Successive variational mode decomposition and blind source separation based on salp swarm optimization for bearing fault diagnosis

In this paper, we are interested in developing a new approach that combines successive variational mode decomposition and blind source separation based on salp swarm optimization for bearing fault diagnosis. Firstly, vibration signals are pre-processed using successive variational mode decomposition to increase the signal-to-noise ratio. Then, the dynamic time-warping algorithm is adopted to select the most effective modes which will be considered mixture signals. In the second step, we apply the salp swarm algorithm (SSA) for estimating the de-mixing matrix to extract independent components from mixture signals. However, SSA suffers from the problem of population diversity. Consequently, it offers somewhat different independent sources at every execution of the program. To overcome this shortcoming, the SSA-based source estimation will be executed several times with different ranges of initial positions. Then, a fuzzy C-mean algorithm is introduced to select the reliable independent components. The suggested method is tested based on two experiments and compared to state-of-the-art methods. The obtained results demonstrate the effectiveness of the suggested method in recovering reliable independent components and extracting the fault frequency of bearings.


Introduction
Bearings represent the most crucial element in rotating machines. To decrease the process shutdown, monitoring the bearing state is an essential step. Several methodologies have been developed for bearing fault diagnosis [1][2][3][4]. Although it has several advantages, vibrations signals are always contaminated with noise that should be reduced as possible as we can. Several signal-processing methods have been developed to overcome this shortcoming such as the following: wavelet analysis, empirical mode decomposition, and variational mode decomposition (VMD) [5][6][7][8].
Wavelet analysis needs a good choice of the mother wavelet and the maximum level of decomposition [5]. The main disadvantage of empirical mode decomposition is mode mixing [9,10]. Variational mode decomposition requires the right choice of the penalty parameter and the decomposition number of the intrinsic mode function [7]. In order to ameliorate these disadvantages and increase the convergence rate without extracting the useless modes, a new algorithm called successive variational mode decomposition (SVMD) is developed. It has the advantages that it does not require to precise the number of modes, and it has a lower computational complexity than VMD [11].
Selecting relevant modes is another important task. The dynamic time warping algorithm (DTW) has been successfully employed to select salient modes, it provides better results than the correlation factor [12].
A blind source separation algorithm has been recently applied for bearing fault diagnosis [13][14][15]. Researchers have employed an independent component algorithm (ICA). This algorithm is based on the estimation of a de-mixing matrix in order to extract the source signals [16,17]. However, the shortcoming of ICA is that it requires the calculation of the derivative of the search space. In other words, it suffers from the fact that it is based on a gradient algorithm in which several parameters affect the obtained results [16,18].
Several swarm optimization algorithms, such as PSO and Bat, have been employed for blind source separation. Even though they have the advantage of not requiring the calculation of the derivative of the search space, they cannot ensure the global minimum and suffer from the problem of population diversity [19][20][21][22].
For all the above reasons, it is necessary to use a robust algorithm that can calculate the de-mixing matrix without being affected by the choice of the initial positions of the population.
Salp swarm algorithm (SSA) is a recent swarm optimization algorithm that has been successfully employed for solving various optimization problems [23][24][25][26]. It has the advantage of simplicity and efficiency [27,28]. In addition, SSA has fewer initial parameters compared to other swarm algorithms [29,30]. However, like other swarm algorithms, SSA suffers from the problem of population diversity [31,32].
In this paper, the separation matrix is estimated based on the Salp swarm algorithm. However, the suggested algorithm offers somewhat different independent sources at every execution of the program because of the initial positions of the population.
In order to overcome this shortcoming, the separation matrix based on the salp swarm algorithm will be executed several times with different ranges of initial positions of the population. As a result, several independent components will be obtained. An unsupervised classifier, based on a fuzzy C-mean algorithm as a simple clustering method, is then adopted to select the reliable independent sources.
To validate the suggested approach, two experimental setups have been considered. In these experiments, vibration signals are collected from tri-axial accelerometers based on low and high-speed rotating frequency.
The rest of this paper is organized as follows: Sections 2 and 3 describe successive variational mode decomposition and the principal of blind source separation, respectively. Section 4 introduces the basics of the salp swarm algorithm. Section 5 briefly reviews the fuzzy c-means clustering algorithm. The suggested methodology is detailed in Section 6. In Section 7, the experimental results and comparison are given and discussed. Finally, the main conclusions and future work are provided in Section 8.

