*Corr. Author’s Address: Henan University of Technology, No. 100 Lianhua Street, Zhengzhou, China, wanzhenshuai@haut.edu.cn 531
Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 531-542 Received for review: 2024-05-21
© 2024 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2024-08-03
DOI:10.5545/sv-jme.2024.1047 Original Scientific Paper Accepted for publication: 2024-08-09
Acceleration Harmonic Estimation and Suppression  
for Hydraulic Load Simulator Based on Artificial Bee Colony  
with Chaotic Search Strategy
Wan, Z. – Yue, L. – Wang, Y . – Zhao, P.
Zhenshuai Wan
1,2,*
 – Longwang Yue
2
 – Yanfeng Wang
2
 – Pu Zhao
2
1 
Henan University of Technology, Key Laboratory of Superhard Abrasives and Grinding Equipment, China 
2 
Henan University of Technology, School of Mechanical and Electrical Engineering, China
This paper presents a new hybrid algorithm based on artificial bee colony (ABC) and chaotic search strategy (CSS), known as ABC-CSS to 
solve the harmonic estimation problems in the case of time varying acceleration response signal. The ABC algorithm simulates the intelligent 
behavior of bees in nature to find high-quality nectar sources according to different divisions of labor. Aiming at the lack of diversity, poor global 
and local search ability, as well as slow convergence speed of ABC, an improved ABC based on CSS is proposed. The algorithm generates new 
neighborhood points by the transformation of chaotic and decision variables by carrier mapping, which provides a broader search space and 
better location of nectar sources for bees and recruited observation bees, and enhances the diversity of bee colonies. At the same time, the 
investigation of a local honey bee search better solves the algorithm problem of the local minimum and improves the convergence property 
of the ABC. Simulation and experimental studies are conducted to validate the superiority of the proposed ABC-CSS in harmonic estimation.
Keywords: artificial bee colony, chaos-decision variable, harmonic distortion, harmonic estimation, harmonic suppression
Highlights
• 	 The mathematical model of acceleration harmonic estimation is constructed.
• 	 The chaotic search strategy is adopted to the improve the global and local search ability as well as convergence speed of the 
artificial bee colony.
• 	 The flowchart of the artificial bee colony and chaotic search strategy algorithm for acceleration harmonic estimation and 
elimination is proposed.
• 	 The simulation and experiment results validate the harmonic estimation effectiveness of the artificial bee colony and chaotic 
search strategy.
0  INTRODUCTION
From simple parts to complicated equipment, 
vibration testing is required in the manufacturing 
process [1]. The purpose of vibration testing is to 
simulate the force and movement of equipment in 
the working process. Compared with field testing, 
load simulator testing under laboratory conditions 
is an indispensable part of product development and 
final mass production, and possesses the advantages 
of repeatability, low cost and safety [2]. As vibration 
environment simulation equipment, the load simulator 
can simulate sinusoidal, random, typical shock signal 
and self-set time domain waveform, such as the real 
load borne by the aircraft in the air [3], the structural 
stability of the car in the process of driving [4], and 
the seismic performance of large buildings under the 
excitation of seismic waves [5].
According to the drive types, load simulators 
are mainly divided into electric load simulators and 
hydraulic load simulators. Hydraulic load simulators 
are widely used in various industrial scenarios, such 
as exoskeleton robot and fatigue test devices due to 
their advantages of fast response, large load capacity 
and high tracking accuracy. However, the inherent 
nonlinear characteristic and external disturbance of 
hydraulic system parameters result in the serious 
waveform distortion. Such waveform pollution may 
lead to adverse effects, such as the deterioration of 
vibration quality, the losses of vibrational energy, and 
structural failure. Thus, accurate harmonic parameter 
estimation is the first step to suppress waveform 
distortion [6].
To address the harmonic estimation problem, 
various methods have been proposed to estimate the 
amplitude and phase of harmonics. The fast Fourier 
transform (FFT) is widely used for stationary signal 
harmonic analysis, but aliasing, leakage, and picket 
fence phenomena restrict its application under time 
varying frequency condition [7]. As an important time-
domain analysis tool, wavelet analysis overcomes the 
defect that FFT is completely localized in frequency 
domain and has no localization in time domain [8]. 
The orthogonal wavelet decomposition is carried 
out for the signals containing harmonics, and the 
multi-resolution concept is used to treat the results 
on the high scale as the fundamental component 
without harmonics. However, the algorithm needs 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 531-542
532 Wan, Z. – Yue, L. – Wang, Y. – Zhao, P.
software to realize the large amount of calculation, 
which influences the real-time performance and 
the estimation accuracy [9]. In addition, the real-
time performance of wavelet analysis is difficult to 
guarantee, and the estimation accuracy is relatively 
low.
Recently, several intelligent optimization 
algorithms, such as simulated annealing (SA) 
[10], genetic algorithm (GA) [11], particle swarm 
optimization (PSO) [12], gravitational search 
algorithm (GSA) [13], and artificial bee colony (ABC) 
[14], have been applied to estimate the harmonic 
components presented in power system voltage 
waveforms. Bečirović et al. [10] developed SA to 
correctly estimate the harmonics state, where root 
mean square and angle values of voltage harmonics are 
used as important indexes to measure the identification 
accuracy. Milovanović et al. [11] presented the GA 
approach to mitigate harmonics in power distribution 
systems with linear and nonlinear loads, where the 
harmonic identification problem is converted to solve 
the multi-objective optimization problem with equality 
and inequality constraints. Waqas et al. [12] proposed 
the PSO method to estimate the harmonics polluted 
by time-varying noisy power signals, where PSO 
was hybridized with least square (LS) to effectively 
detect both the linear amplitude and nonlinear phases 
of harmonics. Singh et al. [13] discussed power 
system harmonic estimation based on the GSA with 
the aim of reduction of total harmonic distortion 
(THD) and improvement in waveform. The above-
mentioned intelligent optimization methods have a 
parallel processing ability that enhances their search 
capability when applied to complex optimization 
problems [14]. Due to the fact that voltage signal is 
nonlinear in phase and linear in amplitude, phase 
of the harmonic components is obtained by using 
intelligent optimization method and the amplitude of 
the harmonic is acquired by LS [15]. The separation 
identification of amplitude and phase intensifies the 
complexity of the algorithm, which is not conducive 
to the real-time suppression of harmonics. In addition, 
Liu et al. [16] to [18] proposed improved ant colony 
optimization algorithm and multi-search strategy to 
solve many engineering problems including three-
dimensional pipe routing design and path planning, 
which broadens the application of the intelligent 
optimization algorithm.
In recent years, ABC has been developed for 
solving the limited numerical optimization problem, 
neural network training, digital filter design and so 
on, which achieves good test results [19]. Biswas et 
al. [20] combined ABC and recursive least square 
(RLS) to solve the harmonic estimation problems 
even that the number of harmonics is unknown. 
Hillis et al. [21] concluded that the ABC has better 
optimization performance than other intelligent 
optimization algorithms in iterative error curves by 
quantitative comparison and verification of various 
intelligent optimization algorithms. Zhou et al. 
[22] combined Markov property and convergence 
criterion of random search algorithm to prove the 
global convergence property of ABC. Chahkandi et 
al. [23] improved and optimized the ABC from three 
aspects: colony initialization, neighborhood search, 
and following behavior, to effectively balance global 
search and local search, where the adaptive learning 
mechanism is used to solve complex problems in 
different forms. In fact, the ABC algorithm obtains 
the optimal honey source by searching the location 
information, so neighborhood search is particularly 
important. At present, the improvement methods of 
ABC are mostly in convergence speed, which maybe 
miss a part of the neighborhood search range. What 
is more, the limitation of neighborhood search will 
directly lead to the reduction of bee colony diversity 
and a tendency to fall into a local optimal value [24]. 
It is possible to fail to find the global optimal solution 
for some special optimization problems. Therefore, an 
improved ABC, based on CSS, is proposed to estimate 
the harmonic information of hydraulic load simulator. 
The CSS enhances the diversity of population and 
improves the global search ability by introducing 
carrier mapping. Compared with other intelligent 
optimization algorithms, the proposed ABC-CSS 
has been proven to have faster convergence, higher 
capability to escape from local optima and obtain 
better quality of solutions. In addition, the proposed 
hybridized method gets better estimation performance 
and less computational burden in presence of noise 
components.
1  MATHEMATICAL MODEL OF  
ACCELERATION HARMONIC ESTIMATION
The sine acceleration response signal of a hydraulic 
load simulator is usually made up of a fundamental 
component plus some harmonic components and 
noise. Mathematically, the sine acceleration response 
can be modelled as the sum of sine or cosine functions 
with higher-order frequencies. Thus, the output of 
acceleration response can be represented as
 at At vt
nn n
n
N
() sin( )( ),  


