2.2. Adaptive Variational Modal Decomposition
Compared with EEMD and other adaptive decomposition algorithms, the VMD algorithm has better sparse modal components, but the decomposition result of the VMD algorithm is affected by multiple parameters, among which the modal number and penalty factor have a great impact on the VMD algorithm. Improper parameter selection will cause over-decomposition, modal aliasing and false modes. Therefore, the parameter optimization of modal number and penalty factor is the focus of scholars’ discussion.
This article chooses the method of combining the correlation coefficient with VMD to optimize the value of parameter
in VMD. The correlation coefficient method is one of the most important methods to process and analyze signals. In the segmentation method proposed in this study, the correlation coefficient is used to calculate the correlation coefficient between the modal component obtained after VMD and the original signal. The mathematical model of the correlation coefficient is as follows.
In the formula, is the mean value of the IMF component, is the mean value of the original signal and is the correlation coefficient of the two sets of signals, the range is between [−1,1]. The weaker the correlation, the closer gets to 0, indicating that the component signal has little or no correlation with the original signal.
Therefore, due to the smaller , this study chooses to set a correlation coefficient threshold as the critical value of the correlation coefficient in the proposed AVMD algorithm and use it to optimize the value of the modal number .
The penalty factor is one of the parameters that must be adjusted manually in VMD. Too small a penalty factor will increase the probability of mode aliasing and too large a penalty factor will weaken the effect of noise reduction. According to the spectral distribution characteristics of the fault vibration signal of rotating machinery, the mid-low frequency region is mainly composed of the harmonics of the rotating frequency and its related characteristic frequencies (such as the bearing fault characteristic frequency and gear meshing frequency, etc.), while the fault impact and noise interference are mostly located in the high frequency area. At the same time, the harmonic signal has the characteristics of longtime domain duration and relatively compact frequency domain, while the impulse signal has the characteristics of a short time domain and wide frequency domain. Therefore, in order to better separate the inherent harmonic signal, fault impact and noise signal, the value of the penalty factor is set to verify the decomposition effect of AVMD in this study.
VMD and the correlation coefficient are combined to optimize the selection of the modal number. The initial modal number
. Perform VMD on the signal and obtain the correlation coefficient between each mode after decomposition and the original signal. If the minimum value of the correlation coefficient between each mode and the original signal after decomposition is less than the threshold, the decomposition will stop. Otherwise, the mode will increase the number and continue to decompose until the stopping condition is met, in order to determine the modulus
, and finally store the optimal value of
. The AVMD flow chart is shown in
Figure 1.
In order to check the effectiveness of the algorithm, this section takes the simulated signal as an example, and uses the AVMD algorithm proposed in this article to verify the decomposition effect. For periodic simulation signal:
, , the actual signal contains three modes: cosine with amplitude of 1 at 4 Hz, cosine with amplitude of 1/4 at 48 Hz and cosine with amplitude of 1/16 at 288 Hz. Perform AVMD on it, preset different
values, and the penalty factor
. Then calculate the correlation coefficient between each modal component and the original signal under different
values, and the results are shown in
Table 1.
It can be seen from the correlation coefficient between each modal component and the original signal that when
, the correlation coefficient between the modal com-ponent
and the original signal is 0.1083. When
, the correlation between the modal component
and the original signal is 0.1083. The coefficient is only 0.0878 (less than the set threshold
). Then, the selected mode number
meets the iterative stop condition and makes the VMD diagram when
, as shown in
Figure 2.
As shown in
Figure 2, when
, the three modes are well separated. However, when
, it can be seen that the 4 Hz cosine signal and the 48 Hz cosine signal in the original signal are superimposed together, and the phenomenon of “modal aliasing” appears; when
,
,
represents the cosine signal of 4 Hz and 48 Hz in the original signal, and
represents the cosine signal of 288 Hz, but a false mode
also appears.
According to the sampling frequency of the periodic simulation signal,
, take
; take
,
,
,
respectively, and make the spectrum of each modal component after VMD under different penalty factor
values distribution map, as shown in
Figure 3.
From the spectrogram of each modal component after VMD under different values, it can be seen that when , , each modal signal component experiences a serious modal aliasing phenomenon; when , , the phenomenon of component modal aliasing is weakened, but there are still some frequency components that have not been decomposed. This is mainly because the bandwidth of the Wiener filter is narrow at this time, which belongs to some frequencies of the original signal. Components will be filtered, resulting in missing information, and some frequency components will appear in multiple components at the same time.
Therefore, from the above analysis of the simulated signal, it can be seen that the default penalty factor , proposed in this study, and the adaptive variational modal decomposition method selected by optimizing the modal number through the correlation coefficient, can well separate the signal from high frequency to low frequency and avoid the phenomenon of modal aliasing. The decomposition algorithm has a good effect on the decomposition of the signal.
2.4. PSO-Optimized Multiscale Fuzzy Entropy
The Particle Swarm Optimization (PSO) algorithm is one of the evolutionary algorithms. It starts from a random solution, finds the optimal solution through iteration, and evaluates the quality of the solution through fitness. However, it is simpler than the genetic algorithm rule, and finds the global optimum by following the currently searched optimal solution. This algorithm has the advantages of easy implementation, high precision and fast convergence. Additionally, it has demonstrated its superiority in solving practical problems. Therefore, this study chooses to optimize the parameters of MFE with a PSO algorithm.
The PSO is initialized to a group of random particles (random solutions), and then the optimal solution is found through iteration. At each iteration, the particle updates itself by tracking two “extreme values” (
). After finding these two optimal values, the particle updates its velocity and position by using the formula below.
In order to improve the performance of the algorithm, the weight
is introduced to
of the above formula, as shown in the following formula.
The above two formulas form the standard form of the PSO algorithm.
It is necessary to determine a fitness function when a PSO algorithm is used to find the optimal parameter of MFE. At this point, the Ske of the data can be obtained. The larger the absolute Ske is, the more problematic the efficiency of the mean is, or the smaller the absolute Ske is, the more reliable the mean is.
In this study, the square function of Ske of MFE is selected as the objective function to find its minimum value. In this way, the parameter values of the embedding dimension M, the scale factor S and the time delay T of the MFE are optimized.
where
is the mean value of sequence
;
is the standard deviation of sequence
;
is the expectation of finding the sequence.