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Abstract

Due to complicated transfer paths and strong background noise interference, the fault pattern information deeply hides in common features of the vibration signal at the engine surface. In this study, the refined composite multiscale fuzzy entropy (RCMFE) used to measure the irregularity and self-similarity of time series is proposed to quantify the feature of various fault patterns. Followed by RCMFE, the features dug out are recognized by a parameter-adaptive support vector machine based on the firefly algorithm (FASVM). After putting forward the diagnosing schematics, the RCMFE-FASVM is applied to a fault diagnosis case of a diesel engine on a test rig. A comparative analysis of the four methods to extract features and the four methods to recognize fault patterns are conducted. Results indicate the proposed method has superior recognition performance and can effectively identify the working states of the diesel engine, contrasting the existing methods. Under the small samples and features task of identifying the working states of a diesel engine, the recognition rate of the proposed method with more stability can reach 98.2%, which is larger than other methods. Given the superior performance of the proposed method, the number of input features and training samples should vary from 8 to 20 and from 35 to 50.

Keywords

Diesel engine fault diagnosis vibration signal refined composite multiscale fuzzy entropy firefly algorithm parameter-adaptive support vector machine

Introduction

Due to the superiority over dynamic performance, working stability and security, and economy, diesel engine, as one significant power plant, has been widely applied to some specific domains, for example, engineering machinery, agricultural machinery, and military machinery in particular. Meanwhile, the complicated structure, interaction of various exciting forces, and poor working conditions frequently lead to the unexpected abnormalizes and faults of diesel engines, which have a significant influence on the practical application of a diesel engine.^1,2 Therefore, it is curial importance that the unexpected states were detected timely or as early as possible.

With enhanced research on the fault diagnosis of diesel engines, various methods had been developed, which include the thermal parameters analysis-based methods,³ the vibration analysis-based methods,⁴ and the acoustic analysis-based methods.⁵ Among these methods, due to the abundant working information about the reciprocating and rotating motion, cyclic impacts, and gas-liquid-solid interaction can be provided, the vibration analysis-based methods have been widely used and proved to be excellent diagnosis performance.⁶ Thus, the vibration analysis-based methods for diesel engine faults diagnosis are focused on in the paper.

When the faults diagnosis of a diesel engine is conducted, there are two crucial steps as follows: feature extraction and faults pattern recognition.⁷ The former is that the features were effectively extracted to reflect the prominent pattern information hidden in the vibration signal of diesel engines and input to the classifiers. The latter is that the extracted and determined features are input to the fault classifiers. Given the distribution and statistic of the extracted features under different states of diesel engine, the predetermined parameters in the fault classifiers are determined. After that, the trained faults classifier could be used to recognize the unknown state of the vibration signal of diesel engine. Therefore, the extracted features have a significant influence on the recognition performance of fault classifiers. Based on this, this paper focuses on the feature extraction for the default diagnosis of diesel engines.

In recent decades, various methods to extract features have been developed for the fault diagnosis for rotational machinery, which can be divided into four methods as follows: the based signal process methods,⁸ the based information entropy methods,⁹ the based sparsity measuring methods,^10,11 and the based deep learning methods.¹² In the above methods to extract features hidden in the complicated signal, the third method has been widely used to monitor the working trend and state change of the equipment and is hardly treated as the features used to classify multiple states of equipment;¹³ the fourth methods are based on the amount of historical working data of equipment, meanwhile, the obtained feature parameters by fourth methods are based on the optimization problem and hard to be interpreted.^14,15 Due to that the extracted features are difficult to be interpreted and a larger amount of working data of rotational machinery is hard to be obtained, the third and fourth methods are ignored in this paper.

For the signal processing-based methods, given that the interference of strong background noise and complicated frequency components existing in the vibration signals of diesel engines, various signal processing methods are commonly used to reduce the noise and extract the prominent frequency components, hidden in the vibration signal of diesel engines.⁵ After that, the processed vibration of diesel engine was calculated to feature the fault pattern information of the vibration signal of diesel engines, including time-, frequency- and time-frequency- domain statistic parameters, including means, standard deviation, kurtosis, skewness, gray values, and waveform factor. These commonly signal processing methods include empirical mode decomposition (EMD),^16,17 intrinsic time scale decomposition (ITD),^18,19 wavelet packet transform (WPD),²⁰ variational mode decomposition (VMD),^21,22 empirical wavelet transform (EWT),²³ and their improved algorithms.^24,25 Although, the based signal processing methods to extract the fault features of the diesel engine vibration signals have been proved to be successfully applied to diagnose the faults of diesel engines by various optimized methods. However, there are existing two shortcomings. One is that the above signal processing methods have some limitations:^25,26 mode mixture in ITD and EMD, selection of wavelet basis functions in WPD, determination of central frequencies in EWT and predetermined number of modes and determination of central frequencies in VMD, which lead to the complicated and time-consuming procedure of extracting features. The other one is that the above statistic parameters are used to assess the various characteristics of the whole vibration signal of diesel engines, by which, the internal characters of the vibration signal are ignored and the optimized selection of features is needed.

To settle those drawbacks, various entropy-based methods have been introduced and developed into the feature representation of rolling bearing and gear,^27–29 including sample entropy (SE) and³⁰ fuzzy entropy (FE)³¹.^32–35 Those entropy-based methods in a single time scale are used to be measuring the complexity and irregularity of time series by assessing the internal self-similarity of time series, by which, the fault features, that is, periodicity, cyclic impacts, and cyclostationarity, hidden in the vibration signal of rotational machinery can be deeply mined. The larger entropy value means the more complicated time series. Given the multiscale characteristic of the real-world and complicated signals, Costa et al.³⁶ proposed multiscale sample entropy (MSE) based on the coarse-graining process to strengthen the representation on the complexity and irregularity of time series in multiple time scales. Given the undefined entropy values and the large fluctuations of entropy values in some time scales of MSE, Wu et al.³⁷ proposed an improved MSE algorithm, named refined composite multiscale sample entropy (RCMSE). On the other hand, in the sample entropy algorithm, the step function is used for statistics, which ignores the fuzziness of decision-making in practical engineering. Therefore, Chen et al.³⁸ proposed the fuzzy entropy algorithm (FE) based on fuzzy function and applied it to EEG signal analysis. Based on the superiority of FE compared with SE, and refined composite multiscale coarse-graining methods in RCMSE, Li et al.³⁹ developed an improved entropy method, termed refined composite multiscale fuzzy entropy (RCMFE). The above methods have been proved to effectively resist the disturbance noise and effectively mine the fault pattern information of vibration signal of rolling bearings and gears in the simplified test bench. However, the performance on featuring the vibration signal those entropy-based methods had hardly been validated on complicated equipment, that is, diesel engines, compressors, and so on.