Successive variational mode decomposition
We assume that we have a signal x(t) which is composed of two signals: Lth mode u L (t) and the residual signal x r (t) . x(t) is formulated as follows: where the residual signal contains two parts viz; the sum of the previously obtained modes and the unprocessed part x u (t) as follows: Basically, SVMD is based on the minimization of the four criteria. The first criterion is as follows: Here, (t) and w k denote the Dirac function and the center frequency of the Lth mode, respectively. * represents the convolution operation.
The second criterion is as follows: where L (t) represents the impulse response of the filter given as follows: The third criterion is given by the following equation: Here, i (t) denotes the impulse response of the filter given by the following equation: The last constraint is to guarantee the complete reconstruction of x(t) from the Lth modes and the un-processed part of the signal as follows: The SVMD optimization problem can be defined by the constrained minimization problem given as follows: The International Journal of Advanced Manufacturing Technology (2023) 125:5541-5556

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To transform the constrained minimization problem defined by Eq. (9) into an unconstrained optimization problem, the Lagrangian function can be formulated as follows: where denotes the Lagrangian multiplier.
Based on the Parseval's theorem, Eq. (10) can be transformed to the frequency domain form as follows: Finally, û L , w L , and ̂ are updated iteratively as follows: Here, x(w) and û n L (w) denote the Fourier transform of x(t) and u n L (t) at the n th iteration with the center frequency w n L , respectively. n represents the number of iterations, and denotes the iteration step length.
The flowchart of the SVMD is as follows [11]: The setting of the balancing factor is a very crucial step in the implementation of the SVMD algorithm. A low value of may lead to the mode mixing problem, whereas a high value of may generate some useless modes. To overcome this shortcoming and select the optimal value of , an algorithm of SVMD with varying between min and max has been developed [11]: 1 3

Principle of blind source separation
Blind source separation algorithms recover the independent components based on observed mixtures. Consider that we have a matrix " X " which contains " n " observations, where " X " is given as follows The task of the source separation algorithms is to estimate a mixing matrix " A " in order to find the source signals s = [s 1 , s 2 , ....s l ].
The mixing matrix can be identified as follows: where x i with i = 1...n denotes the i th observed signal and s l with l = 1...m is the l th independent component. The sources signals can be calculated based on the following equation: Here " s " is a matrix that contains our independent sources and " W " is the separation matrix.
The de-mixing matrix is given as follows: As given in Eq. (17), it can be seen that if we have, for example, an observation matrix that contains three mixture signals (n = 3) and we want to recover two independent sources (l = 2), the separation matrix is a 2*3 matrix containing six parameters to be estimated.
The performance of earlier methods for source separation such as ICA is heavily dependent on the right choice of the contrast function that measures the degree of statistical independence of the independent sources and the optimization technique employed to estimate the separation matrix, which cannot ensure the best solution.
The main focus of this paper is to explore the power of the salp swarm algorithm for estimating the separation matrix and resolving the problem of the dependence of the obtained results on the initial positions of the population.

Salp swarm algorithm
Salp swarm algorithm (SSA) is a novel bio-inspired optimization method that simulates the behavior of the salp chains in deep oceans [33].
Salps have a transparent barrel-shaped body which is very similar to jellyfishes. In addition, their movement is highly identical to that of the latter. They pump water through their body in order to change their position. The salp shape is illustrated in Fig. 1a.
In deep oceans, salps always create a swarm named the salp chain. The formed chain is illustrated in Fig. 1b. This behavior is very interesting. Indeed, as interpreted by some scholars, this behavior aims to accelerate the information flow and optimize the search of food.
The salp chain contains two groups: the slap at the head of the chain is called the leader, and the rest of the slaps are named followers. The position of each salp is given using an n-dimensional search space where n represents the number of variables that need to be optimized.
The leader position is updated based on the following equation [33]: Here, x 1 j represents the leader position in the j th dimension, F j indicates the food source position in the j th dimension, and ub j and lb j represent the upper bound and the lower bound of the j th dimension, respectively. c 1 , c 2 and c 3 are random variables. From Eq. (18), it can be seen that only the leader salp updates its position according to the food source. c 1 is the most important coefficient of the salp swarm algorithm; it is used for balancing the exploration and exploitation. It is given as follows [33]: Here, l and L represent the current iteration and the maximum iteration number, respectively. c 2 and c 3 denote random coefficients belonging to [0, 1].
The follower position is updated based on the following equation [33]: Here, x i j denotes the i th follower salp position in j th dimension, and " a " represents the slap's acceleration. Since the step of iteration is equal to 1 and considering v 0 = 0 , Eq. (20) can be reformulated as follow [33]:

Fuzzy C-mean algorithm
The fuzzy c-means (FCM) algorithm was proposed by Dunn [34], and later, it was enhanced by Bezdek [35]. FCM algorithm is based on the principle of fuzzy logic in the definition of clusters. The optimization problem of the FCM algorithm [35] is given by: where X = x 1 , x 2 , … , x n 1 stand for a data set with n data which must be clustered into C C C clusters (1 < C ≤ n) , by calculating the centroids of the clusters v j (j = 1, 2, … , C) and the membership matrix U = [u jk ] . u jk m denotes the degree of membership of x k in the k th clus-ter, m is a fuzziness index, and d 2 x k , v j denotes the Euclidean distance between vector x k and the center x k .The matrix v j and vector v j must be updated via Eqs. (23) and (24),respectively The criterion for stopping iterations, which usually is 6 The suggested methodology Figure 2 outlines the block diagram of the suggested methodology for recovering the independent sources based on the salp swarm algorithm. It is as follows: Step 1. Collection of the vibration signal Step 2. Preprocessing of the measured signal based on successive variational mode decomposition, then selecting the three relevant modes based on the dynamic time warping criterion Step 3. Extract the envelope of each mode by Hilbert transform Step 4. Construction of the input matrix Step 5. Estimation of the separation matrix ' U = [u jk ] ' based on the SSA algorithm. Here: • The position of each salp is given using a dim-dimensional search space where " dim " represents the number of elements in the separation matrix. In other words, it is a vector with a length equal to n .
• The fitness function is the average of the correlation coefficient for all independent sources between each other; the following expression gives it: where: Here, n , n * l , and 0 ≤ k ≤ N denote the s i and the s j independent sources, respectively. " i th " represents the data length, j th , N and R x,s (k) = Step 6. Obtain the independent sources based on Eq. (16) SSA-based sources estimation offers somewhat different independent sources at every execution of the program due to the fact of the initial positions of the population. For this reason, another stage using an unsupervised classifier based on a fuzzy C-mean algorithm is introduced to select the reliable independent sources.
The detail of the suggested algorithm based on fuzzy C-mean for selecting the reliable independent sources is given as follows: Step 1. Definition of the number of cycles Step 2. Estimation of the separation matrix based on SSA as illustrated in Fig. 2. As a result, various independent sources will be obtained.
Step 3. Construction of the " N c " matrix at the n iteration as follows: Here, M denotes the dissimilarity coefficient measuring the distance between Z th and D p,q . It is given as follows: Step 4. Compute the cost function s p as follows: Here, s q denotes the cost function of the C z,i independent component at the C z,i cycle.
Step 5. Since i th jump to step 2. Else, go to the next step Step 6. The fuzzy c-mean clustering algorithm is applied to cluster the whole of the obtained independent sources using the " Z th " matrix as input. Here, the maximum number of iteration of the fuzzy C-mean is equal to 300, and the exponent for the partition matrix is 2. Here, the valid- ity of the fuzzy C-mean algorithm is tested based on two evaluation criteria: (i) variance partition entropy (VPE) [36][37][38][39], (ii) variance partition coefficient (VPC) [36,[38][39][40]. From their definitional formulas, it is given that the best clustering results are obtained when the VPC is close to 1 and the VPE is close to 0.
Step 7. Finally, the independent source with the smallest cost function compared to other independent sources in the same cluster is considered as the reliable independent source.
It should be noted that the initial positions of the population are a function of the number of iterations " Z < N c ," lower band " M ," and upper band " Z ." The initial positions of the population are given as follows:

The first experiment
In the first experiment, vibration signals were gathered from the test bench of the laboratory of electrical engineering of Guelma, as illustrated in Fig. 3.
Here, a tri-axial accelerometer that can simultaneously measure vibration signals through three perpendicular directions is used for data acquisition.
An outer race defect is introduced in a bearing of type UC204. It should be noted that the rotating frequency is 18 Hz, and the characteristic frequency of the fault is approximately 55.13 Hz.
The application of the proposed approach is as follows: Firstly, vibration signals are collected using the tri-axial accelerometer. Then the y axis vibration signal is processed using successive variational mode decomposition. After that, three salient modes are selected based on the dynamic time-warping criterion. Figure 4 illustrates the dynamic time-warping factor of all modes.
As presented in Fig. 4, we can see that the fifth, the third, and the first modes have the highest DTW factors, respectively. After selecting relevant modes, their envelopes are extracted using the Hilbert transform.
The next step consists in estimating the separation matrix based on the suggested methodology explained in Section 6. Here, the population size lb is 30 and the number of cycles Posotion = randn(Na, dim)⋅ * (ub − lb) + lb Where lb = Z * ones(dim, 1) ub = 10 * Z * ones(dim, 1)withZ = 1...N c Na is the population size dim is the search space dimension ub is 20.For the fuzzy c-mean algorithm, the maximum number of iterations is equal to 100, and the exponent for the partition matrix is 2. After the application of the fuzzy c-mean algorithm for clustering all components, we obtained the VPE = 0.1038 and the VPC = 0.8684. Figures 5, 6, and 7 illustrate the obtained results based on SSA-fuzzy C-mean, PSO-fuzzy C-mean, and Bat-fuzzy C-mean, respectively.
From the results in Figs. 5, 6, and 7, we can see that the suggested methodology based on SSA followed by fuzzy C-mean gives good results in which the envelope spectrum of the first independent source contains the rotating frequency (18 Hz). In contrast, the envelope spectrum of the second independent source contains the fault frequency of the outer race (55.13 Hz) and some of its harmonics. It can also be seen that BAT-fuzzy C-mean and PSO-fuzzy C-mean give good results.
To verify the effectiveness of the suggested methodology based on SVMD and SSA blind source separation, several algorithms have been implemented and tested viz: SVMD based on joint approximate diagonalization of eigenmatrices (JADE) blind source separation, SVMD based on second order blind identification (SOBI) blind source separation, wavelet packet (WPT) based on JADE blind source separation, and wavelet based on SOBI blind source separation [41,42]. The obtained results are illustrated in Figs. 8, 9, 10, and 11. From the results in Figs. 8, 9, 10, and 11, it can be seen that the sources are not separated and are not detected except in the case of the application of SVMD followed by SOBI blind sources separation where the sources are well detected and separated.

