Keywords

1 Introduction

Engine valve is an important part of diesel engine [1]. Considering the change of vibration response of diesel engine caused by valve failure, the fault feature of the valve can be extracted by analyzing the vibration response of the diesel engine. The complexity of diesel engine and the complex movement mode of moving parts with complex shapes make it very difficult to diagnose and monitor the engine state accurately and quantitatively.

Fractal is a mathematical set with high geometric complexity, which can simulate many natural phenomena [2]. This is consistent with the idea that multi-scale morphology is used to measure the geometric shape of the object being analyzed at different scales [3]. In order to calculate fractal dimension by using multiscale morphological operators, the improvement and optimization of this method from the angle of improving the computational efficiency, using morphological filtering operators (i.e. erosion and dilation) of fractal dimension estimation [4]. The fractal dimension algorithm based on morphology has the advantages of low computational complexity, good stability and sensitivity to the change of signal geometry, and unaffected by the amplitude of vibration signals [4,5,6]. In practice, because the measuring vibration signals mostly contain noise, the noise components have a great influence on the calculation of fractal dimension, so as to affect the accuracy of the measuring vibration signal characteristics. Vibration signal must be denoised to obtain a fractal dimension.

The ensemble empirical mode decomposition (EMD) can decompose the non-linear and non-stationary signals into finite intrinsic mode components (IMFs) according to the local time characteristics of the signals [8,9,10]. In this paper, multi-scale morphological fractal dimension and ensemble EMD are introduced into vibration signal analysis. First, the vibration signal of the diesel engine is preprocessed by ensemble EMD to eliminate noise interference and obtain fault information of characteristic IMF. Then the fractal dimension of the diesel engine is calculated to quantify the working state of diesel engine accurately, which provides a new method for condition monitoring and fault diagnosis of the diesel engine.

2 Fractal Dimension Estimation Based on Morphological Operations

Morphology is a mathematical method developed on the basis of set theory, integral geometry and topology. It is different from space–time analysis and frequency domain analysis. Erosion and expansion are two basic operations of mathematical morphology. They can remove details smaller than structural elements in the mesoscale of the signal while maintaining the basic characteristics of the signal, so as to obtain the simplified signal data.

This cover can be obtained by using one-dimensional erosion and expansion of \( f(n) \) by a function structural element \( g(m) \), the real function of \( g(m) \) and \( f(n) \) are defined, respectively, in two discrete domains \( F = {\text{\{ 0,1,2,}} \ldots N - 1 {\text{\} }} \) and \( G = {\text{\{ 0,1,2,}} \ldots M - 1 {\text{\} }} \). where \( f (n) \) is a time serial signal, \( g (m) \) is a structural element, by using one-dimensional operation erosion and expansion on \( f (n) \) by a function structure element \( g(m) \), these operations are defined as,

$$ f\varTheta g(n) = \mathop {\hbox{min} }\limits_{m \in G} \{ f(n + m) - g(m)\} $$
(1)
$$ f \oplus g(n) = \mathop {\hbox{min} }\limits_{m \in G} \{ f(n - m) + g(m)\} $$
(2)

Hence, for one-dimensional discrete time series \( f(n) \), the structure element is defined at scales \( \varepsilon \) [4],

$$ \varepsilon g(n) = g \oplus g \oplus \cdot \cdot \cdot \oplus g\,\,\varepsilon {\text{TIMES}} $$
(3)

Taking \( \varepsilon = 1,2, \ldots ,\varepsilon_{\hbox{max} } \), as the range of analysis scale, \( \varepsilon_{ \hbox{max} } \le N/2 \), then the result of erosion and expansion of the signal \( f(n) \) at different scales \( \varepsilon \) is \( f\varTheta \varepsilon g(n) \) and \( f \oplus \varepsilon g(n) \), respectively.

The coverage area of the signal at scale \( \varepsilon \) is defined as

$$ A_{g} (\varepsilon )= \sum\limits_{{{\text{n}} = 1}}^{N} {(f \oplus \varepsilon g(n )} - f\varTheta \varepsilon g(n ) ) $$
(4)

According to [4], the following conditions are satisfied,

$$ \log \frac{{A_{g} (\varepsilon )}}{{\varepsilon^{2} }} = D_{M} \log \frac{1}{\varepsilon } + c $$
(5)

where \( \varepsilon = 1,2, \ldots ,\varepsilon_{\hbox{max} } \).

The fractal dimension \( D_{M} \) of signal is defined as,

$$ D_{M} = \mathop {\lim }\limits_{\varepsilon \to 0} \frac{{\log \left[ {A_{g} (\varepsilon )/\varepsilon^{2} } \right]}}{{\log \left( {\frac{1}{\varepsilon }} \right)}} $$
(6)

In practice, the fractal dimension \( D_{M} \) of signal is equal to the slope of line segment fitted via least squares to \( \log (A_{g} (\varepsilon )/\varepsilon^{2} ) \) and \( \log \left( {\frac{1}{\varepsilon }} \right) \).

The fractal dimension estimation algorithm only uses one-dimensional structure element, so the computational complexity of the algorithm is linear, which greatly improves the computational efficiency. According to the analysis in [4], we use the flat structure element of length 3 as the unit structure element, i.e. \( g(m) = \{ 0,0,0\} \). The planar structure unit is not affected by the amplitude range of the analysis signal, which reduces the amount of calculation and improves the accuracy of fractal dimension estimation. There is no fixed method for the selection of the largest scale, but the choice of scale is too big to bring a lot of calculations. According to the method of determining the maximum grid-scale of periodic signal and combining with the characteristics of diesel engine working cycle vibration signal, the maximum scale \( \varepsilon_{\hbox{max} } \) is 60.