After the faults features were extracted, a multiclasses classifier is utilized to conduct the faults diagnosis of diesel engines. With the extracted features based on RCMFE method, a multi-class classifier is needed to recognize the working state of diesel engines. The commonly used classifiers^40–42 include k-nearest neighbor (KNN),⁴³ extreme learning machine (ELM),⁴⁴ random forest (RF),⁴⁵ and support vector machine (SVM).⁴⁶ Compared with other classifiers, the SVM classifier’s superiority in pattern recognition under a small sample has been validated. In SVM classifier,⁴⁷ the low-dimension data are mapped into high-dimension data, which is prone to recognize the class of data under small sample and feature tasks. However, the predetermined parameters, that is, kernel function, kernel function parameter, and penalty coefficient, have a significant influence on the recognition performance of SVM. Commonly, swarm intelligent search algorithms are used to determine the above parameters of SVM classifier.⁴⁸

In summary, to deeply mine the fault pattern information of the complicated vibration signal of diesel engines, RCMFE is introduced and applied to extract the fault features. After that, the above obtain features are input to the multiclass classifier based on SVM. Considering that the key pre-defined parameters of SVM have an obvious influence on the recognition performance, the firefly algorithm, as an excellent swarm intelligent search method,⁴⁹ is applied to determine the key parameters of SVM. And then, a novel fault diagnosis method based on RCMFE and parameter-adaptive support vector machine based on firefly algorithm (FASVM) is developed in this paper. To illustrate the excellent recognition performance, the proposed method is applied to the experiment data analysis of diesel engines, compared with the based different entropy methods: MFE, MSE, and RCMSE, and other multi-classes classifiers: k-nearest neighbor (KNN), extreme learning machine (ELM), and random forest (RF).

The rest of this paper is organized as follows. First, the detailed calculations of RCMFE, FASVM, and the proposed diagnosis method based on RCMFE and FASVM are shown. Secondly, the conducted experiment and data are illustrated. Thirdly, the experiment results are shown. Finally, the conclusions are summarized.

Method

Refined composite multiscale fuzzy entropy

Fuzzy entropy

In Sample entropy algorithm, Chebyshev distance function and Heaviside function are respectively used to evaluate the similar distance and similarity of two vectors, in which Heaviside function, as a step function, cannot effectively assess the similarity of two vectors.⁵⁰ Based on this, fuzzy entropy based on a fuzzy membership function was proposed.⁵¹ Its calculation procedures are illustrated as follows:⁵¹

(1) Given a time series with data length $N$ , $X (N)$ , ${X (N) = x (1), x (2), \dots, x (N)}$ , it is mapped into high-dimension space based on Taken’s Theorem of Embedding, which is also a special application of phase space reconstruction algorithm when the time delay parameter is equal to 1

S_{i}^{m} = [x_{i}, x_{i + 1}, \dots, x_{i + m - 1}] - u (i), i = 1,2, \dots, N - m + 1

(1)

where

m

is the embedding dimension and

S_{i}^{m}

expresses the m-dimension data of

X (N)

and generalized by removing its average

u (i) = \frac{1}{m} \sum_{j = 0}^{m - 1} x (i + j)

(2) Given the high-dimension shorten data series $S_{i}^{m}$ , its similarity distance of neighboring vector is calculated by Chebyshev distance function:

d_{i, j} [S_{i}^{m}, S_{j}^{m}] = \max_{k = 0,1, \dots, N - m} [| S_{i}^{m} (i + k) - S_{j}^{m} (j + k) |], i, j = 1,2, \dots, N - m + 1 a n d i \neq j

(2)

(3) Fuzzy membership function is introduced to further calculated the fuzzy similarity distance of the neighboring vectors:

A_{i, j} (d_{i, j}) = \exp [- \ln (2) {(\frac{d_{i, j}}{r})}^{n}], d \geq 0, i, j = 1,2, \dots, N - m + 1 a n d i \neq j

(3)

where

n

represents the power of fuzzy function and

r

means similarity tolerance (

r = r \times S D, S D

is the standard deviation of initial time series

X (N)

(4) Given the high-dimension time series $S_{i}^{m}$ , its similarity degree $φ^{m} (n, r)$ is defined as follows:

φ^{m} (n, r) = \frac{1}{N - m} \sum_{i = 1}^{N - m} (\frac{1}{N - m + 1} \sum_{\begin{array}{l} j = 1 \\ j \neq i \end{array}}^{N - m} A_{i j}^{m})

(4)

(5) Given the embedding dimension $m = m + 1$ , the steps (1) ∼ (4) are also repeated and $φ^{m + 1} (n, r)$ is acquired

φ^{m + 1} (n, r) = \frac{1}{N - m} \sum_{i = 1}^{N - m} (\frac{1}{N - m + 1} \sum_{\begin{array}{l} j = 1 \\ j \neq i \end{array}}^{N - m} A_{i j}^{m})

(5)

(6) According to the Shannon entropy theorem, the Fuzzy entropy of the initial time series $X (N)$ is defined as:

F u z z y E n (m, r, N, n) = - \ln (φ^{m + 1} (n, r) / φ^{m} (n, r))

(6)

Multiscale fuzzy entropy with coarse-graining process

Fuzzy entropy is used to measure the complexity of given time series and may fail to account for the multiple time scales pattern information hidden in the complicated real-world signals.³¹ Meanwhile, the fuzzy entropy values may mean arbitrary inconsistent and indistinguishable results. To settle this problem, coarse-graining theory is introduced to measure the complexity of given time series under multiple time scales, termed multiscale entropy. Thirty six multiscale fuzzy entropy (MFE) based on coarse-graining is calculated in two steps:

(1) Given a time series ${X (N) : x (1), x (2), \dots, x (N)}$ , the coarse-graining theory is applied to it and the coarse-grained time series are defined as follows:

P_{j}^{τ} = \frac{1}{τ} \sum_{i = (j - 1) τ + 1}^{j τ} x (i), 1 \leq j \leq ⌊ \frac{N}{τ} ⌋

(7)

where

τ

is a positive integer and represents the time scale factor. This means that the determined time series are processed by a sliding average filter with length

τ

. When

τ = 1

, the coarse-grained time series is the initial time series.