The second experiment
In the second experiment, vibration signals were collected from the test rig developed by the DIRG lab of Politecnico di Torino depicted in Fig. 12 [43]. The test rig comprises a high-speed spindle and three Bearings (B1, B2, and B3).
In this study, the considered bearing is B1, from which accelerations were recorded using three axes accelerometer placed in A1. An inner race fault of 450 µm was introduced in B1. The rotating frequency is 100 Hz with a static load equal to 1800 N and the theoretical characteristic frequency of the inner race fault is 611 Hz.
The application of the proposed approach is as follows: Firstly, vibration signals with 4096 samples are collected from a tri-axial accelerometer positioned in A1. Then vibration signal of the y axis is processed using successive variational mode decomposition. Then, the three most useful modes that have the highest dynamic time-warping factors are selected. Figure 13 illustrates the DTW factor of all the obtained modes.
From the results in Fig. 13, we can observe that the second, the fourth, and the sixth mode have the highest DTW factors, respectively. Following that, envelopes of the three selected modes are extracted based on Hilbert transform. Fig. 6 Envelope spectra of the two independent sources obtained after applying SVMD and PSO based on fuzzy C-mean Fig. 7 Envelope spectra of the two independent sources obtained after applying SVMD and Bat based on fuzzy C-mean 1 3 The next step consists in estimating the separation matrix based on the suggested methodology.
In this experiment, the population size Na is 30 and the number of cycles N c is 20. For the fuzzy C-mean algorithm, the maximum number of iterations is equal to 100, and the exponent for the partition matrix is 2. After the application of the fuzzy C-mean algorithm for clustering all components, we have obtained the VPE = 0.0853 and the VPC = 0.8935. Figures 14, 15, and 16 illustrate the obtained results based on SSA-fuzzy C-mean, PSO-fuzzy C-mean, and Bat-fuzzy C-mean, respectively.
From the results in Figs. 14, 15, and 16, we notice that the suggested methodology based on SSA followed by fuzzy C-mean gives good results. Indeed, the rotating frequency (87.5 Hz) and some of its harmonics appear on the envelope spectrum of the first independent source, and the fault frequency of inner race (537 Hz) and some of its harmonics can be seen on the envelope spectrum of the second independent source.
In addition, we can see that PSO-fuzzy C-mean and Batfuzzy C-mean give bad results because the fault frequency cannot be easily recognized in the envelope spectrum of the second component. This is due to the existence of the rotating frequency that has a magnitude very close to that of the fault frequency.
It should be noted that the rotating frequency and the fault frequency of the inner race are clear at around 87% of the nominal ones due to the applied load. Fig. 8 Envelope spectra of the two independent sources obtained after applying SVMD and JADE Fig. 9 Envelope spectra of the two independent sources obtained after applying SVMD and SOBI Fig. 10 Envelope spectra of the two independent sources obtained after applying WPT and JADE Fig. 11 Envelope spectra of the two independent sources obtained after applying WPT and SOBI Fig. 12 The test rig of the second experiment Fig. 13 The DTW factor of all modes In order to show the effectiveness of the proposed approach based on SVMD and SSA blind source separation, several algorithms have been implemented and tested viz: SVMD based on JADE blind source separation, SVMD based on SOBI blind source separation, WPT based on JADE blind source separation, and WPT based on SOBI blind source separation [42,43]. The obtained results are illustrated in Figs. 17, 18, 19, and 20.
From the results in Figs. 17 and 18, it can be seen that the two methods based on JADE and SOBI cannot separate the two components of the mixture signals obtained using SVMD. In addition, the results in Figs. 19 and 20 clearly show that the application of wavelet packet algorithm at the fourth level using db8 wavelet as mother wavelet followed by JADE and SOBI cannot separate the two sources.
From the obtained results in experiment 1 and experiment 2, it can be concluded that the suggested methodology based on SVMD and SSA followed by fuzzy C-mean provides the best results compared to the other studied approaches. Fig. 14 Envelope spectra of the two independent sources obtained after applying the suggested methodology using SVMD and SSA based on fuzzy C-mean Fig. 15 Envelope spectra of the two independent sources obtained after applying SVMD and PSO based on fuzzy C-mean 1 3

Conclusions
The approach developed in this study concerns the diagnosis of bearing faults. The suggested methodology consists of applying the successive variational mode decomposition algorithm to preprocess vibration signals and reduce noise. Then, relevant modes that have the highest dynamic timewarping factors were employed to construct an input mixed matrix. In addition, the salp swarm algorithm is adopted for the extraction of independent sources by estimating the separation matrix. However, SSA-based source estimation could not ensure good results due to the problem of the initial positions of the population. Therefore, to address the issue of the dependence of the obtained sources on the initial positions of the population, the SSA-based source estimation will be executed several times with different Fig. 16 Envelope spectra of the two independent sources obtained after applying SVMD and Bat based on fuzzy C-mean Fig. 17 Envelope spectra of the two independent sources obtained after applying SVMD and JADE ranges of initial positions, then a fuzzy C-mean algorithm is adopted to select the reliable independent components.
The suggested methodology was tested and evaluated over two experimental setups, and compared to several algorithms such as SVMD based on JADE blind source separation, SVMD based on SOBI blind source separation, WPT based on JADE blind source separation, and WPT based on SOBI blind source separation. The comparative experiments and evaluations demonstrate the effectiveness and the good performance of the suggested methodology for bearing fault diagnosis.
In future work, more research is needed to apply swarm algorithms for selecting the optimal balancing parameter of SVMD. In addition, improved swarm algorithms that do not have the shortcoming of population diversity can be employed for estimating the separating matrix. Future work could also establish if the suggested methodology can be applied for extracting combined faults. Fig. 18 Envelope spectra of the two independent sources obtained after applying SVMD and SOBI Fig. 19 Envelope spectra of the two independent sources obtained after applying WPT and JADE Fig. 20 Envelope spectra of the two independent sources obtained after applying WPT and SOBI