3 Ensemble EMD

EMD is used to analyze the non-linear and non-stationary signals [7]. It can decompose any complex signal into finite IMFs to represent the natural oscillation modes embedded in the non-linear and non-stationary signals, and acting as the basis function determined by the original signal. However, schema mixing is one of the fundamental drawbacks of EMD. It will not only cause serious deterioration of time-frequency distribution but also produce vague meanings of a single IMF. In order to eliminate these shortcomings of EMD, an ensemble EMD, which incorporates white noise into the analysis signal, is proposed as a noise-aided data analysis method [8]. The ensemble EMD program is as follows:

  1. (1)

    The time series \( N(t) \) of white noise is added to the signal \( y(t) \).

    $$ S(t) = y(t) + N(t) $$
    (7)
  2. (2)

    The data series \( S(t) \) is decomposed into finite IMFs.

    $$ S(t) = \sum\limits_{j = 1}^{n} {c_{j} (t)} + r_{n} (t) $$
    (8)
  3. (3)

    Repeat steps 1 and 2, but use different white noise series each step, \( i = 1 \sim m \),

    $$ S_{i} (t) = y(t) + N_{i} (t) $$
    (9)
    $$ S{}_{i}(t) = \sum\limits_{j = 1}^{n} {c_{ij} (t)} + r_{in} (t) $$
    (10)
  4. (4)

    The average value of IMF is obtained and considered as the final decomposition result.

    $$ c_{j} (t) = \frac{1}{m}\sum\limits_{i = 1}^{m} {c_{ij} (t)} $$
    (11)

In order to verify the effectiveness of ensemble EMD, EMD and integrated EMD are used to decompose the simulation signal, which consists of low-frequency sinusoidal signal and small impulse component. Figure 1 shows time-domain waveform of the simulation signal. Figure 2 shows decomposition results by using EMD. Because of abnormal interference of noise, EMD generates mode mixing, and emerges the pseudo IMFs which cannot meet requirements of fault characteristic extraction. Sinusoidal signal and impact signal are decomposed into the same IMF C1. In addition, the sinusoidal signal is decomposed into the different IMFs C1 and C2, decomposition results emerged a very severe warping, EMD generates the mode mixing, and leads to the pseudo IMFs which cannot meet the requirements of fault characteristic extraction [11].

Fig. 1
figure 1

Simulation signal

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Fig. 2
figure 2

Decomposition results by using EMD

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The simulation signal is decomposed by the ensemble EMD method. We determine that the decomposition ensemble number is 100 and the additional noise amplitude is 0.01 times the standard deviation of the signal. Figure 3 shows the decomposition results, the component C1 corresponding impact signal components, the component C2 corresponding to sine signal, small impact component and sine signal can be accurately decomposed.

Fig. 3
figure 3

Decomposition results by using ensemble EMD

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4 Analysis of Engine Fault Signal

A four-cylinder two-stroke diesel engine under normal operating conditions and varying degrees of exhaust valve leakage under the conditions. We measured the vibration response signal of the diesel engine on the same cylinder surface, which, respectively, represents the engine in normality, slight leakage and severe leakage state. The sampled frequency is 25.6 kHz. The rotational speed of diesel engine crankshaft is 1100 r/min. The vibration signals of diesel exhaust valves under three operating conditions are shown in Fig. 4.

Fig. 4
figure 4

Vibration signal of the diesel engine in three conditions

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The method presented in this paper is applied to identify the leakage of diesel exhaust valves. Using the fractal dimension estimation algorithm, the fractal dimension of the vibration signal in Fig. 4 is calculated directly, as shown in Fig. 5. In these three cases, the response of background noise is similar in morphology, so the fractal dimension is very close, and it is hard to identify the state of the exhaust valve.

Fig. 5
figure 5

Morphological fractal dimension of vibration signal in three conditions

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By using ensemble EMD, the vibration response of diesel engine can be decomposed into several IMFs, which represent different frequency components of the diesel engine vibration signal. According to the fault mechanism of exhaust valve leakage and the fault information of high-frequency components contained in the vibration component, the fractal dimension is only calculated and decomposed by IMF1 component, which can quantify the working state of the exhaust valve. Therefore, the IMF1 of diesel engine vibration response under three conditions is estimated by using a morphological fractal dimension. As shown in Fig. 6, the fractal dimension can be clearly separated from the three regional conditions of diesel engine exhaust valve leakage.

Fig. 6
figure 6

Morphological fractal dimension of characteristic IMF in three conditions

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5 Conclusion

Mathematical morphology provides a new fast analysis method for digital signal processing. This paper presents a fractal dimension calculation method based on mathematical morphology. Ensemble EMD can effectively separate characteristic components from non-linear non-stationary vibration signals and the noise measured from the diesel engine, thereby improving noise–signal ratio of signal fractal analysis. By morphological fractal dimension analysis of characteristic components, morphological fractal dimension can quantitatively describe the geometric characteristics of diesel engine vibration signals. By analyzing the fractal dimension of the measured vibration signal, it is shown that the method can effectively identify fault feature information of diesel engine vibration signal.