⌊ \frac{N}{τ} ⌋

is a round number.

(2) Fuzzy entropy is applied to calculate the entropy of each coarse-grained time series, termed MFE.

M F E (τ, m, r, N, n) = F u z z y E n (P^{τ})

(8)

Refined composite multiscale fuzzy entropy

There are two shortcomings existing in the above coarse-graining in the MFE. First, the continuous sliding window with the stride equaling to time scale factors is used in the coarse-graining process, which leads to the breakage of some pattern information in the initial time series.⁵² When time scale factors are two, the similarity of two vectors $[x_{1}, x_{2}]$ and $[x_{3}, x_{4}]$ is measured, but that of the two vectors $[x_{1}, x_{2}]$ and $[x_{2}, x_{3}]$ is effectively and reasonably measured. Second, with the increase of time scale factors, the length of the coarse-graining time series decrease, which results in the significant fluctuation of entropy values in the large time scale factors.^6,9

To alleviate the above drawbacks, refined composite multiscale fuzzy entropy (RCMFE) is developed. RCMFE is also obtained in two steps.⁵³

(1) Given a time series ${X (N) : x (1), x (2), \dots, x (N)}$ , the improved coarse-graining process is applied to it and the coarse-grained time series are obtained as:

P_{j, k}^{τ} = \frac{1}{τ} \sum_{i = (j - 1) τ + k}^{j τ + k - 1} x (i), 1 \leq j \leq ⌊ \frac{N}{τ} ⌋, 1 \leq k \leq τ

(9)

for a determined time scale factor,

τ

coarse-graining time series is obtained, and only one time series is generated by the coarse-graining in MFE, which is shown in Figure 1.

Figure 1.

Coarse-graining procedure: (a) Coarse-graining procedure in MFE and (b) Coarse-graining procedure in RCMFE.

(2) When the time scale factor is determined, the fuzzy entropy of $τ$ coarse-graining time series is computed. Then, $φ_{k, τ}^{m + 1} (n, r)$ and $φ_{k, τ}^{m} (n, r)$ are also obtained. Finally, the RCMFE is calculated as

R C M F E (τ, m, r, N, n) = - \ln ({\bar{φ}}_{k, τ}^{m + 1} (n, r) / {\bar{φ}}_{k, τ}^{m} (n, r))

(10)

where,

{\bar{φ}}_{k, τ}^{m + 1} (n, r)

and

{\bar{φ}}_{k, τ}^{m} (n, r)

are respectively the average of

φ_{k, τ}^{m + 1} (n, r)

and

φ_{k, τ}^{m} (n, r)

The compared flowcharts of both MFE and RCMFE are shown in Figure 2. Totally, in RCMFE, the improved coarse-graining method and the entropy averaging method are applied to surmount the shortcomings, existed in MFE, by which the reasonable and stable entropy results are obtained.

Figure 2.

Flowcharts of calculating entropy value: (a) that of MFE and (b) that of RCMFE.

Firefly algorithm-based support vector machine classifier

Given the superiority of SVM in the state recognition of rotation machinery, it was applied to achieve the state identification of diesel engines. Factually, in the application of SVM, various kernel functions should be predetermined, such as polynomial function, sigmoid function, and radial basis function (RBF) in particular. Here, the RBF is selected to map the data into a higher dimension space.²⁸ As known that the two parameters, $c$ and $g$ , termed penalty coefficient and kernel function parameters have significant influence on the recognition performance of SVM. Given the effects of artificial preset these parameters and the superiority on intelligent optimization ability of firefly algorithm,⁵⁴ it was used to determine the above parameters, $c$ , and $g$ . The flowchart of the parameter-adaptive support vector machine based on firefly algorithm (FASVM) are shown in Figure 3. The FASVM are summarized as follows:

(1) The collected samples based on extracted features are divided into respectively the training samples sets and the testing samples sets, both of which would be input into the SVM classifier. Meanwhile, $c$ and $g$ change from $2^{- 2}$ to $2^{2}$ .

(2) The parameters of the firefly algorithm are predetermined (shown in Table 1), that is, number of iterations, number of search agents, randomization parameter $α$ , attractiveness $β$ , and absorption coefficient $γ$ , among which, randomization and attractiveness parameters are used to update the location of fireflies.

(3) The locations of all the fireflies, which mean $c$ and $g$ , are initialized, under which the SVM classifiers are respectively trained and tested.

(4) Given the best results under the above calculations, the locations of all the fireflies are updated.

(5) According to the updated $c$ and $g$ , the results under all the fireflies are repeatedly computed.

(6) Steps (4) and (5) are repeated until the number of iterations reaches the max number of iterations.

Figure 3.

Flowchart of FASVM.

Table 1.

Parameters for firefly algorithm.

Parameters	Number of iteration	Number of search agents	$α$	$β$	$γ$
Values	100	300	0.5	0.2	1

The proposed framework based on RCMFE and FASVM

Based on the above analysis, the based RCMFE and FASVM diagnosis method for diesel engines is developed, where RCMFE and FASVM are respectively applied to feature the fault pattern information hidden in the vibration signal of a diesel engine and recognize the complicated state of a diesel engine. The whole procedure of the proposed framework based on RCMFE and FASVM is exhibited in Figure 4. The further detailed steps are summarized as follows:

(1) The vibration experiment of a diesel engine with normal or abnormal states is separately conducted. Meanwhile, the vibration acceleration sensor is fixed on the body of the diesel engine and the vibration signal of the diesel engine under different states was collected with the determined speed, load, and sampling frequency.

