Variational mode decomposition (VMD) is a new method of signal adaptive decomposition. In the VMD framework, the vibration signal is decomposed into multiple mode components by Wiener filtering in Fourier domain, and the center frequency of each mode component is updated as the center of gravity of the mode’s power spectrum. Therefore, each decomposed mode is compact around a center pulsation and has a limited bandwidth. In view of the situation that the penalty parameter and the number of components affect the decomposition effect in VMD algorithm, a novel method of fault feature extraction based on the combination of VMD and particle swarm optimization (PSO) algorithm is proposed. In this paper, the numerical simulation and the measured fault signals of the rolling bearing experiment system are analyzed by the proposed method. The results indicate that the proposed method is much more robust to sampling and noise. Additionally, the proposed method has an advantage over the EMD in complicated signal decomposition and can be utilized as a potential method in extracting the faint fault information of rolling bearings compared with the common method of envelope spectrum analysis.

Rolling bearing is one of the most commonly used parts in rotating machinery to support rotating shafts. Due to the fact that its health state is directly related to the safety and a stable operation of the machine, the research of rolling bearing fault diagnosis has a great significance in actual application [

Currently, there are many methods used to fault diagnosis for rolling bearings, but these methods have some inherent limitations. For instance, wavelet transform (WT) [

Lately, Dragomiretskiy and Zosso proposed a new variational mode decomposition (VMD) [

The flowchart about the proposed method.

The rest of the paper is organized as follows. In Section

Variational mode decomposition (VMD) is a new method of signal decomposition based on Wiener filtering, one-dimensional Hilbert transform, and heterodyne demodulation proposed lately by Konstantin Dragomiretskiy. Different from EMD, it defines the mode component as amplitude-modulated-frequency-modulated (AM-FM) signals as follows:

The input signal is decomposed by VMD method into the mode component of a specified scale according to the subjective setting scale assuming that each mode is a finite bandwidth signal with a pulse as the center. In order to evaluate the bandwidth of each mode and construct the constraint model of the variational problem, VMD firstly uses the Hilbert transform to obtain the single spectrum of each mode and then transfers them to the fundamental frequency by exponential correction. The bandwidth of each mode is obtained through Gauss smooth demodulation signal finally, which is called as

Therefore, the Lagrange multipliers, modal functions, and their corresponding central frequency are iteratively updated by using alternating direction multiplier method (ADMM) to obtain the saddle point in the expression. Specific algorithm of classical VMD is given as the flowing expression.

Initialize

The value of

Repeat the iterative process of (

Particle swarm optimization (PSO) is an intelligent algorithm to imitate birds’ foraging behavior proposed by Kennedy, through referring to the characteristics of all the individual bird in the process of feeding, which is widely used in solving nonlinear problems.

Each particle in the PSO algorithm is used as the solution of the optimization problem, which has a position and the corresponding speed determined by the optimization function. The algorithm evaluates the pros and cons of all particles by setting the appropriate fitness function. In each iteration, the particles constantly update their speed and position according to the fitness value of individual and the group. The updated particles continue to search the optimal value in the search space.

Specific configuration steps of PSO are as follows.

Establish the appropriate fitness function according to the actual problem. The iteration calculation is carried out by setting the number of iterations, population number, the initial position, and velocity of the particles.

Calculate the optimal values

Update the position and speed of all particles in the population according to the formula

Repeat Steps

In the traditional algorithm of variational mode decomposition, the user needs to set the penalty parameter and the number of the components before processing the signal because of the theory limitation. From the theoretical study of VMD, it can be known that the larger penalty parameter indicates the smaller bandwidth of each component decomposed by source signal and vice versa. Similarly, inappropriate setting number of components will also result in some unacceptable mode compositions. Therefore, selecting the appropriate parameter group of the component number and the penalty parameter is the key to accurately extract the fault information.

PSO algorithm is a widely used intelligent optimization algorithm compared with other optimization algorithms such as genetic algorithm and artificial fish algorithm. It is suitable for the optimal selection of parameters in consideration of its simple principle and mechanism, fast convergence speed, and, meanwhile, the good performance of global search. The key part of the PSO algorithm based on the variable mode decomposition is the selection of fitness function. Because of the incorrect settings of the penalty parameter and the number of components, some artifact components will generate, which are independent with the source signal. It is acceptable that the artifact components have less similarity with the source signal. Therefore, the cross-correlation coefficient between the decomposed mode component and the original signal is regarded as an evaluation index, which is defined in the following formula:

From the above analysis, it can be seen that the cross-correlation coefficient

On the basis of the above theory analysis, VMD based on PSO algorithm is applied to the analysis of simulated signal and the fault feature extraction of rolling bearing experiment system to verify the validity of the method in fault diagnosis.

