1. Introduction
Rotating machinery, such as electric motors, centrifugal pumps, and turbine engines, represent the most widely used mechanical equipment in industrial processes [
1]. The mechanical equipment usually operate under unstable loads and harsh working conditions, thus various failures of their critical components, such as bearing damage and impeller damage, are inevitable. The operating states of rotating machinery directly affect the productivity and safety of the industrial sector. Therefore, accurate and reliable fault diagnosis of rotating machinery is of great practical significance [
2].
The key to fault diagnosis of rotating machinery is to extract fault features from vibration signals. Vibration signals are nonlinear and nonstationary [
3], and are easily interfered by noise, thus it is difficult to extract hidden features. Therefore, it is necessary to combine the appropriate time–frequency analysis method with the entropy measurement method to extract the hidden tiny fault features. The first step is to choose the appropriate signal processing method. Studies have shown that when the fault signal is disturbed by noise, traditional time–frequency analysis techniques, such as Fourier transform (FFT) and Wavelet Transform (WT) cannot accurately extract fault features [
4,
5]. The more commonly used method is the Empirical Mode Decomposition (EMD) method proposed by Huang et al. in 1998 [
6]. The EMD can adaptively decompose the signal into the sum of finite intrinsic mode functions (
IMF), each
IMF component represents a set of characteristic scale signals, and the feature extraction of each component can better reveal the fault information intrinsic characteristics. However, EMD suffers from modal aliasing, end-point effects, and a lack of rigorous mathematical framework for using envelopes in an iterative manner [
7]. Although the Ensemble Empirical Mode Decomposition (EEMD) [
8] optimized on the basis of EMD can effectively improve the problem of mode aliasing, and the Fast Ensemble Empirical Mode Decomposition (FEEMD) [
9] further improves the calculation speed, neither escape the drawbacks of using envelopes in an iterative fashion without a rigorous mathematical framework. Subsequently, Dragomiretskiy K et al. proposed a new adaptive Variational Mode Decomposition (VMD) method. The method is a non-recursive variational decomposition model, and the optimal solution of the variational model is iteratively searched by the alternating direction multiplier method, thereby determining the center frequency and bandwidth of each mode. It avoids mode mixing in EMD, and has better robustness to noise [
10]. However, VMD suffers from relatively slow computational efficiency, and its performance depends heavily on its two input parameters, namely the penalty factor and the number of decomposition modes [
11]. The Iterative Filtering (IF) method proposed by Lin et al. and its derivatives [
12], such as the Adaptive Local Iterative Filtering (ALIF) method [
13], the Fast Iterative Filtering (FIF) method [
14] can produce results similar to EMD-based algorithms, with the important advantage that their convergence and stability are guaranteed. Moreover, the FIF method uses a fixed low-pass filter function to replace the envelope mean curve in the EMD method, which solves the problem of EMD lacking a strict mathematical framework. Meanwhile, the FIF method is unaffected by mode aliasing, and mode splitting can be easily avoided by adjusting the value of the stopping criterion parameter [
4]. Furthermore, FIF greatly improves the calculation speed on the basis of ensuring decomposition accuracy, with small decomposition error, good noise robustness, and can achieve efficient and accurate signal decomposition [
15]. Therefore, this paper adopts the FIF method to decompose the vibration signal of rotating machinery.
