13292


FACTA UNIVERSITATIS  
Series: Mechanical Engineering Vol. 23, No 4, 2025, pp. 881 - 907  

https://doi.org/10.22190/FUME241130008L 

© 2025 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND 

Original scientific paper 

A ROLLING BEARING FAULT DIAGNOSIS METHOD BASED 

ON EXTREME LEARNING MACHINE OPTIMIZED BY 

IMPROVED WHALE OPTIMIZATION ALGORITHM 

Xin Li1, Shiliang Guo2, Dejie Sun1, Lijun Cao1, Cong Li1,  

Shuyao Tian3, Peng Liu2, Yadong Qi2 

1School of Mathematics and Information Science & Technology,  
Hebei Normal University of Science & Technology, Qinhuangdao, China 

2School of Electrical Engineering, Yanshan University, Qinhuangdao, China 
3College of Electronics and Control Engineering,  

North China Institute of Aerospace Engineering, Langfang, China 

ORCID iDs: Xin Li  https://orcid.org/0009-0009-3792-4083 
 Shiliang Guo  https://orcid.org/0000-0003-2033-6807 
 Dejie Sun  https://orcid.org/0009-0006-4882-1053 
 Lijun Cao  https://orcid.org/0009-0004-5759-9031 
 Cong Li  https://orcid.org/0009-0005-8217-3785 
 Shuyao Tian  https://orcid.org/0009-0006-1853-1151 
 Peng Liu  https://orcid.org/0009-0000-9345-4260 
 Yadong Qi  https://orcid.org/0009-0003-8268-3369 

Abstract. Rolling bearing is one of the most commonly used components in rotating 
machinery, and researching fault diagnosis techniques for it has important practical 
significance. In this paper, a fault diagnosis method based on extreme learning machine 
optimized by improved whale optimization algorithm (IWOA-ELM) is proposed for 
rolling bearing vibration signals. Firstly, Variational Mode Decomposition (VMD) is 
used to decompose the vibration signal of the bearing, and the energy entropy is 
calculated to form the eigenvector. Secondly, based on the original whale optimization 
algorithm, a hybrid initialization population strategy is adopted to generate an initial 
population with a certain quality. Selecting convergence factors based on reinforcement 
learning to improve global search capability, and using adaptive weights and random 
jumps to update individual positions. In this process, the t-distribution-levy flight 
variation strategy is introduced to avoid being attracted by local extremum. Then, the 
improved whale optimization algorithm is used to optimize the input weights and hidden 
layer thresholds of the Extreme Learning Machine (ELM). Finally, the feature set is input 
into an improved ELM model for training and testing. Experiments on fault diagnosis of 
rolling bearings of different types and degrees have shown that the model proposed in 
this paper can effectively improve the accuracy of fault classification.  

Key words: Fault diagnosis, Variational mode decomposition, Whale optimization 

algorithm, Reinforcement learning 

 
Received: November 30, 2024 / Accepted February 10, 2025   
Corresponding author: Shiliang Guo 
School of Electrical Engineering, Yanshan University, Street Address No.438, Hebei Avenue, Haigang District, 
Qinhuangdao 066004, China 
E-mail: guosl0112@ysu.edu.cn 



882 X. LI, S. GUO, D. SUN, L. CAO, C. LI, S. TIAN, P. LIU, Y. QI 

1. INTRODUCTION 

Rotating machinery plays an important role in modern society, and rolling bearings are 

critical components that impact its performance. However, the working environment of 

most rolling bearings is relatively harsh, which makes them prone to failure, resulting in 

significant economic losses and even catastrophic events. The fault vibration signal of 

rolling bearings has nonlinear and non-stationary characteristics. How to extract fault 

information from vibration signals and accurately locate bearing faults are essential for 

fault diagnosis and normal machinery operation [1-4]  

The application of machine learning methods in fault diagnosis can minimize manual 

involvement and greatly improve the accuracy of fault diagnosis, such as support vector 

machines (SVM) [5-7], backpropagation neural networks (BPNN) [8-10], and 

convolutional neural networks (CNN) [11-13]. However, these models face limitations: 

SVM needs to store large datasets, making it difficult to solve multi classification problems; 

BPNN and CNN have slow convergence speed and high computational cost, particularly 

with deeper layers. 

Extreme Learning Machine (ELM) has the characteristics of fast training speed, few 

learning parameters, strong generalization ability, and is widely used in the field of fault 

diagnosis [14-16]. Wang et al. [17] proposed a bearing fault diagnosis method based on ELM 

for adaptive parameter optimization, and introduced spectral cross-correlation to help select the 

optimal penalty factor, improving classification accuracy. In order to further enhance the 

generalization ability of ELM, Zhang et al. [18] used the Multitask Beetle Antennae Swarm 

Algorithm (MBAS) to optimize the input weights and biases of ELM, which can simultaneously 

reduce the number of conditions and regression errors. From the above description, it can be 

seen that optimizing the parameters of ELM based on optimization algorithms can further 

improve the performance of the model. Whale Optimization Algorithm (WOA) is a meta-

heuristic algorithm developed in recent years, which is used to handle various optimization 

problems in different fields due to its advantage of fewer adjustable parameters [19-21]. 

However, as the dimensionality of the search space increases, WOA faces problems such as 

easily falling into local optima and slow convergence speed.  

Based on the above analysis, we propose a rolling bearing fault diagnosis method based 

on extreme learning machine optimized by improved whale optimization algorithm 

(IWOA-ELM), as shown in Fig. 1. Firstly, the variational mode decomposition (VMD) 

method is used to extract the fault signal features of rolling bearings, which are then divided 

into a training set and a testing set. Then, based on the WOA, a hybrid initialization 

population strategy, a convergence factor selection based on reinforcement learning, an 

adaptive weight and random jump update strategy, and a t-distribution-levy flight variation 

strategy are introduced to solve the problem of the algorithm falling into local optimum. 

Finally, the IWOA was used to optimize the parameter settings of the extreme learning 

machine, thereby improving the speed and accuracy of bearing fault diagnosis. The IWOA-

ELM algorithm was compared with WOA-ELM, PSO-ELM, GWO-ELM, and the 

experimental results verified the effectiveness of the improved rolling bearing fault 

diagnosis method proposed in this paper. 



 A Rolling Bearing Fault Diagnosis Method based on Extreme Learning Machine... 883 

 

Fig. 1 Methodological framework of this study 

2. SIGNAL DECOMPOSITION AND FEATURE EXTRACTION 

2.1. VMD Theory 

VMD is an innovative and adaptive signal processing method [22] that can solve the 

common endpoint effects and mode component mixing problems in Empirical Mode 

Decomposition (EMD). VMD decomposes the initial signal into k discrete sub-signals, 

ensuring that each decomposition represents a finite bandwidth modal component centered 

at a specific frequency. At the same time, the sum of estimated bandwidths for each 

modality is minimized, and the sum of all modes is equal to the original signal as a 

constraint. The constraint variational expression can be written as: 

    
 

2
j

2,

1

min ( ( ) j / )* ( ) e

s.t.

k

k k

t

t k
u

K

kk

k

t t u t

u f




 −

=

  
 +   

  
 

= 


 (1) 

where, K is the number of decomposed modes. The symbols {uk} and {ωk} refer to the kth 

mode component and center frequency after decomposing, respectively. (t) is the Dirac 

function where t represents time. * is the convolution operator. 