The measured rolling bearing vibration signal is always consisted of the amplitude-modulated-frequency-modulated (AM-FM) signals, harmonic signal, and noisy signal in actual application. In order to verify the validity of the VMD based on PSO, the fault signal model is built by the following simulated signal:

The simulated signal

The VMD based on particle swarm optimization algorithm is applied to decompose the above simulated signal. The number of iterations and the particles is 20, the inertia weight linear decrease in the iterative process of the initial value is 0.9, and the final value is 0.4. The penalty parameter and the number of components optimized and selected by particle swarm optimization (PSO) search algorithm are a collection of (2064, 3), and the results of the decomposition are shown in Figure

The result of VMD decomposition.

The fitness value during the iteration process

The original and reconstructed sinusoidal signal

The original and reconstructed cosine signal

The original and reconstructed FM signal

In the three decomposed components shown in Figures

The cross-correlation coefficient of reconstruction signal.

Sinusoidal signal | Cosine signal | FM signal | |
---|---|---|---|

Cross-correlation coefficient | 0.9962 | 0.9985 | 0.9974 |

Empirical mode decomposition (EMD) is used to decompose the above simulated signals, and the results of EMD decomposition are shown in Figure

The result of EMD decomposition.

Similarly, the decomposition result also carried out similarity analysis with the original component. Since the similarity of the three signal components (sinusoidal signal, cosine signal, and FM signal) compared with mode components after IMF3 is basically close to zero, so there is only a list of the previous three components shown in Table

The result of EMD method.

Sinusoidal signal | Cosine signal | FM signal | |
---|---|---|---|

IMF1 | 0.2021 | 0.6132 | 0.5921 |

IMF2 | 0.7983 | 0.2180 | 0.0032 |

IMF3 | 0.0276 | 0.0014 | 0 |

From the decomposition results of EMD, it can be known that sine, cosine, and FM signal are partly mixed in IMF1 and IMF2, which cannot be separated well. Compared with the proposed method in this paper, the decomposition effect of VMD is obviously better than EMD, which can well separate the components from the original signal.

The above experimental results show that the proposed method can almost completely separate the components from the simulation signal free of noisy signal. However, the actual measured bearing vibration signal in the runtime is often affected by the strong noise background. Thus, the fault feature information is usually submerged in the noise environment. In order to prove the validity of the method, we discuss the feature extraction effect of this method under the different noise level. In the simulation signal, the Gauss white noise with standard variance being 0.1, 0.2, 0.4, 0.3, 0.5, 0.7, and 0.9 is added in turn. The cross-correlation coefficient between the decomposed mode components and the original components in different noise conditions is calculated. Results of signal reconstruction by the proposed method are shown in Figure

The results of VMD decomposition with varied noise level.

The similarity of sinusoidal component with different noise level

The similarity of cosine component with different noise level

The similarity of FM component with different noise level

The noise of different intensity is added in the simulation signal, and then the signal is processed by using the proposed method. From Figure

In order to verify that the proposed method is effective in the experiment, the vibration data of rolling bearing experiment system is used to be analyzed. The experimental system is shown in Figure

The experimental parameters and fault frequency.

Rotating speed |
Rotating frequency/Hz | Sampling frequency/Hz | Sampling time/s | Outer fault frequency/Hz |
---|---|---|---|---|

1450 | 24.17 | 16384 | 1 | 87.01 |

The rolling bearing experiment system.

The time-domain graph of measured bearing fault signal is shown in Figure

The time-frequency diagram of the measured signal.

Time-domain graph of the measured signal

Envelope spectrum graph of the measured signal

From Figure

The fitness value during the iteration process.

The correlated kurtosis of different component.

The envelope spectrum analysis of the eighth decomposed component.

It can be seen from the decomposition results of optimized VMD algorithm that the identification accuracy of fault feature frequency is improved compared with the traditional envelope spectrum analysis shown in Figure

The decomposition results of EMD.

A novel method of particle swarm optimization in variational mode decomposition method was introduced in faint fault feature extraction of rolling bearing. The main conclusions of this paper include the following. (1) The particle swarm optimization algorithm was applied to the parameter selection of the optimal penalty parameter and the number of components, which largely depends on the suitable fitness function determined by the maximum ratio between the mean value and the variance of cross-correlation coefficient. Moreover, the maximum correlated kurtosis is used to select the optimal component. It is significant that the proposed method can avoid the interference of human experience and the diagnostic results are more reasonable. (2) Simulated signal and measured fault bearing signal measured from the rolling bearing experiment system were used to verify the validity of the method. The result demonstrated that the proposed method has an advantage over the traditional EMD method and envelope spectrum analysis in faint fault signal processing for rolling bearings, which make it possible for the proposed method to be a powerful tool in solving the problem of signal channel bind source separation.

The authors declare that there are no competing interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (nos. 51475339 and 51405353) and the Key Laboratory of Metallurgical Equipment and Control of Education Ministry, Wuhan University of Science and Technology (2015B11).