(2) The RCMFE values of all the raw vibration signals of the diesel engine under different states are calculated with the predetermined parameters and time scale factor in particular.

(3) The whole sample set of RCMFEs under the vibration signal of different states was divided into the training sample sets and the testing sample sets. The training samples sets and the testing samples sets were respectively input into the based FASVM multi-faults classifiers. The former sets were used to train the classifier and determine the predetermined parameters in SVM, and the latter sets were applied to test the trained classifier using FASVM. With the above steps, the fault diagnosis of the diesel engine was conducted.

Figure 4.

Flowchart of the proposed diagnosis framework based on RCMFE and FASVM of diesel engine.

Experiment and data collection

In this subsection, the vibration test of one diesel engine was carried out on the test platform. As shown in Figure 5, the test platform mainly was composed of a diesel engine, a dynamometer, a measurement, and control system used to control the diesel and measure the heat parameters, and a data acquisition system mainly used to collect the vibration signal.

Figure 5.

Test platform.

The main parameters of one diesel engine are shown in Table 2. An acceleration sensor and a photoelectric coder were separately mounted as the first cylinder head and the free end of the diesel engine. The vibration signal and rotation speed signal were collected at a sampling frequency of 51,200 Hz. As shown in Table 3, 10 working states in the diesel engine were considered in this article, including one normal state, and two components faults (blocking air intake filter and abnormal intake valve clearance). Compared with the abnormal inlet valve clearance, the blocking inlet gas filter had the similar effect on the diesel engine; the former resulted in the change of the total air inflow at the front of the suction manifold, and the former led to the change of that at the front of one cylinder (it is fixed on the first cylinder closing to the dynamometer).

Table 2.

Main parameters.

Parameter	Value
Number of cylinder	6
Order of fire	1-5-3-6-2-4
Engine displacement	9.5 L
Maximum torque/speed	1900 N.m/1200–1300 r/min
Rated power/speed	294 kW–1900 r/min
Compression ratios	17.5:1
Economic speed	1000–1400 r/min
Inlet valve clearance	0.4 ± 0.03 mm

Table 3.

Working state of the diesel engine in the experiment.

Number	Location of fault	Description	Label
1	No (clearance 0.39 mm)	Normal	1
2	Air intake filter	Blocking 20%	2
3		Blocking 40%	3
4		Blocking 60%	4
5	Intake valve	Clearance 0.19 mm	5
6		Clearance 0.25 mm	6
7		Clearance 0.34 mm	7
8		Clearance 0.44 mm	8
9		Clearance 0.52 mm	9
10		Clearance 0.59 mm	10

For each working state, the running condition with rotating speed at 1200 r/min and 75% load was considered in the experiment. Under the above running condition, a 10 s signal was collected in each working state. Considering the working cycle of a diesel engine, a sample with 5120 data points of each working state under 1200 r/min and 75% load was segmented and 100 samples of each working state were acquired. The time domain vibration signal of every working state is shown in Figure 6. It is difficult to recognize the states of the diesel engine from the time domain vibration signals.

Figure 6.

Time domain signal of each working state.

Experiment result

Features analysis of the vibration signal of the diesel engine

As the above description, 100 samples with 5120 points of each state of diesel engine were applied to test the performance of RCMFE in vibration signal analysis of diesel engine. RCMFEs, MFEs, RCMSEs, and MSEs of all the above samples were separately calculated. According to Ref. [26,29], the parameters of RCMFE and MFE are set as $τ = 20, m = 2, r = 0.15 * S D, n = 2$ . In particular, $τ$ , $m$ , and $r$ are also set in RCMSE and MSE.

In Figure 7, the mean and standard deviation of RCMFEs, MFEs, RCSEs, and MSEs are exhibited. It can be seen that RCMFEs, MFEs, MSEs, and RCMSEs of vibration signal of diesel engine decrease with the increasing of time scales, which indicates that the pattern information hidden in the vibration signal of diesel engine are represented at different time scales and the coarse-graining process is beneficial to feature the complex vibration signal of diesel engine. Second, RCMFEs of vibration signal of the diesel engine with little standard deviation fluctuate slighter than that of MFEs and RCMSEs, which have close fluctuation and fluctuate slighter than that of MSEs and this benefit for promoting the classification performance. Third, as exhibited in Figure 7, RCMFEs, MFEs, RCMSEs, and MSEs of vibration signal of each state of diesel engine differed insignificantly, due to that the above states have a similar influence on the working of diesel engine. Finally, the above feature analysis indicates the superiority of RCMFE to feature the fault pattern information of the diesel engine. Compared with MFEs, RCMSEs, and MSEs, RCMFEs are prone to be more stable, which benefits for the fault recognition of the vibration signal of the diesel engine.

Figure 7.

Mean and standard deviation of different methods of the vibration signal under each working state: (a) mean and standard deviation of RCMFE; (b) mean and standard deviation of MFE; (c) mean and standard deviation of RCMSE; and (d) mean and standard deviation of MSE.

Next, to illustrate the superiority of RCMFE, RCMFEs, as well as MFEs, RCMSEs, and MSEs are visualized by t-SNE (t-Distributed Stochastic Neighbor Embedding). The results are shown in Figure 8, in which the number of the dimensionalities of all RCMFEs, MFEs, RCMSEs, and MSEs are reduced to two. The distribution of total samples has a significant influence on the recognition results, that the concentration of same-class samples and the dispersion of different-class samples in particular, which benefits the recognition of the diesel engine. As illustrated in Figure 8, compared with MSEs, RCMSEs, and MFEs, RCMFEs express the better dispersion of the different-class samples and the aggregation of the same-class samples, with the mixtures of small samples under different classes. The distribution of RCMFEs is beneficial to the further fault recognition of the diesel engine.

Figure 8.

Features distribution of four methods: (a) RCMFE; (b) MFE; (c) RCMSE; and (d) MSE.