Furthermore, we introduce the Lagrange multiplication operator  and the quadratic 

penalty factor  to find the optimal solution for the constrained variational problem, and 

obtain the augmented function: 



884 X. LI, S. GUO, D. SUN, L. CAO, C. LI, S. TIAN, P. LIU, Y. QI 

 

   ( )  
2

j

2

2

2

, , ( ( ) j / )* ( ) e

                           ( ) ( ) ( ), ( ) ( )

k t

k k t k

k

k k

k k

L u t t u t

f t u t t f t u t

   



−
=  + 

+ − + −



 
 (2) 

where the mentioned quadratic penalty factor α is used to reduce the interference of 

Gaussian noise. The alternate direction method of multipliers (ADMM) iterative algorithm 

combined with Parseval/Plancherel and Fourier isometry transform is used to optimize 

each eigenmode component and center frequency, and to search for saddle points of the 

extended Lagrangian function. The expressions for uk, ωk, and λ change after alternating 

optimization iteration is as follows: 

 
1

2

ˆ ˆˆ
ˆ

( ) ( ) ( ) / 2
( )

1 2 ( )

in i k

k

k

f u
u

   


  

+ 
− +


+ −


 (3) 

 

2
1

1 0

2
1

0

ˆ ( ) d

) dˆ (

n

kn

k
n

k

u

u

  


 


+

+


+





 (4) 

 1 1ˆˆ ˆ ˆ( ) ( ) ( ) ( )n n n

k

k

f u      + + 
 + − 

 
  (5) 

where, γ is the noise margin, 
1ˆ ( )n

ku +
, ˆ ( )iu  , ˆ( )f   and ˆ( )   correspond to the Fourier 

transforms of uk
n+1(t), ui(t), f(t) and λ(t) respectively. The main iterative VMD solution process 

is as follows: 

(1) Initialize 
1 1 1ˆˆ , ,k ku    and set the maximum number of iterations N, n←0; 

(2) Using Eqs. (3) and (4), update ûk and ωk; 

(3) Update ̂  using Eq. (5); 

(4) Accuracy convergence criterion ε>0, if n<N and
2 2

1

2 2
ˆ ˆ ˆ/n n n

k k k

k

u u u + −   are not 

satisfied, return to the second step. Otherwise, the iteration is completed.  

2.2. Extraction of Feature: Energy Entropy 

Yu [23] proposed the theory of energy entropy, and evaluated the fault information of 

rolling bearings based on the difference in energy by different frequency bands. The energy 

for any part of the waveform is defined as: 

 
2 2

1

) d(
G

i i ik

k

E c t t x
+

−
=

= =  (6) 

where, G is the number of sample points of the ith IMF, and xik is the amplitude. 



 A Rolling Bearing Fault Diagnosis Method based on Extreme Learning Machine... 885 

The original vibration signal is decomposed into n IMF vibration components, and the 

energy of each IMF component is Ei. The energy of the original vibration wave is 

total 

1

n

i

i

E E
=

= . Introducing Pi as the fraction of total signal energy in the i-th IMF part: 

 /i i totalP E E=  (7) 

the corresponding energy entropy equation is given as: 

 
1

( log )
n

EN i i

i

H P P
=

= −  (8) 

3. IWOA-ELM MODEL 

3.1. ELM theory 

ELM is used to train single hidden layer feedforward neural networks (SLFN). Fig. 2 

illustrates the network structure of ELM. 

 

Fig. 2 The network structure of ELM 

Assuming there are F random samples (xi, ti), i = 1, 2,…, F, where xi = [xi1,xi2,…,xin]T  Rn 

and ti = [ti1,ti2,…,tim]T  Rm. For a SLFN with L hidden layer nodes and an activation 

function g(x) can be calculated: 

 
i i j i i i j i j

1 1

( ) ( , j 1,2) , ,
L L

i i

g x g x b t n  
= =

= + = =    (9) 

where, ωi, bi, and βi represent the input weight, deviation, and output weight of the ith node 

in the hidden layer. We can use a matrix to represent it: 𝐻𝛽 = 𝑇, where H is the output of 

the hidden layer node, β is the output weight, and T is the output matrix.  



886 X. LI, S. GUO, D. SUN, L. CAO, C. LI, S. TIAN, P. LIU, Y. QI 

 

1 1 1 1

1 1 1

1 1

( ) ( )

 H( , , , , , , , , )

( ) ( )

L L

L L L

N L N L N L

g x b g x b

b b x x

g x b g x b

 

 

 


+



 + 
 

=

 +  +











 (10) 

 

TT

11

TT

22

TT

 , T

NL N m

t

t

t









  
  
  = =
  
  

      


 (11) 

Thus, the output weight is: 

 H T +=  (12) 

where, H+ is the Moore-Penrose generalized inverse of H. 

3.2.WOA Theory 

In WOA algorithm, each whale represents a set of solutions and can be optimized by 

its location and speed. When a whale detects its prey, it performs a spiral motion around it 

and releases a bubble net to trap the prey. The whale individuals can interact with each 

other in various ways, including "encircling prey," "bubble net attack," and "search for 

prey," which are described as follows:  

3.2.1. Encircling Prey 

During hunting, whales will continuously update their position and swim towards their 

prey after determining its location. In the solution space of the algorithm, assume that the 

position of the individual whale close to the prey is the optimal position. Other individual 

whales follow the best whale to update their position and gradually approach their prey. 

The above behavior is expressed as: 

 
*

( 1) ( )X t X t A D+ = − 


 (13) 

 
*

( )( )D C tX X t=  −


 (14) 

where, t is the current iteration number, *X


 is best position, X


 is current position of an 

individual whale, D stands for distance. A


 and C


 are coefficient vectors, which are 

calculated as follows:  

 
12A r =  −


 (15) 

 
22C r= 


 (16) 

 
2

2
t

a
M

= −  (17) 



 A Rolling Bearing Fault Diagnosis Method based on Extreme Learning Machine... 887 

where, a is the convergence factor with its value decreasing linearly from 2 to 0 during the 

iteration, M is the maximum number of iterations. 
1r


 and 
2r


 
are both random vectors 

in [0,1]. 