The components of the vibration signal following decomposition by FIF contain rich fault information. Moreover, the components of vibration signals in different states of rotating machinery show different complexity, so the entropy parameter can be used to extract the fault information [
16]. Approximate Entropy (ApEn), Sample Entropy (SampEn), Fuzzy Entropy (FE), and Permutation Entropy (PE) are widely used in the field of rotating machinery fault diagnosis to measure the complexity of vibration signals [
17,
18,
19]. However, ApEn and Multiscale Approximate Entropy include the comparison of their own data segments in the calculation process, and their calculation depends on the data length. If the data length is short, the obtained value is usually smaller than the actual value. The SampEn is an improvement on the approximate entropy. It does not include the comparison of its own data segments, and has higher calculation accuracy and better consistency. However, SampEn and its improvements also have clear shortcomings: Firstly, SampEn and its improvements use Heaviside functions to measure the complexity of time series, resulting in inaccurate estimates in practical applications [
20]. Secondly, SampEn and its improvements are computationally inefficient, especially for long time series. FE and its improvements replace the Heaviside function with a fuzzy membership function that is insensitive to background noise and highly sensitive to dynamic changes, but it is computationally inefficient [
16]. PE is a method to measure the complexity of chaotic time series. PE has high computational efficiency, can be used to calculate huge datasets, and exhibits good anti-noise performance. However, the main disadvantage of PE is that it is prone to generating undefined entropy values for short-term time series and cannot classify well-defined patterns for a specific design [
21]. In order to overcome the above problems, Hamed Azami et al. proposed a nonlinear time complexity evaluation method of Dispersion Entropy (DE). DE can generate reliable entropy values, is insensitive to noise interference, can accurately capture signal characteristics, and calculate with high efficiency [
22]. Subsequently, in order to improve the extraction ability of hidden fault features, Hamed Azam et al. continued to propose the Refined Composite Multiscale Fluctuation-based Dispersion Entropy (RCMFDE), which can more accurately analyze the complexity of nonlinear time series under various scale factors, with more stable entropy values [
23].
However, in the RCMFDE method, there are two key parameters (i.e., embedding dimension and class number) that need to be manually selected in advance. Furthermore, the parameter setting of the RCMFDE algorithm will affect the final processing result. If the parameter settings are unreasonable, the hidden tiny fault features may not be accurately extracted, resulting in misclassification. Aiming at the determination of the embedding dimension m and the class number c in the RCMFDE algorithm, this paper proposes a Parameter Adaptive Refined Composite Multiscale Fluctuation-based Dispersion Entropy (PARCMFDE). The method takes skewness as the objective function, and uses a Genetic Algorithm (GA) to optimize parameters of RCMFDE. PARCMFDE can automatically and effectively determine the important parameters of RCMFDE, so as to describe the complexity and uncertainty of time series more accurately, and achieve the purpose of extracting the features of hidden faults. In view of the shortcomings of existing methods, relevant research is carried out, and the main contributions are as follows:
- (1)
PARCMFDE based on GA is proposed, which overcomes the insufficiency of experience-based parameter selection. PARCMFDE can more accurately extract tiny fault features hidden in vibration signals of rotating machinery.
- (2)
A fault diagnosis method for rotating machinery based on FIF, PARCMFDE and Fuzzy C-Means (FCM) is proposed, which can classify rotating machinery faults accurately and automatically without depending on the length of data samples.
- (3)
The effectiveness of the method is verified by the bearing data of Case Western Reserve University and the experimental data of centrifugal pumps obtained by building a water circulation experimental system. Compared with other methods, it shows that feature extraction of PARCMFDE is more accurate and stable, and the rotating machinery fault diagnosis method based on FIF, PARCMFDE and FCM exhibits better classification effect.
This paper is mainly divided into the following sections:
Section 2 briefly introduces the basic principles and characteristics of the FIF algorithm. In
Section 3, the principle of PARCMFDE is introduced and compared with RCMFDE and Multiscale Sample Entropy (MSE) and Multiscale Fluctuation-based Dispersion Entropy (MFDE).
Section 4 briefly introduces the principle and evaluation index of FCM.
Section 5 presents the method of fault diagnosis of rotating machinery.
Section 6 verifies the effectiveness of the method and compares it with other vibration signal fault diagnosis methods through the bearing data of Case Western Reserve University and experimental data from centrifugal pumps obtained by building a water circulation experimental system.
Section 7 provides the conclusion.
2. Fast Iterative Filtering
The key idea of Fast Iterative Filtering is to iteratively subtract the simple oscillatory components contained in the signal from the signal itself, the so-called IMFs, by approximating the moving average of the signal, thereby separating the simple oscillatory components in the signal [
14]. The approximate moving average is computed by convolution with the window/filter function
w. Consider a raw vibration signal
, define a window/filter function
w is a non-negative even function in the range of
. The Fokker–Plank filter is used here, and
,
denotes the Fourier transform of
s,
denotes the discrete Fourier transform, and
denotes the inverse discrete Fourier transform. The specific implementation process of FIF is as follows:
- (1)
Calculate the length
L of the corresponding filter
w of the signal
:
where
N is the total number of sampling points of the signal
,
k is the number of its extreme points, and
is a tuning parameter, which is usually fixed around 1.6 for the Fokker–Plank filter.