Result analysis of the proposed method

The proposed method based on RCMFE and FASVM was applied to recognize the states of the diesel engine with the above collected vibration data, in which RCMFE was calculated to mine the working states pattern information of the diesel engine and FASVM was used to recognize the working states of diesel engine. All 50 samples of each working state were respectively set into training samples sets and testing samples sets. To illustrate the performance of RCMFE, MFE, MSE, and RCMSE were compared, and their recognition rates are shown in Figure 9.

Figure 9.

Recognition result of the different methods: (a) the proposed method based on RCMFE and FASVM; (b) the method based on MFE and FASVM; (c) the method based on RCMSE and FASVM; and (d) the method based on MSE and FASVM.

In Figure 9, the proposed method based on RCMFE and FASVM has the best performance on the fault recognition of the diesel engine, comparing the methods based on RCMSE-FASVM, MSE-FASVM, and MFE-FASVM, and its accuracy recognition rate reaches 98%.

In Figure 9(a), a little number of samples of each state are recognized as other states by the method based on RCMFE and FASVM and there is little misclassification between the different fault types or the different degrees of fault, all the samples under the normal state is correctly recognized. As shown in Figure 9(b) and (c), the methods based on MFE-FASVM and RCMSE-FASVM have a close performance on fault recognition and their recognition is 93.80 and 94.00%, respectively.

Their recognition results of total samples indicate that none of the samples under normal state are recognized as other states, but more fault samples are misclassified as other fault states. As illustrated in Figure 9(d), the method based on MSE and FASVM has the worst performance on the fault recognition of the diesel engine, and its recognition rate is 89%; by this method, most of the samples, including normal and fault states are misclassified. The above experiment analysis results indicate that the superiority of recognition performance of the proposed method based on RCMFE and FASVM has been validated. Compared with the methods based on RCMSE-FASVM, MFE-FASVM, and MSE-FASVM, the method based on RCMFE and FASVM can correctly and effectively recognize the working states of the diesel engine.

Comparison of the different methods for extracting fault features

To further illustrate the superiority of the proposed method based on RCMFE and FASVM, comparing with MFE-FASVM, RCMSE-FASVM, and MSE-FASVM, RCMFEs, MFEs, RCMSEs, and MSEs with different time scales are used to feature the fault pattern information of vibration signal of each state of diesel engines and input to the FASVM parameter-adaptive multi-classifier for multi-fault state recognition of diesel engine. The recognition results are exhibited in Figure 10, in which both the number of training samples and testing samples are 50 and the number of input feature values varies from 1 to 20.

Figure 10.

Recognition rate comparison of the methods based on RCMFE, MFE, RCMSE, MSE, and FASVM under the different number of feature values.

First, under the determined number of training samples, with the increase in the number of input eigenvalues, the accuracy recognition rate of all the methods based on RCMFE, MFE, RCMSE, and MSE increases (when the number of the input eigenvalues is smaller than nine), and tend to be a constant with slight fluctuation, which indicates the entropies of RCMFE, MFE, RCMSE, and MSE under different time scale have different performance on representing the fault patter information of vibration signal of diesel engine. Second, when the number of input eigenvalues is one, all the recognition rates of the recognition methods based on RCMFE, MFE, RCMSE, and MSE have the worst performance and mean about 45%; when the number of input eigenvalues is larger than one, it is obvious that the recognition rate of the recognition method based on RCMFE tends to be 98%, which are always larger than that of the recognition methods based on MFE and RCMSE. In particular, when time scale factor $τ = 1$ , all the diagnosis methods based on RCMFE, MFE, RCMSE, and MSE express the worst recognition results, which means the effectiveness of the coursing-graining process in RCMFE, MFE, RCMSE, and MSE. Third, the coarse-graining process and similarity algorithm (Heaviside function) in MSE was respectively promoted by refined composite coarse-graining in RCMSE and fuzzy membership function in MFE. In the fault recognition of vibration signal of diesel engine, the recognition rate of the methods based on RCMSE-FASVM and MFE-FASVM differed insignificantly and tends to be 94%, which is always larger than that of the recognition method based on MSE-FASVM. Particularly, when the number of input eigenvalues varies from 11 to 17, the recognition rate of the methods based on RCMSE and MFE differed significantly. The recognition method based on MSE has the worst recognition performance and its recognition rate tends to be 89%. Finally, the above experiment data analysis indicates that RCMFE with slight standard deviation fluctuation, compared with MFE, RCMSE, and MFE, has the best performance on the representation of the fault pattern information of vibration signal of diesel engine. When the number of input eigenvalues is larger than one, the recognition rate of the proposed method based on RCMFE and FASVM always expresses superiority recognition performance and tends to be 98%.

To further illustrate the superiority of the proposed method based on RCMFE-FASVM, comparing with MFE-FASVM, RCMSE-FASVM, and MSE-FASVM, RCMFEs, MFEs, RCMSEs, and MSE under twenties time scale are used to feature the pattern information of vibration signal of each state of diesel engine and input to the FASVM parameter-adaptive multi-classifier for multi-fault recognition of diesel engine. The recognition results are exhibited in Figure 11, in which the number of testing samples is 50, and the number of training samples varies from 5 to 50 for each kind of diesel engine states. The recognition results are shown in Figure 11.

Figure 11.

Recognition rate comparison of the methods based on RCMFE, MFE, RCMSE, MSE, and FASVM under the different number of training samples.

First, with the increase in the number of training samples, all the recognition rates of the methods based on RCMFE, MFE, RCMSE, and MSE tend to be constant. Second, when the number of training samples is determined, the proposed recognition method based on RCMFE and FASVM has the best recognition performance, and its recognition rate is larger than that of the recognition methods based on MFE-FASVM and RCMSE-FASVM, which are larger than that of the recognition methods based on MSE-FASVM. Third, with the increase in the number of training samples, the recognition rate of the method based on RCMFE is always larger than that of the methods based on MFE and RCMSE and tends to be about 98%. And, when the number of training samples varies, the recognition rates of the methods based on MFE and RCMSE fluctuate significantly, which indicates both the methods, as one promoted method of MSE, have different performance on representing the fault pattern information of vibration signal of diesel engine. The recognition method based MSE has the worst performance of fault recognition of vibration signal of diesel engine. Finally, the above analysis validates that the proposed method based on RCMFE and FASVM has the best recognition performance, compared with other methods based on MFE-FASVM, RCMSE-FASVM, and MSE-FASVM. When the number of training samples is suitable, the recognition rate of the proposed method based on RCMFE-FASVM tends to be 98%.