3.2.2. Bubble-Net Attacking Method (Exploitation Phase) 

At this stage, the shrinking encircling mechanism of the WOA algorithm is shown in 

equation (13). In addition, after determining the position of the prey, the humpback whale 

will move along the spiral towards the prey and update its individual position. The 

mathematical model is as follows:   

 
' *

( 1 cos(2 )) ( )blX e lt D X t+ = +


 (18) 

 
' *( ) ( )D X t X t= −

 
 (19) 

where, 
'

D


 is the distance between the current whale's position and the best whale's 

position, b is a logarithmic spiral shape constant, and l is a random value between [-1,1]. 

The two hunting behaviors of shrinking encircling and spiral movement are performed 

simultaneously. These two actions are selected by setting the parameter p, where p is a 

random number between [0,1], and the probability of p<0.5 and p ≥ 0.5 is 50%. The 

mathematical model is as follows:  

 

*

' *

     0.5

t
( 1)

  

( )

cos(2 ) ( ) pbl

D p
X

e
t

D

X t X

l X

  
+ = 

    +

−







 



 (20) 

3.2.3. Search for Prey (Exploration Phase) 

At this stage, the behavior of whales is controlled by coefficient vectors A


 to expand 

the search range and improve search capabilities. When the value of A is not within the 

range of [-1,1], individual whales will randomly select other whales as guides to update 

their position. The mathematical model is as follows: 

 ( 1) randX t X A D+ = − 


 (21) 

 ( )randD C X X t=  −


 (22) 

where,
randX


is the vector of random individual positions. 

3.3. Improved Whale Optimization Algorithm 

The WOA algorithm was briefly summarized in the previous section. In order to 

improve the algorithm's ability and efficiency in finding the global optimal solution, this 

section proposes IWOA. A hybrid initialization population scheme is designed based on 

the original WOA. Meanwhile, we select convergence factor based on reinforcement 



888 X. LI, S. GUO, D. SUN, L. CAO, C. LI, S. TIAN, P. LIU, Y. QI 

learning and design adaptive weight. The random jump update strategy and t-distribution-

levy flight variation strategy are implemented to improve the search ability of the algorithm. 

3.3.1. Hybrid Initialization 

In this section, we use improved Tent chaos mapping and opposition-based learning to 

initialize the population, and design a random exchange strategy for disadvantaged 

populations. 

(1) Improved Tent chaos mapping 

Tent chaotic mapping is a piecewise linear mapping that has more uniform traversal 

and faster search speed than Logistic chaos mapping. The expression of Tent chaos 

mapping is: 

 
1

1
        0

2 2

2(1 ) 1
           1

2

i

i

i

x
x

x
x

x

+

 


= 
−  

 (23) 

The expression after Bernoulli shift transformation is: 

 
1 (2 )mod1i ix x+ =  (24) 

To prevent the Tent chaotic sequence from getting stuck in unstable periodic points 

during the iteration, Zhang [24] introduced a random variable to the original Tent chaotic 

mapping equation as follows: 

 1

1 1
2 (0,1)         0

2

1 1
2(1 ) (0,1)      1

2

i

i

i

x rand x
N

x

x rand x
N

+


+   

= 
 − +   


 (25) 

where N represents the initial population size. 

The expression after Bernoulli shift transformation is: 

 1

1
(2 ) mod1 (0,1)i ix x rand

N
+ = +   (26) 

(2) Opposition-based learning 

Opposition-based learning mechanism can effectively improve the quality of the 

population and is applied to the improvement of various optimization algorithms. The 

expression of OBL is as follows: 

 ( )ix lb rand ub lb= + −  (27) 

 
'

i ix lb ub x= + −  (28) 

where rand is a random value on the interval [0,1], lb and ub are the upper and lower bounds of 

the search space, respectively. Applying the OBL idea to the randomly generated N initial 

solutions  1 2, , ,N nX x x x=   to obtain N reverse solutions  1 2, , ,N nX x x x   =  . In the 



 A Rolling Bearing Fault Diagnosis Method based on Extreme Learning Machine... 889 

process of searching for the optimal solution, the current solution and the reverse solution are 

simultaneously searched, and the better solution among them is taken as the problem solution, 

which can expand the search scope and improve the search efficiency of the algorithm. The 

initial solutions generated by chaotic mapping and opposition-based learning are mixed into a 

population with 2N initial solutions, and then the N initial solutions with smaller fitness values 

are selected according to the ranking of fitness values. This move can reduce the loss of good 

solutions. 

(3) Random exchange of inferior populations 

In order to further improve the quality of the initial population, two different individuals 

were randomly selected from the N initial solutions with poor fitness generated in the 

previous step to form a group (total N/2 groups). The algorithm randomly select K non 

repeating positions for each individual in each group, and replace the K positions of 

individual 1 with those of individual 2. Similarly, perform the same operation on individual 

2 to generate N new initial solution individuals. Then the fitness value of each individual 

is calculated and compared with the previous N smaller initial solutions, and the N initial 

solutions with smaller fitness values are selected. The random exchange operation for the 

inferior groups is shown in Fig. 3. 

 

Fig. 3 Random exchange 

3.3.2. Improved Hunting Patterns 

(1) Adaptive weight 

In order to improve the ability of traditional WOA algorithms to find optimal solutions, 

we introduce an exponentially changing adaptive weight during the algorithm's prey 

encirclement stage. The algorithm uses larger weights in the early stage to improve its 

global search ability, but when approaching the optimal solution, the weight values 

decrease exponentially to enhance the algorithm's local optimization ability. The weight 

expression can be written as: 

 
( )10 /

e
t T


−

=  (29) 

Taking the above equation into Eq. (20), the individual position update model for the 

encircling prey stage is obtained: 

 

*

*'

                        ( )

cos(2 ) ( )

0.5
( 1)

       0.5bl

X A D p
X t

Xl p

t

D e t

 −  
+ = 

   + 




  (30) 

where T is the highest number of iterations and t is the current number of iterations. 



890 X. LI, S. GUO, D. SUN, L. CAO, C. LI, S. TIAN, P. LIU, Y. QI 

(2) Random jump update strategy 

The superiority of WOA in the population iteration process depends on the search agent 

selected from the previous generation of the population. From Eqs. (21) and (22), it can be 

derived that the search agent position update during exploration is based on a randomly 

selected individual, and the new population convergence speed and accuracy are 

necessarily affected by the random individual. To address this phenomenon, a random 

jump update strategy is used to select the search agent more efficiently. The improved 

formula is as follows: 

 ( )randD XC X t= −
 

 (31) 

 ( ) ( )1X t C D C X t+ = • + •


 (32) 

(3) Reinforcement learning based convergence factor selection 

Whales act ‘nonlinearly’ in the real hunting process, and the linear descent strategy used 

by traditional algorithms cannot fully reflect the true situation of WOA. Therefore, this paper 

adopts different nonlinear control strategies to optimize the convergence factor a. 