- (2)
Calculate the discrete Fourier transform of the signal and the corresponding filter w, denoted as and ), respectively.
- (3)
- (4)
Calculate
and
:
where
> 0, represents the required precision;
represents the number of iterations required to achieve the required precision
when calculating a specific
;
represents the
kth element of the Fourier transform of the signal
s;
represents the
kth eigenvalue;
is the
kth eigenvector;
I is the identity matrix;
D is the diagonal matrix, whose diagonal is the eigenvalue.
- (5)
Judgment of inner loop stop condition: if the stop standard
is met, then stop the inner loop, otherwise let
repeat steps (3)–(5), the stop standard
is calculated by the following formula:
- (6)
Calculate the
component and the new
s:
- (7)
Judgment of outer loop stop condition: Calculate the extreme point of s, if there is only one extreme point of s or less, the outer loop stops, otherwise repeat steps (1)–(7).
- (8)
Extract the final
IMF component
In short, the FIF method includes two processes: inner loop and outer loop. The purpose of the inner loop is to filter out the IMF components of each order. The purpose of the outer loop is to end the process of extracting the IMF component of the inner loop. When the residual obtained by removing all IMF components from the original signal contains only one or less extreme points, the outer loop stops.
4. Fuzzy C-Means Clustering
Fuzzy C-means (FCM) clustering algorithm is the most widely used fuzzy clustering algorithm based on objective function. It obtains the membership degree of each sample point to all class centers by optimizing the objective function, so as to determine the class of the sample point to achieve the purpose of automatically classifying the sample data [
28].
Let
be the set of data samples, and
n is the number of samples.
is the cluster center vector, and
t is the total number of clusters. The FCM clustering algorithm minimizes the objective function shown in Equation (
15) through continuous iteration of the least squares method, and its constraints are shown in Equation (
16).
where
is the
kth sample point to be clustered,
is the degree of membership of
to the
ith cluster center
,
m is the weight index of the degree of membership, generally
.
The cluster center
and the membership matrix
are randomly selected initially. Then iteratively calculate through Equations (
17) and (
18), and stop until the change of the objective function is less than the threshold.
The average fuzzy entropy
E, classification coefficient
S and classification accuracy
are used to analyze and evaluate the clustering effect of the fuzzy C-means, which are, respectively, defined as:
where
and
denote the actual class and the class assigned by FCM on the test dataset,
n is the number of samples in the test dataset.
The ambiguity of clustering is represented by the average fuzzy entropy
E, which reflects the distribution characteristics of the clustering dataset, so it can be used as an index to judge the clustering effect and correctness. The smaller the ambiguity, the higher the order of the system. The classification coefficient
S measures the overlap between clusters, and the closer it is to 1, the more effective the clustering result [
29]. Therefore, the closer
E is to 0, the closer
S is to 1, and the closer
is to
, the better the sample clustering effect is.
7. Conclusions
To overcome the shortcomings of traditional feature extraction methods that bear difficulty in extracting tiny fault features hidden in vibration signals, and the shortcomings of RCMFDE to select parameters based on experience, a PARCMFDE is proposed. PARCMFDE takes the skewness of RCMFDE as the objective function, and uses genetic algorithm to optimize parameters. PARCMFDE can more accurately extract tiny fault features hidden in vibration signals of rotating machinery. At the same time, a new fault diagnosis method for rotating machinery based on FIF-PARCMFDE-FCM is proposed, which can classify rotating machinery faults accurately and automatically without depending on the length of data samples. FIF quickly decomposes the original vibration signal, and selects components with large correlation coefficients for reconstruction. The reconstructed signal features are extracted by PARCMFDE, and the feature vector is formed into FCM for automatic label-free classification. The bearing experiments with clear fault characteristics prove that the classification performance of this method is superior to other methods. Experiments on centrifugal pumps with weak fault features demonstrate that this method can extract hidden weak fault features from vibration signals and perform accurate and reliable automatic classification. Therefore, the proposed diagnostic method can achieve reliable diagnosis performance for rotating machinery.
However, the proposed method only identifies single faults of rotating machinery, and does not consider the identification of compound faults. Furthermore, PARCMFDE is slower than RCMFDE. Therefore, the identification of compound faults in rotating machinery and the improvement of the computing speed of PARCMFDE will be regarded as the focus of our future work.