Next, to generally illustrate the superiority of the method based on RCMFE and FASVM, the comparing analysis is further conducted. The recognition results are illustrated in Figure 12, in which the number of training samples under each state varies from 5 to 50, and the number of the input features changes from 2 to 20. First, with the change in the number of input features and training samples, the recognition rate of all the methods, that is, RCMFE-FASVM, MFE-FASVM, RCMSE-FASVM, and MSE-FASVM increases. In particular, the best recognition rate of the RCMFE-FASVM-based method reaches 98.2%, which is larger than that of the MFE-FASVM method and RCMSE-FASVM-based method, tending to be about 94%. The MSE-FASVM-based method expresses the worst recognition result, which tends to be 90%. Second, with the increase in the number of input features and training samples, the recognition rate of the RCMFE and FASVM-based method fluctuates slightly and increases gradually. By contrast, the methods based on RCMSE-FASVM, MFE-FASVM, and MSE-FASVM fluctuate significantly and increase discontinuously. Finally, the above analysis indicates that due to the superiority of RCMFE, the RCMFE-FASVM-based method has been to be effective, and the number of input features and training samples has a smaller influence on the RCMFE-FASVM-based method, compared with the methods based on MFE-FASVM and RCMSE-FASVM and MSE-FASVM. To effectively identify the states of the diesel engine, the number of input features and training samples should be determined between [10, 20] and [35 50]. When the number of input features and training samples are suitable, its recognition rate tends to be about 98%.

Figure 12.

Recognition rates under different number of input features and training samples: (a) RCMFE-FASVM; (b) MFE-FASVM; (c) RCMSE-FASVM; and (d) MSE-FASVM.

Comparison of the different classification methods

For illustrating the superiority of the multi-classifier FASVM in the proposed method, the above RCMFEs are also input to other classifiers, that is, Extreme Learning Machine (ELM), k-Nearest Neighbors (KNN), and Random Forest (RF). The results are shown in Figure 13 and Figure 14, where the influence of the number of input eigenvalues and training samples on the fault identification performance of the above four classifiers are expressed.

Figure 13.

Comparing results of RCMFE and four classifiers under different numbers of eigenvalues.

Figure 14.

Comparing results of RCMFE and four classifiers under different numbers of training samples.

As shown in Figure 13, RCMFEs are taken as the input features, both the number of training samples and test samples are fifty for four multi-classifiers, and the number of input features varies from 1 to 20. First, with the increase of the number of input eigenvalues, the recognition rates of four classifiers, FASVM, ELM, KNN, and RF, increase, and fluctuate slightly when the number of input features varies from 9 to 16, and tend to be a constant. This indicates that the recognition performance of four classifiers can be promoted with the increase of the number of input features and not all the input features can be suitable for representing the fault pattern information of diesel engine. Second, when the number of input features is larger than two, the recognition rates of the proposed method based on FASVM are always larger than that of the methods based on ELM, KNN, and RF, whose recognition rates show ELM > KNN > RF. Finally, the best recognition performance of the proposed method based on FASVM has been validated by the above analysis. When the number of the input eigenvalues varies, the recognition rates of the proposed method based on FASVM always retain a better performance, and with the increase of the number of the input features, that tends to be 98.00%, which is larger than that of other methods, that is,. ELM, KNN, and RF.

As shown in Figure 14, RCMFEs are also taken as the input features, the number of test samples and input features are separately 50 and 20 for the above four multi-classifiers, and the number of training samples varies from 5 to 50 for each kind of diesel engine states.

First, with the increase in the number of training samples, the recognition rates of the proposed method based on FASVM increase, tend to be constant, and the recognition rates of the methods based on ELM, KNN, and RF increase and decrease when the number of training samples is larger than 35, which expresses the more stability of the proposed method based on FASVM, compared with other three methods. Second, when the number of training samples increases, the recognition rates of the proposed fault recognition method based on FASVM, almost are larger than that of those methods, based on ELM, KNN, and RF, which express greater diversity under the smaller number of training samples. Finally, the above analysis also validates the superiority of the proposed fault recognition method based on RCMFE and FASVM. When the number of training samples varies, it has little influence on the recognition rate of the method based on RCMFE and FASVM, compared with that of those methods based on RCMFE-ELM, RCMFE-KNN, and RCMFE-RF. When the number of training samples is larger than a suitable number, the recognition rates of the method based on FASVM, tend to be constant at 98.00%, which is much larger than that of those recognition methods based on ELM, KNN, and RF.

Figure 15.

RCMFE and different classifiers: (a) RCMFE-FASVM; (b) RCMFE-ELM; (c)RCMFE-KNN; and (d) RCMFE-RF.

First, with the change in the number of input features and training samples, the recognition rates of all the methods, that is, RCMFE-FASVM, RCMFE-ELM, RCMFE-RF, and RCMFE-KNN increase. In particular, the best recognition rates of those methods indicate that RCMFE-FASVM (98.2%)> RCMFE-ELM (97.4%)> RCMFE-KNN (88.2%)> RCMFE-RF (79.2%). Second, with the increase in the number of input features and training samples, compared with the recognition rate of the RCMFE and FASVM-based method, the methods based on RCMFE-ELM, RCMFE-KNN, and RCMFE-RF fluctuate significantly and increase discontinuously, which means that the RCMFE-FASVM-based method expressed the strong robustness and stability. Finally, the above analysis indicates that due to the SVM classifier’s superiority in the recognition task of small features and samples, the RCMFE-FASVM-based method has been to be effective in the experimental analysis of diesel engine, and are less influenced by the number of input features and training samples have smaller influence, compared with the methods based on RCMFE-ELM, RCMFE-KNN, and RCMFE-RF.