Ref [25] uses a nonlinear convergence factor based on sinusoidal function to improve 

whale optimization algorithm, and its mathematical model is: 

 ( )( )sin / 2max max mina a a a t T= − −  . (33) 

On this basis, several adjustment strategies for control parameter a were proposed in 

references [26, 27], such as: 

Nonlinear decreasing strategy based on quadratic function: 

 2( )( / )max max mina a a a t T= − −  (34) 

Nonlinear decreasing strategy based on logarithmic form: 

 ( )( )ln 1 (e 1) /max max mina a a a t T= − − + −  (35) 

Through numerous experiments in references [26, 27], it can be seen that the 

convergence factor adjustment strategy has different effects on the utility of test functions. 

In order to investigate the experimental effect, the various adjustment strategies are 

considered as actions, and a suitable action is chosen to control the position of the whale at 

each step of the whale hunt. In addition, the possibility of making the right decision is 

greater if one can consider a few more steps forward before making a decision. Therefore, 

when whales choose actions, the effectiveness of multiple evolutionary steps can be judged 

by evaluating the advantages and disadvantages of different actions, which increases the 

whales' ability to find the best. 

Reinforcement learning can gradually learn the optimal action policy by selecting the 

action that maximizes the agent’s cumulative return with discount. Q-learning is a model 

independent reinforcement learning algorithm that obtains the optimal action strategy 

through multi-step learning of the agent to find the action with the maximum discounted 

return. If the adjustment strategy of the convergence factor is regarded as an action, the 

individual whale’s choice of the optimal adjustment strategy is transformed into the agent’s 

choice of the optimal action. Based on this, the standard whale optimization algorithm, Eq. 



 A Rolling Bearing Fault Diagnosis Method based on Extreme Learning Machine... 891 

(33), Eq. (34), and Eq. (35) four convergence factor adjustment strategies are selected as 

the set of actions in this paper. 

The basic format of Q-learning reads 

 ( , ) ( , ) max ( , )
a

Q s a r s a Q s a


= +  (36) 

where, r(s,a) is the immediate reward of the agent after taking action a in state s. s and s 

represent the two states of the agent, while a and a represent the two actions of the agent. 

γ(0≤γ<1) is the discount factor, and max Q(s,a) denotes the agent the maximum value of 

the reward that can be obtained by choosing a different action in the next state s. 

Assume that the set of operations available to the agent in any state is A = {a1,a2,…,an},. 

If the first action taken by the agent is a, then the revenue when the agent chooses the action 

a after m+1 steps is calculated as: 

 (1) 2 (2) ( )( ) ( ) ( ) ( ) ( )m mQ a r a Q a Q a Q a   = + + + +  (37) 

where, a, ( )ia A , and 1 i m  , with the parameter m controlling the number of steps 

to calculate the Q value looking forward. 

With the above Q-learning algorithm, the mapping between WOA and Q-learning 

algorithm can be achieved by considering the whale as an agent and various distance 

control parameter adjustment strategies as a set of agent actions. For any individual whale 

in the WOA, looking forward m+1 steps, the gain Q(a) when the whale chooses action a 

can be calculated by Eq. (37). Define the individual immediate gain, r(a) = fp(a)- fo(a). 

Here, fp(a) is value of fitness function corresponding to parent whale individual and fo(a) 

is value of fitness function about offspring whale individual generated after selecting action 

a. If the number of optional actions of whales is n, then nm whales will be generated after 

m times of evolution. To calculate the optimal evolutionary strategy for the whales, an 

exponential calculation is required, which is too computationally intensive. For this 

purpose, the probability of each offspring being retained is calculated using the Boltzmann 

distribution: 

 ( )/ ( )/

1
( ) /i i

nr a T r a T

i i
p a e e

=
=   (38) 

Furthermore, the highest probability of the n whale offspring generated will be retained. 

After this simplification, only one of the n offspring produced by a whale after one 

evolution is preserved during the evolution process, which greatly reduces the complexity 

of the algorithm. The convergence of the evolutionary planning algorithm based on Q-

learning is proved in [28]. The proof process of WOA is similar, and its convergence will 

not be proved in this paper due to the limitation of space. 

3.3.3. T-Distribution-Levy Flight Variation Strategy 

(1) T-distribution variation strategy 

T-distribution has a degree of freedom parameter of n and its probability density 

function is: 



892 X. LI, S. GUO, D. SUN, L. CAO, C. LI, S. TIAN, P. LIU, Y. QI 

 

1
2 2

1

2
( )

2

 1

n

t

n
Г

x
p x

n n
Г

x

n

+
−

+ 
    

= + −   + 
     
 

 (39) 

For each whale, the algorithm generates a random number in the [0,1]. If the variation 

probability is greater than the generated random number, t-distribution perturbation is 

applied to it. The specific location update method is as follows: 

 ( )j j

new best

j

bestx x t iteration x+ =  (40) 

where, t(iteration) is the t-distribution with degree of freedom parameter as the number of 

iterations.  

j

bestx  and  

j

newx  are the positions of the optimal whale individual in the jth 

dimension before and after variation perturbation, respectively. If the new solution after t-

distribution perturbation has a better fitness value, the old solution is replaced; otherwise, 

the old solution is retained. 

(2) Levy flight strategy 

Levy flight is a search method that obeys levy distribution [29]. This paper introduces 

the Levy flight strategy into the whale optimization algorithm after T-distribution 

perturbation. By conducting small-scale exploration around the optimal solution, the search 

area of the algorithm is further increased to avoid the population falling into local optima. 

The location update formula is 

( )k k

new bestx x levy = +
                           (41) 

where, levy(β) is a method to generate random numbers that obey the levy distribution. In 

this paper, we take advantage of the normal distribution to resolve random numbers [30]. 

The steps to generate random numbers submitted to levy distribution are as follows: 

 
1

~ ( )
u

s levy

v 

 =  (42) 

where, u obeys a normal distribution, i.e. u  N(0,2), and v obeys a standard normal 

distribution, i.e. v  N(0,1). The value of σ is taken as follows: 

 

1

1

2

sin
2

2
1

2











−

  
(+ )  

   =
 + 

   
  

 (43) 

Generally speaking, the range of values for β is 1≤β≤3, which is taken as 1.5 in this 

paper. 



 A Rolling Bearing Fault Diagnosis Method based on Extreme Learning Machine... 893 

3.3.4. IWOA Algorithm Performance Test 

In order to verify the feasibility of the algorithm, 8 benchmark test functions are chosen 

to evaluate the IWOA algorithm. Table 1 displays the test functions. For each benchmark 

function, IWOA is compared with WOA, OBCWOA [31], REMWOA [32], GWO [33] and 

GOA [34], SSA [35]. The population size is set to 30 and the maximum number of iterations 

is 500.The parameters of the OBCWOA method are configured as follows: a=4, k=1. The 

relevant parameters of GOA algorithm are cmax=1 and cmin=0.00004. The SSA algorithm is 

set to ST=0.8, the proportion of producers is 20% and the proportion of scouts is 10%. 

