Page 110 - IJOCTA-15-4

Zhang et al. / IJOCTA, Vol.15, No.4, pp.649-669 (2025)
In this context, n represents the IMF compo- To rectify this deficiency, this study presents a
nent index; m represents the index of the equal- comprehensive evaluation factor-based algorithm
length interval used in calculating the envelope for identifying sensitive IMF components. 26
entropy; p n (m) is the probability of the n-th IMF The correlation coefficient is an effective met-
component falling within the m-th equal-length ric for assessing the similarity between signals.
interval; and a n (m) is the count of points from Therefore, computing the correlation coefficient
the n-th IMF component within the m-th inter- in the frequency domain can effectively suppress
val. noise interference and improve calculation accu-
The steps for optimizing VMD using the GJO racy. In addition, frequency-domain analysis re-
algorithm are as follows: veals the characteristics of signals across different
a. Initialize GJO parameters: Begin by set- frequencies, providing richer information for sig-
ting the parameters for the GJO algorithm nal processing and feature extraction. Moreover,
and defining the range for the VMD pa- when a rolling bearing experiences a fault, the
rameters K and α. probability density of high-amplitude vibrations
b. Calculate fitness values: Assess the suit- increases, causing the amplitude distribution to
ability of every member within the group. deviate from a normal distribution. This results
The two jackals are selected among the in skewness or dispersion in the normal curve and
elite as the best-fit individuals with the an increase in kurtosis.
highest fitness, serving as both the jackal To identify sensitive components that effec-
pair and the prey. tively represent signal characteristics, this study
c. Update prey position through exploration, computed both the frequency-domain cross-
encirclement, and attack: The golden jack- correlation coefficients and the kurtosis for each
als update the prey’s position by exploring mode component resulting from VMD. By apply-
and approaching it, surrounding it, stimu- ing a weighted factor to these values, the IMFs
lating it, and finally attacking it. that optimally capture the signal’s key features
d. Replace individuals based on prey position were selected. This method provides a more ro-
updates: Update the positions of the best bust approach to IMF selection, ensuring that im-
and second-best individuals by replacing portant diagnostic information is retained.
them with the positions of the current prey. Under the action of the Coati Optimization
e. Repeat Steps b–d: Iterate the process until Algorithm (COA)–VMD, the vibration signal of
the maximum number of cycles is reached. the rolling bearing (x[t], t = 1, 2, . . . , n) is de-
Should the termination criterion be met, composed into IMF components (K; u i [t], i =
the algorithm outputs the optimal param- 1, 2, . . . , K). The global criterion function con-
eter combination [K, α]. If the expected stitutes the kernel of the sensitive IMF discrimi-
optimal effect is not achieved, the process nation algorithm, enabling accurate computation.
returns to Step b and repeats the whole (i) Compute the frequency-domain cross-
process until the preset termination condi- correlation coefficient (ρ) for each IMF
tions are met. This procedure allows for an component relative to the original signal.
effective optimization of VMD parameters
by leveraging the cooperative and compet-
R
f a ¯ ¯
itive dynamics within the GJO framework. 0 (G u i − G u )(G x i − G x )df
ρ i = q q
2
2
Figure 1 illustrates the key parameters of the R f a − G u ) df · R f a − G x ) df
¯
¯
0 (G u i 0 (G x i
VMD algorithm employing the GJO method.
(3)
In this equation, G u and G x represent
2.3. Sensitive intrinsic-mode-function
signal power spectra u i (t) and x(t), respec-
selection algorithm based on
comprehensive evaluation factors tively; G u and G x denote the correspond-
ing mean power spectra; and f a indicates
Effectively identifying informative components the analysis frequency.
from the IMF decomposition of VMD is critical (ii) Calculate the kurtosis of each IMF compo-
for the efficacy of signal processing. Conventional nent:
methods for excluding and reconstructing false
IMFs often fall short, particularly when analy- 4
E[u i (t) − ¯u(t)]
ses are confined to either the time domain or the kur i = (4)
frequency domain alone. Such approaches tend to σ 4
overlook the signal’s combined time–frequency at- In the given equation, ¯u(t) and σ rep-
tributes, leading to the loss of critical information. resent the mean values of the i-th IMF
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