	42.129 61.439 38.538 57.03 83.566 21.63	0.1685	0.83373	0.69054	0.65008	Yes
4	43.602 60.672 38.925 57.362 83.221 21.193	0.13576	0.80099	0.6578	0.61734	Yes
5	43.069 61.293 46.175 56.321 84.483 13.946	0.39394	0.79008	0.91598	0.59123	Yes
6	43.617 60.528 45.748 57.65 84.216 14.27	0.32607	0.74988	0.8481	0.54133	Yes
7	35.746 66.207 40.551 49.158 87.889 20.51	0.64785	0.60968	0.46026	0.81138	Yes
8	40.218 63.833 39.124 53.446 85.512 21.893	0.40407	0.73676	0.68734	0.88564	Yes

In order to observe the effectiveness of this method, we used wavelet decomposition to extract features of the signals. The vibration signals were decomposed by wavelet in 4 layers resolution, and then the wavelet coefficients S1, S2, S3, S4, and S5 from low to high frequency were obtained. Based on these coefficients, the frequency band energy was calculated. After standardization, we obtained the features E1, E2, E3, E4, and E5. Parts of the features from wavelet decomposition are shown in Table 4. Using these features as the input parameters of FCM method, the samples can be classified.

Table 4:

Parts of features from wavelet decomposition.

Wear degree of crankshaft bearing	Group of signals	E1 (%)	E2 (%)	E3 (%)	E4 (%)	E5 (%)
Normal	1	9.6582	16.177	46.369	16.633	12.164
	2	8.5646	19.88	46.948	14.464	10.142
	3	10.1676	18.362	44.799	14.497	12.174
Slight	1	13.354	18.705	24.419	18.623	24.899
	2	14.36	18.603	25.148	17.857	24.032
	3	13.041	19.019	25.118	18.854	23.968
Moderate	1	14.168	9.3078	41.685	16.983	17.857
	2	16.275	8.3216	43.144	14.334	17.926
	3	17.758	6.6254	40.363	17.24	18.013
Severe	1	32.662	10.72	24.791	11.108	20.719
	2	32.693	10.562	26.387	10.576	19.782
	3	30.29	9.317	26.271	10.882	23.24

Through statistics in 200 test samples, the classification accuracy was 86% in our method but was 77% in wavelet decomposition and that demonstrated the good effectiveness of this method.

6. Conclusions

A method of fault detection in diesel engine has been presented, based on symmetrical polar coordinates and image recognition. A type of diesel engine crankshaft bearing wear fault has been taken as an example to explore the method. By this method, vibration signal of diesel engine can be transformed to mirror symmetry snowflake image by symmetrical polar coordinate method without frequency transform. The extracted features from the image include the position of centroid and direction angle can reflect the different wear conditions of the crankshaft bearings. Although the boundaries of normal, mild, and severe wear in crankshaft bearing are fuzzy, FCM clustering can be used to recognize the images from the features. The experimental results show that the method can work well for the fault detection in diesel engine. And the method is simple and effective.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.