Table 1 Functions for testing  

Fig. 4 illustrates the iterative optimization process of several algorithms. It can be seen 
that the original WOA has a slow convergence rate. Although the convergence speed of 
OBCWOA and REMWOA has improved, it is easy to fall into local optimum in different 
test functions. IWOA improves the original whale optimization algorithm by using 
multiple optimization strategies and improves the optimization capability of WOA. The 
IWOA algorithm can jump out of local optima and find the optimal solution of the function 
in a timely manner even when encountering local optima, which improves the optimization 
performance of the algorithm. 

Function Dim Range fmin Peak 

2

1

1

( )
n

i

i

f x x
=

=  30 [-100,100] 0 Single 

2

1 1

( )
nn

i i

i i

f xx x
= =

= +   30 [-10,10] 0 Single 

2

3

1 1

( )
n i

j

i j

f x x
= −

 
=  

 
   30 [-100,100] 0 Single 

4

4

1

( ) [0,1)
n

i

i

f x ix random
=

= +   30 [-1.28,1.28] 0 Single 

2

5

1

( ) 10cos(2 ) 10
n

i i

i

f x x x
=

 = − +   30 [-5.12,5.12] 0 Multiple 

2

6

1

1

1
( ) 20exp 0.2

1
exp cos(2 ) 20

n

i

i

n

i

i

f x x
n

x e
n



=

=

 
= − − 

 
 

 
− + + 

 





 30 [-32,32] 0 Multiple 

2

7

1 1

1
( ) cos 1

400

nn
i

i

i i

x
f x x

i= =

= − +   30 [-600,600] 0 Multiple 

2
211

1 2

8 2
1 3 4

)
( )

( i i

i

i i i

x b b x
f x a

b b x x=

 +
= − 

+ + 
   4 [-5,5] 0.0003 Multiple 



894 X. LI, S. GUO, D. SUN, L. CAO, C. LI, S. TIAN, P. LIU, Y. QI 

(a)  

(b)  

(c)  

(d)  



 A Rolling Bearing Fault Diagnosis Method based on Extreme Learning Machine... 895 

(e)  

(f)  

(g)  

(h)  

Fig. 4 Convergence curves of 8 test functions:  

(a) f1(x); (b) f2(x); (c) f3(x); (d) f4(x); (e) f5(x); (f) f6(x); (g) f7(x); (h) f8(x) 



896 X. LI, S. GUO, D. SUN, L. CAO, C. LI, S. TIAN, P. LIU, Y. QI 

All algorithms were executed 30 times on the 8 benchmark functions, and the average 

values were computed. The results of experiment are given in Table 2. 

Table 2 Optimization performance of 8 benchmark functions 

Table 2 illustrates that the WOA is particularly obvious disadvantaged when solving 

for the optimal value of high dimensional functions. The reason for this result is that there 

are three ways to update it, but only one way of obtaining the exploration, and the 

probability of getting it is low. Therefore the WOA is prone to falling into local optima. 

Compared with WOA, the algorithm used in this paper improves the convergence accuracy 

of the algorithm in later iterations through the choice of convergence factors and the 

variational strategy. Meanwhile, compared with other improved WOA, IWOA can also 

effectively solve the problem of algorithm falling into local extremes. It shows significant 

advantages over several other algorithms about convergence accuracy and stability. 

3.4. IWOA-ELM Fault Diagnosis Model 

In this section, the IWOA algorithm is selected to optimize the input weights and biases 

of the ELM model, and use the training set error rate and validation set error rate of the 

extremum learning machine as fitness values, as described in equation (44), in order to 

improve the accuracy and generalization ability of the ELM model classification. The 

feature set is divided into training set, validation set and testing set as the input of ELM, 

and the minimum fitness value Fmin and population extreme value Gbest and individual 

extreme value Pbest are obtained. The optimal solution, individual extreme value Pbest and 

population extreme value Gbest are updated iteratively until the best fitness value is obtained 

or the maximum number of iterations is reached, and the optimal hidden layer weights and 

bias are obtained.  

 2 train verifyfitne Accuracyss Accuracy −= −  (44) 

where, Accuracytrain and Accuracyverify represent the correct classification ratio of training 

set and validation set, respectively. The procedure of IWOA-ELM model is shown in Fig. 5. 

Function WOA REMWOA OBCWOA IWOA GOA SSA GWO 

f1(x) 5.97×10-72 1.26×10-290 2.06×10-262 0 93.63 3.99×10-60 1.00×10-27 

f2(x) 6.11×10−52 1.26×10−147 6.55×10−135 0 77.88 1.79×10−30 7.39×10−17 

f3(x) 43722.24 1.27×10−233 6.4×10−225 0 2007.86 2.63×10−24 3.92×10−5 

f4(x) 0.003952 8.29×10−5 9.39×10−5 7.25×10-5 0.463665 0.001386 0.002197 

f5(x) 0 0 0 0 189.4202 0 2.3886621 

f6(x) 4.32×10−15 8.88×10−16 8.88×10−16 8.88×10-16 8.60 8.88×10−16 1.05×10−13 

f7(x) 0.014453 0 0 0 2.535341 0 0.002539 

f8(x) 0.000584 0.000352 0.000714 0.000397 0.009826 0.000332 0.005038 



 A Rolling Bearing Fault Diagnosis Method based on Extreme Learning Machine... 897 

 

Fig. 5 Fault diagnosis flowchart 

4. EXPERIMENTAL STUDY 

4.1. Equipment and Data  

In this paper, we used data from both normal and faulty rolling bearings, which were 

provided by Case Western Reserve University (CWRU). Fig. 6 illustrates the experimental 

simulation platform of CWRU. The experimental setup includes a 2 hp motor, a torque 

transducer, a dynamometer, and the corresponding control electronics. During the 

experiment, vibration signals were collected using an accelerometer with a sampling 

frequency of 12 kHz. 

Case 1: Vibration signals for different fault conditions were diagnosed under varying 

loads, with the fault diameter in the rolling bearings kept constant. The data that collected 

from the 6 o'clock position at the 12K drive end serves as the primary source, including 

fault damage diameter of 0.36mm inner and outer rings, rolling elements, and normal 

bearing data. The specifics of the bearing data are provided in Table 3. 

Case 2: Diagnose vibration signals under different fault diameters while keeping the 

load on the rolling bearing constant. The data source acquisition method remains 

unchanged, including the normal state of the bearing inner ring and the fault damage 

diameters of 0.18 mm, 0.36 mm, and 0.53 mm, respectively. The bearing data are presented 

in Table 4. 