Conclusion

In this paper, the fault features extracted from vibration signals of power machinery are quantified by RCMFE, and the fault patterns are recognized by FASVM. The proposed algorithm is applied to a fault test of a diesel engine. The main conclusions are summarized as follows:

(1) The RCMFE in multiple time scales can effectively reflect and mine the fault pattern information of vibration signal of diesel engine. Compared with commonly used MFE, RCMSE, and MSE, the proposed RCMFE shows the lowest fluctuation in a specific fault pattern and the greatest distinctions between multiple fault patterns.

(2) The proposed method based on RCMFE-FASVM, has better recognition performance and more stability than the used existing methods, that is, MFE-FASVM, MSE-FASVM, and RCMSE-FASVM. When the number of input features and training samples are suitable, the recognition rate of the proposed method can reach 98.2% which is larger than that of the method based on MFE-FASVM, RCMSE-FASVM, and MSE-FASVM whose recognition rates are 93.8, 94, and 89%.

(3) The comparison experiment data analysis also indicates that compared with the used existing method, that is, RCMFE-ELM, RCMFE-KNN, and RCMFE-RF, the proposed method based on RCMFE-FASVM expressed the best recognition performance with more stability. When the number of input features and training samples are suitable, the recognition rate of the proposed method can reach 98.2% which is larger than that of the method based on RCMFE-ELM, RCMFE-KNN, and RCMFE-RF whose recognition rates are 97.4, 88.2, and 79.2%.

(4) To effectively recognize the fault states of the diesel engine, the number of input features and training samples should vary from 8 to 20 and from 35 to 50.

The entropy methods have not been commonly applied to extract the fault features of diesel engines and other specific equipment. Further research could be conducted, that is, further improved methods combined the working cycle of diesel engine, the multi-frequency components, and multiple isomorphic- or heterogeneous-information fusion.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was funded by the Open Project of State Key Laboratory of Engine Reliability in China (SKLER:202009).

ORCID iD

Xiaolong Zhu

References

Junhong

Fengrong

, et al. A fault diagnosis approach for diesel engine valve train based on improved ITD and SDAG-RVM. Meas Sci Technol 2015; 26: 1–14.

Sun

Yin

, et al. Experimental investigation on the characteristics and performance of homogeneous charge induced ignition(HCII) on a multi-cylinder heavy-duty engine. Fuel 2020; 280: 118610.

Cai

Weng

Zhang

. A novel approach for marine diesel engine fault diagnosis. Cluster Comput 2017; 20: 1691–1702.

Mathew

Zhang

. Acoustic-based engine fault diagnosis using WPT, PCA and bayesian optimization. Appl Sci (Switzerland) 2020; 10: 1–18.

Delvecchio

Bonfiglio

Pompoli

. Vibro-acoustic condition monitoring of internal combustion engines: a critical review of existing techniques. Mech Syst Signal Process 2018; 99: 661–683.

Tharanga

KLP

Liu

Zhang

, et al. Diesel engine fault diagnosis with vibration signal. J Appl Math Phys 2020; 08: 2031–2042.

Kumar

Lokesha

Kumar

, et al. Vibration based fault diagnosis techniques for rotating mechanical components: review paper. IOP Conf Ser Mater Sci Eng 2018, 376: 012109.

Tian

, et al. A feature extraction and visualization method for fault detection of marine diesel engines. Meas J Int Meas Confederation 2018; 116: 429–437.

Zhao

Wang

, et al. Twenty years of entropy research: a bibliometric overview. Entropy 2019; 21: 1–27.

10.

Hurley

Rickard

. Comparing measures of sparsity. IEEE Trans Inf Theor 2009; 55: 4723–4741.

11.

Hou

Wang

Xia

, et al. Investigations on quasi-arithmetic means for machine condition monitoring. Mech Syst Signal Process 2021; 151: 107451.

12.

Zhang

Dai

. Mechanical fault diagnosis methods based on convolutional neural network: a review. J Phys Conf Ser 2021; 1750: 012048.

13.

Hou

Wang

Xia

, et al. Generalized Gini indices: complementary sparsity measures to Box-Cox sparsity measures for machine condition monitoring. Mech Syst Signal Process 2022; 169(August 2021): 108751.

14.

Wang

Kuen

, et al. Recent advances in convolutional neural networks. Pattern Recognition 2018; 77: 354–377.

15.

Zhao

, et al. Deep learning algorithms for rotating machinery intelligent diagnosis: An open source benchmark study. ISA Trans 2020; 107: 224–255.

16.

Huang

. Review of empirical decomposition. Wavelet Appl VIII 2001 2001; 4391: 71–80.

17.

Wang

, et al. Diesel engine gearbox fault diagnosis based on multi-features extracted from vibration signals. IFAC-PapersOnLine 2021; 54: 33–38.

18.

Frei

Osorio

. Intrinsic time-scale decomposition: time-frequency-energy analysis and real-time filtering of non-stationary signals. Proc R Soc A: Math Phys Eng Sci 2007; 463: 321–342.

19.

Zhang

Liu

. Application of complete ensemble intrinsic time-scale decomposition and least-square SVM optimized using hybrid DE and PSO to fault diagnosis of diesel engines. Front Inf Technol Electron Eng 2017; 18: 272–286.

20.

Jiang

, et al. Fault diagnosis of diesel engines based on wavelet packet energy spectrum feature extraction and fuzzy entropy feature selection. J Vibration Shock 2020; 39: 273–298.

21.

Dragomiretskiy

Zosso

. Variational mode decomposition. IEEE Trans Signal Process 2014; 62: 531–544.

22.

Liu

, et al. Knock detection based on the optimized variational mode decomposition. Meas J Int Meas Confederation 2019; 140: 1–13.

23.

Gilles

. Empirical wavelet transform. IEEE Trans Signal Process 2013; 61: 3999–4010.

24.

Qiao

Jin

, et al. A novel fault diagnosis method for diesel engine based on mvmd and band energy. Shock and Vibration 2020: 2020–2117.

25.

Feng

Zhang

Zuo

. Adaptive mode decomposition methods and their applications in signal analysis for machinery fault diagnosis: a review with examples. IEEE Access 2017; 5: 24301–24331.

26.