898 X. LI, S. GUO, D. SUN, L. CAO, C. LI, S. TIAN, P. LIU, Y. QI 

 

Fig. 6 Baring fault simulation test-bed 

Table 3 Overview of the data sets for Case 1 

Table 4 Overview of the data sets for Case 2 

4.2. Proposed Model Evaluation 

4.2.1. Experiment 1 Validation 

Among the collected data, we selected 120,000 vibration signal samples and 10 seconds 

sampling time. The vibration signals were divided into 400 segments, each containing 300 

points. The time-domain waveform of the initial signal is shown in Fig. 7. From Fig. 7, it can 

be seen that identifying the fault type from the time-domain waveform is challenging and can 

be time-consuming even when successful, highlighting the urgent need for the proposed 

diagnostic method. We use VMD to decompose the vibration signal of rolling bearings, with 

VMD decomposition K and penalty factor α values of 4 and 1989, respectively [36]. Fig. 8 

illustrates the modal components along with their corresponding frequency domain diagrams 

for various fault types. It can be seen from Figure 8 that the energy in the low-frequency range 

is more pronounced for the normal state, while the inner and outer ring failures exhibit the 

highest energy in their respective frequency bands. The rolling element fault shows significant 

energy distribution in both low and high-frequency ranges. Short-duration low-frequency pulses 

in the vibration signal indicate shock events due to partial damage in the bearing. Additionally, 

the signal contains resonant components generated by these shock events, which excite high-

frequency intrinsic vibrations, modulated by their amplitude. 

Bearing status Fault diameter(mm) Acronym Label 

Normal 
Ball fault 
Inner race fault 
Outer race fault 

- N 1 

0.36 BF 2 

0.36 IRF 3 

0.36 ORF 4 

Bearing status Fault diameter(mm) Acronym Label 

Normal 
Inner race fault 

- N 1 

0.18 

0.36 

0.53 

IRF-0.18 

IRF-0.36 

IRF-0.53 

2 

3 

4 



 A Rolling Bearing Fault Diagnosis Method based on Extreme Learning Machine... 899 

Finally, the IMFs constituents obtained from the signals of the four fault types are 

decomposed according to the VMD and the energy entropy is calculated. The characteristic 

sample sets are obtained by VMD decomposition and calculation of energy entropy, with 

300 sets per sample set and a total of 1200 sets. From each group, 250 groups are randomly 

selected, of which 600 groups are used as the training sample set and 400 groups are used 

as the test sample set. Set the population size of IWOA N=20, the maximum value of 

iterations M=50, amax=2, amin=0, Q-learning discount factor γ=0.5, m=3, the value of nodes 

of ELM hidden layer is set to 100. The parameters of the ELM network weights and 

thresholds are optimized by the IWOA. Table 5 shows the results of diagnosing four fault 

states using the IWOA-ELM method. The accuracy of fault diagnosis for the training 

samples is 98.83%, and for the test samples, it is 97.5%. These findings indicate that the 

method successfully identify all four fault states of rolling bearings. 

(a)   

(b)  

Fig. 7 The time-domain waveforms of rolling bearing fault types: (a) Case 1; (b) Case 2 



900 X. LI, S. GUO, D. SUN, L. CAO, C. LI, S. TIAN, P. LIU, Y. QI 

(a)  

(b)  

(c)  



 A Rolling Bearing Fault Diagnosis Method based on Extreme Learning Machine... 901 

(d)  

Fig. 8 Modal components and frequency distribution diagrams obtained after fault 

decomposition: (a) Normal; (b) Ball fault; (c) Inner race fault; (d) Outer race fault 

Table 5 Diagnostic performance results for samples in diverse fault scenarios, divided into 

training and test groups. 

4.2.2. Experiment 2 Validation 

In the data collected, we selected 120,000 vibration signal samples from normal and 

three bearing inner rings with different degrees of failure (sampling time: 10 seconds). The 

vibration signals were divided into 400 segments, each containing 300 points. Fig. 7 shows 

the time-domain waveform of the raw vibration signal. Decompose the vibration signal of 

rolling bearings using VMD, with the decomposition number K and the penalty factor α 

value of 4 and 1992, respectively. Fig. 9 displays the modal components and their 

frequency plots for inner ring of the bearing after fault decomposition at damage levels of 

0.18 mm and 0.53 mm. Since the normal state and the 0.36 mm fault diameter have been 

discussed, we will not elaborate further here. As shown in Fig. 9, the energy associated 

with the inner ring failure is primarily focused on in the mid-frequency range. Furthermore, 

for the same section of the bearing, the amplitude corresponding to specific frequencies 

increases with the degree of damage. This comparison shows that as a bearing transitions 

from a normal state to a failed state, the primary energy in its spectrum is gradually 

transferred from low frequency band to middle frequency band. Subsequently, we 

calculated the energy entropy of IMF based on VMD decomposition of normal state signals 

and three types of fault vibration signals in the inner ring. 

Bearing 

status 

Training Number/ 

testing samples 
N BF IRF ORF 

Training 

accuracy 

Testing 

accuracy 

N 150/100 149/99 1/1 0/0 0/0 99.33% 99% 

BF 150/100 1/1 147/97 1/2 1/0 98% 97% 

IRF 150/100 0/0 0/2 148/95 2/3 98.67% 95% 

ORF 150/100 0/1 0/0 1/1 149/98 99.33% 98% 



902 X. LI, S. GUO, D. SUN, L. CAO, C. LI, S. TIAN, P. LIU, Y. QI 

(a)  

(b)  

Fig. 9 Frequency domain diagrams show inner ring damage with diameters of: (a) 0.18 mm 

and (b) 0.53 mm 

Finally, 1200 sets of feature samples were obtained through VMD decomposition and 

energy entropy calculation, including 300 sets for each of the three different fault degrees 

and normal states. Randomly select 250 groups from each group, with 600 groups as 

training samples and 400 groups as testing samples. Set the population size of IWOA to 

N=20, the maximum iterations number M=50, amax=2, amin=0, Q-learning discount factor 

γ=0.5, m=3, the ELM activation function is a sigmoid function, and the number of hidden 

layer nodes is 100. The weights and thresholds of ELM are optimized through IWOA. The 

results of using IWOA-ELM to diagnose normal state and three different degrees of fault 

samples in the bearing inner race are shown in Table 6. The diagnosis accuracy reached 

99% for the training samples and 96.75% for the test samples. This discrepancy may be 

attributed to certain failure feature parameters being less distinct at varying degrees of 



 A Rolling Bearing Fault Diagnosis Method based on Extreme Learning Machine... 903 

failure damage, making it challenging to differentiate between states. However, with the 

increase of the degree of failure, salient characteristics of the characteristic parameters of 

the failure state become more obvious. 

Table 6 Comparison diagnostic results on training and testing samples across multiple fault 

severity levels 

4.3. Comparison of Different Fault Diagnosis Models 

In order to verify the effectiveness of optimization algorithm in improving the accuracy of 

fault diagnosis, we compared it against several other methods, including SVM, ELM, PSO-

ELM, GWO-ELM, and WOA-ELM. The population sizes for IWOA, WOA, and GWO were 

all set to 30, with a maximum of 100 iterations. The PSO algorithm was configured with a 

population size of 30 and executed for 100 iterations, setting the maximum particle velocity to 

0.8 and the minimum particle velocity to -0.8, and the inertia weight to 0.9. For extreme learning 

machines, g(x) was the sigmoid function and there were 100 hidden layer nodes. In the case of 

the SVM, the penalty coefficient was chosen as C=2 and the kernel width coefficient was set as 

g=0.2. The diagnostic results of different fault diagnosis methods for Case 1 and Case 2 are 

shown in Table 7 and Table 8, respectively. The convergence curves and comparison between 

predicted and actual results are shown in Fig. 10 and Fig. 11, respectively. 

Comparing the results in Tables 7 and 8, the IWOA-ELM model achieved the highest 

fault identification accuracy for both different types and levels of bearing fault diagnosis  

Table 7 Diagnosis results of different fault diagnosis methods for Case 1 

Model Training accuracy rate Testing accuracy rate 

IWOA-ELM 98.83% 97.5% 

WOA-ELM 98.67% 95.5% 

GWO-ELM 98.33% 95.0% 

PSO-ELM 98.17% 95.0% 

ELM 98.17% 90.5% 

SVM 92.00% 91.25% 

Table 8 Diagnosis results of different fault diagnosis methods for Case 2 

Model Training accuracy rate Testing accuracy rate 

IWOA-ELM 99.00% 96.75% 

WOA-ELM 98.17% 96.00% 

GWO-ELM 98.00% 94.75% 

PSO-ELM 98.17% 94.75% 

ELM 98.00% 92.75% 

SVM 96.17% 94.50% 

Bearing 

status 

Training Number/ 

testing samples 
N 

IRF-

0.18 

IRF-

0.36 

IRF-

0.53 

Training 

accuracy 

Testing 

accuracy 

N 150/100 149/99 0/0 1/0 0/1 99.33% 99% 

IRF-0.18 150/100 0/1 145/90 5/9 0/0 96.67% 90% 

IRF-0.36 150/100 0/0 0/0 150/100 0/0 100% 100% 

IRF-0.53 150/100 0/1 0/0 0/1 150/98 100% 98% 



904 X. LI, S. GUO, D. SUN, L. CAO, C. LI, S. TIAN, P. LIU, Y. QI 

in the training set samples. This suggests that the model effectively leverages the training set, 

resulting in optimal training outcomes. The test set samples further validate this conclusion, 

as the IWOA-ELM  model also achieves the highest diagnostic accuracy. Furthermore, 

comparing the fault diagnosis results with and without optimization shows that the optimized 

method consistently outperforms the unoptimized approach in terms of accuracy. In addition, 

while the ELM diagnostic model surpasses the SVM model on the training set, the SVM 

model performs better on the test set. This discrepancy highlights the differing applicability 

of these methods to various forms of fault diagnosis, leading to varying diagnostic results. 

However, this does not affect the diagnostic conclusions. The convergence curves of the 

different diagnostic models in Figs. 10 and 11 show that the IWOA-ELM diagnostic model 

gets better fitness values relative to the optimized ELM diagnostic model by other algorithms 

at the same number of iterations. 

(a)  

(b)  

Fig. 10 Comparison of different fault diagnosis models under Case 1: 

(a) The convergence curves; (b) Comparison of test set prediction results 



 A Rolling Bearing Fault Diagnosis Method based on Extreme Learning Machine... 905 

(a)  

(b)  

Fig. 11 Comparison of different fault diagnosis models under Case 2:  

(a) The convergence curves; (b) Comparison of test set prediction results 

5. CONCLUSION 

The setting of parameters has a significant impact on the performance of recognition 

algorithms for rolling bearing fault diagnosis. In response to this issue, the paper introduces 

an improved whale optimization algorithm into rolling bearing fault detection and proposes 

a fault diagnosis model based on the IWOA-ELM algorithm. Firstly, the VMD method is 

used to decompose the vibration signals of rolling bearings, and the energy entropy values 

of IMF decomposed for different scenarios are calculated. Furthermore, multiple 

optimization strategies are used to improve the performance of the WOA algorithm, and it 

is combined with ELM to optimize the input layer weights and hidden layer thresholds. 

Comparing the improved algorithm with SVM, ELM, GWO-ELM, PSO-ELM, and WOA-



906 X. LI, S. GUO, D. SUN, L. CAO, C. LI, S. TIAN, P. LIU, Y. QI 

ELM algorithms, the diagnostic results show that the rolling bearing fault diagnosis method 

based on IWOA-ELM can effectively improve the convergence speed and have a higher 

recognition rate. The idea proposed in this paper of optimizing ELM related parameters 

through improved optimization algorithms to enhance the performance of rolling bearing 

fault diagnosis can provide some reference for future research on rolling bearing fault 

diagnosis. 

Acknowledgement: This paper was funded by the Science and Technology Project of Hebei 

Education Department (Grant No.: ZD2022087); the Hebei Innovation Center for Smart Perception 

and Applied Technology of Agricultural Data (Grant Nos.: ADIC2023Y002, ADIC2024T001); the 

Central Government Guides the Local Science and Technology Development Fund Project (Grant 

No.: 246Z4418G); and the Doctoral Foundation Project of North China Institute of Aerospace 

Engineering (Grant No.: BKY-2022-05). 

REFERENCES  

1. Rai, A., Upadhyay, S. H., 2016, A review on signal processing techniques utilized in the fault diagnosis of 

rolling element bearings, Tribology International, 96, pp. 289-306.  
2. Singh, J., Azamfar, M., Li, F., Lee, J., 2020, A systematic review of machine learning algorithms for 

prognostics and health management of rolling element bearings: fundamentals, concepts and applications, 

Measurement Science and Technology, 32(1), 012001.  
3. El-Thalji, I., Jantunen, E., 2015, A summary of fault modelling and predictive health monitoring of rolling 

element bearings, Mechanical Systems and Signal Processing, 60, pp. 252-272.  

4. Kamat, P., 2024, Fault precognition system for remaining useful life estimation in bearing systems using 
autoencoder-LSTM and clustering techniques, Journal Européen des Systèmes Automatisés, 57(6), pp. 

1721-1728.  

5. Araste, Z., Sadighi, A., Jamimoghaddam, M., 2023, Fault diagnosis of a centrifugal pump using electrical 
signature analysis and support vector machine, Journal of Vibration Engineering & Technologies, 11(5), 

pp. 2057-2067. 

6. Islam, M. M., Kim, J. M., 2024, Time–frequency envelope analysis-based sub-band selection and 
probabilistic support vector machines for multi-fault diagnosis of low-speed bearings, Journal of Ambient 

Intelligence and Humanized Computing, 15, pp. 1527–1542. 

7. Shi, Q., Zhang, H., 2020, Fault diagnosis of an autonomous vehicle with an improved SVM algorithm 
subject to unbalanced datasets, IEEE Transactions on Industrial Electronics, 68(7), pp. 6248-6256.  