Das

Prusty

Bingi

. Review of adaptive decomposition- based data preprocessing for renewable generation rich power system applications review of adaptive decomposition-based data preprocessing for. J Renew Sustain Energ 2021; 13: 062703.

27.

Wang

Liu

, et al. The entropy algorithm and its variants in the fault diagnosis of rotating machinery: a review. IEEE Access 2018; 6: 66723–66741.

28.

Zhu

Zheng

Pan

, et al. Time-shift multiscale fuzzy entropy and laplacian support vector machine based rolling bearing fault diagnosis. Entropy 2018; 20: 602.

29.

Wang

Yao

Cai

. Rolling bearing fault diagnosis using generalized refined composite multiscale sample entropy and optimized support vector machine. Meas J Int Meas Confederation 2020; 156: 107574.

30.

Wang

Zhao

, et al. Fault diagnosis method based on modified multiscale entropy and global distance evaluation for the valve fault of a reciprocating compressor. J Mech Eng 2019; 65: 123–135.

31.

Zheng

Pan

Cheng

. Rolling bearing fault detection and diagnosis based on composite multiscale fuzzy entropy and ensemble support vector machines. Mech Syst Signal Process 2017; 85: 746–759.

32.

Pincus

. Approximate entropy as a measure of system complexity. Proc Natl Acad Sci United States America 1991; 88: 2297–2301.

33.

Wang

Liu

. Concentric diversity entropy: a high flexible feature extraction tool for identifying fault types with different structures. Mech Syst Signal Process 2022; 171: 108934.

34.

Wang

. Multiscale diversity entropy: a novel dynamical measure for fault diagnosis of rotating machinery. IEEE Trans Ind Inform 2021; 17: 5419–5429.

35.

Wang

Yang

, et al. Multiscale symbolic fuzzy entropy: an entropy denoising method for weak feature extraction of rotating machinery. Mech Syst Signal Process 2021; 162(May 2021): 108052.

36.

Costa

Goldberger

Peng

. Multiscale entropy analysis of biological signals. Phys Rev E 2005; 71: 021906.

37.

S De

Lin

, et al. Analysis of complex time series using refined composite multiscale entropy. Phys Lett Section A Gen At Solid State Phys 2014; 378: 1369–1374.

38.

Chen

Wang

Xie

, et al. Characterization of surface EMG signal based on fuzzy entropy. IEEE Trans Neural Syst Rehabil Eng 2007; 15: 266–272.

39.

Miao

Zhang

, et al. Refined composite multiscale fuzzy entropy: localized defect detection of rolling element bearing. J Mech Sci Technol 2019; 33: 109–120.

40.

Zhang

, et al. A rotating machinery fault diagnosis method based on multi-scale dimensionless indicators and random forests. Mech Syst Signal Process 2020; 139: 106609.

41.

Jafarian

Mobin

Jafari-Marandi

, et al. Misfire and valve clearance faults detection in the combustion engines based on a multi-sensor vibration signal monitoring. Meas J Int Meas Confederation 2018; 128: 527–536.

42.

Liu

Yang

Zio

, et al. Artificial intelligence for fault diagnosis of rotating machinery: a review. Mech Syst Signal Process 2018; 108: 33–47.

43.

Zhao

Zhang

Jiang

, et al. A new fault diagnosis method for a diesel engine based on an optimized vibration mel frequency under multiple operation conditions. Sensors (Switzerland) 2019; 19: 2590.

44.

Kowalski

Krawczyk

Woźniak

. Fault diagnosis of marine 4-stroke diesel engines using a one-vs-one extreme learning ensemble. Eng Appl Artif Intelligence 2017; 57(October 2016): 134–141.

45.

Zhao

Mao

Wei

, et al. Fault diagnosis of diesel engine valve clearance based on variational mode decomposition and random forest. Switzerland: Applied Sciences, 2020, p. 10.

46.

Liu

. Fault diagnosis of valve clearance in diesel engine based on BP neural network and support vector machine. Trans Tianjin Univ 2016; 22: 536–543.

47.

Jing

Liu

, et al. Diesel engine valve clearance fault diagnosis based on features extraction techniques and fastICA-SVM. Chin J Mech Eng (English Edition) 2017; 30: 991–1007.

48.

Yan

Jia

. A novel optimized SVM classification algorithm with multi-domain feature and its application to fault diagnosis of rolling bearing. Neurocomputing 2018; 313: 47–64.

49.

Sharma

Zaidi

Singh

, et al. Optimization of SVM classifier using Firefly algorithm. 2013 IEEE 2nd International Conference on Image Information Processing, IEEE ICIIP 2013 2013, Shimla, India, 09–11 December 2013: 198–202.

50.

Azami

Fernández

Escudero

. Refined multiscale fuzzy entropy based on standard deviation for biomedical signal analysis. Med Biol Eng Comput 2017; 55: 2037–2052.

51.

Chen

Zhuang

, et al. Measuring complexity using FuzzyEn, ApEn, and SampEn. Med Eng Phys 2009; 31: 61–68.

52.

Minhas

Singh

, et al. A novel method to classify bearing faults by integrating standard deviation to refined composite multi-scale fuzzy entropy. Meas J Int Meas Confederation 2020; 154: 107441.

53.

Zhang

. Bearing fault diagnosis based on refined composite multi-scale dispersion entropy and extenics. IEEJ Trans Electr Electron Eng 2021; 17: 70–485.

54.

Kumar

. A systematic review on firefly algorithm: past, present, and future. Arch Comput Methods Eng, 2021; 28: 3269–3291.

Refined composite multiscale fuzzy entropy based fault diagnosis of diesel engine

Abstract

Keywords

Introduction

Method

Refined composite multiscale fuzzy entropy

Fuzzy entropy

Multiscale fuzzy entropy with coarse-graining process

Refined composite multiscale fuzzy entropy

Firefly algorithm-based support vector machine classifier

The proposed framework based on RCMFE and FASVM

Experiment and data collection

Experiment result

Features analysis of the vibration signal of the diesel engine

Result analysis of the proposed method

Comparison of the different methods for extracting fault features

Comparison of the different classification methods

Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

ORCID iD

References