8. Seghiour, A., Abbas, H. A., Chouder, A., Rabhi, A., 2023, Deep learning method based on autoencoder 

neural network applied to faults detection and diagnosis of photovoltaic system, Simulation Modelling 
Practice and Theory, 123, 102704.  

9. Ghorvei, M., Kavianpour, M., Beheshti, M. T., Ramezani, A., 2023, Spatial graph convolutional neural 
network via structured subdomain adaptation and domain adversarial learning for bearing fault diagnosis, 

Neurocomputing, 517, pp. 44-61.  

10. Lee, C. Y., Ou, H. Y., 2021, Induction motor multiclass fault diagnosis based on mean impact value and 
PSO-BPNN, Symmetry, 13(1), 104.  

11. Thomas, J. B., Chaudhari, S. G., Shihabudheen, K. V., Verma, N. K., 2023, CNN-based transformer model 

for fault detection in power system networks, IEEE Transactions on Instrumentation and Measurement, 72, 
2504210.  

12. Huang, T., Zhang, Q., Tang, X., Zhao, S., Lu, X., 2022, A novel fault diagnosis method based on CNN and 

LSTM and its application in fault diagnosis for complex systems, Artificial Intelligence Review, 55(2), pp. 
1289-1315.  

13. Janssens, O., Slavkovikj, V., Vervisch, B., Stockman, K., Loccufier, M., Verstockt, S., Van Hoecke, S., 

2016, Convolutional neural network based fault detection for rotating machinery, Journal of Sound and 
Vibration, 377, pp. 331-345.  

14. Huang, G. B., Zhu, Q. Y., Siew, C. K., 2006, Extreme learning machine: Theory and applications, 

Neurocomputing, 70(1-3), pp. 489-501.  



 A Rolling Bearing Fault Diagnosis Method based on Extreme Learning Machine... 907 

15. Zhao, Y. P., Chen, Y. B., 2022, Extreme learning machine based transfer learning for aero engine fault 
diagnosis, Aerospace Science and Technology, 121, 107311. 

16. Jayapal, P. K., Muvva, V. R., Desanamukula, V. S., 2023, Stacked Extreme Learning Machine with Horse 

Herd Optimization: A Methodology for Traffic Sign Recognition in Advanced Driver Assistance Systems, 
Mechatronics and Intelligent Transportation Systems, 2(3), pp. 131-145.  

17. Wang, J., Zhang, Y., Zhang, F., Li, W., Lv, S., Jiang, M., Jia, L., 2021, Accuracy-improved bearing fault 

diagnosis method based on AVMD theory and AWPSO-ELM model, Measurement, 181, 109666.  
18. Zhang, X., Yang, Z., Cao, F., Cao, J., Wang, M., Cai, N., 2020, Conditioning optimization of extreme 

learning machine by multitask beetle antennae swarm algorithm, Memetic Computing, 12, pp. 151-164. 

19. Wu, X., Zhang, S. E. N., Xiao, W., Yin, Y., 2019, The exploration/exploitation tradeoff in whale 

optimization algorithm, IEEE Access, 7, pp. 125919-125928. 

20. Shim, E. S., Kim, H. D., Lee, S. 2024, Optimal multi-impulse elliptic rendezvous using stagnation escaping 

whale optimization algorithm, International Journal of Aeronautical and Space Sciences, 25, pp. 1092-1104.  
21. Abdel-Basset, M., Manogaran, G., El-Shahat, D., Mirjalili, S., 2018, A hybrid whale optimization algorithm 

based on local search strategy for the permutation flow shop scheduling problem, Future Generation 

Computer Systems, 85, pp. 129-145.  
22. Dragomiretskiy, K., Zosso, D., 2013, Variational mode decomposition, IEEE Transactions on Signal 

Processing, 62(3), pp. 531-544.  

23. Yu, Y., Yu, D. J., Cheng, J. S., 2006, A roller bearing fault diagnosis method based on EMD energy entropy 
and ANN, Journal of sound and vibration, 294(1-2), pp. 269-277.  

24. Zhang, N., Zhao, Z. D., Bao, X. A., Qian, J., Wu, B., 2020, Gravitational search algorithm based on 

improved Tent chaos, Control and Decision, 35(4), pp. 893-900.  
25. Jiang, T., Zhang, C., Zhu, H., Gu, J., Deng, G., 2018, Energy-efficient scheduling for a job shop using an 

improved whale optimization algorithm, Mathematics, 6(11), 220.  

26. Lv, H., Feng, Z., Wang, X., Zhou, W., Chen, B., 2021, Structural damage identification based on hybrid 
whale annealing algorithm and sparse regularization, J. Vib. Shock, 40, pp. 85-91.  

27. Teng, X., Luo, Y., Zheng, T., Zhang, X., 2022, An improved whale optimization algorithm based on 

aggregation potential energy for QoS-driven web service composition, Wireless Communications and 
Mobile Computing, 2022(1), 9741278.  

28. Zhang, H. X., Lu, J., 2008, An adaptive evolutionary programming algorithm based on Q learning, Acta 

Automatica Sinica, 34(7), pp. 819-822.  
29. Motamarri, R., Bhookya, N., 2020, JAYA algorithm based on Lévy flight for global MPPT under partial 

shading in photovoltaic system, IEEE Journal of Emerging and Selected Topics in Power Electronics, 9(4), 

pp. 4979-4991.  
30. Mantegna, R. N., 1994, Fast, accurate algorithm for numerical simulation of Levy stable stochastic 

processes, Physical Review E, 49(5), 4677.  

31. Chen, H., Li, W., Yang, X., 2020, A whale optimization algorithm with chaos mechanism based on quasi-
opposition for global optimization problems, Expert Systems with Applications, 158, 113612.  

32. Liu, J., Shi, J., Hao, F., Dai, M., 2022, A reinforced exploration mechanism whale optimization algorithm 
for continuous optimization problems, Mathematics and Computers in Simulation, 201, pp. 23-48.  

33. Mirjalili, S., Mirjalili, S. M., Lewis, A., 2014, Grey wolf optimizer, Advances in engineering software, 69, 

pp. 46-61.  
34. Saremi, S., Mirjalili, S., Lewis, A., 2017, Grasshopper optimisation algorithm: theory and application, 

Advances in Engineering Software, 105, pp. 30-47.  

35. Xue, J., Shen, B., 2020, A novel swarm intelligence optimization approach: Sparrow search algorithm, 
Systems science & Control Engineering, 8(1), pp. 22-34.  

36. He, C., Wu, T., Gu, R., Jin, Z., Ma, R., Qu, H., 2021, Rolling bearing fault diagnosis based on composite 

multiscale permutation entropy and reverse cognitive fruit fly optimization algorithm–extreme learning 
machine, Measurement, 173, 108636.