 
1
 (1)

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 531-542
533 Acceleration Harmonic Estimation and Suppression for Hydraulic Load Simulator Based on Artificial Bee Colony with Chaotic Search Strategy  
where a(t) is the acceleration response signal; A
n
, ω
n
, 
Φ
n
 are amplitude, angular frequency and phase of n
th
 
the harmonic component; v is measurement noise. The 
angular frequency is given as ω
n
 = 2πf
n
 = 2πnf
1
, where 
f
1
 is the fundamental frequency.
The discrete model of the acceleration response 
can be expressed as
ak An kA nk
nn
n
N
nn
() coss in() sinc os() . 


 
1
 (2)
The parametric representation of acceleration 
response signal can be written as
 ak WkXk () () () , = (3)
Define X
n1
 = A
n
 cosΦ
n
, X
n2
 = A
n
 sinΦ
n
, then the 
harmonic vector X(k) and parameter vector W(k) are 
defined as
 Xk
kk kk
Nk Nk
()
sin( ),cos( ),sin( ),cos( ),
,sin() ,cos()

 

22

 





T
, (4)
 Wk xxxx xx
NN
() [,,,,, ,] , =
11 12 21 22 12
 (5)
where the harmonic vector X(k) is derived from the 
given reference frequency, the parameter vector W(k) 
is updated by the proposed algorithm.
The estimation model of the acceleration response 
signal is given by
 ak ak ek () () () , 

 (6)
where ak ak ek () () () , 

 is the estimation of a(k), e(k) denotes the 
estimation error.
The goal of harmonic estimation is that it will try 
to find the best value of W(k) so that e(k) becomes the 
minimum value. After obtaining the optimal value of 
parameter vector, the amplitudes and phases of the 
fundamental and nth harmonic are derived as
 
Ax x
xx
in n
in n







1
2
2
2
21
 arctan(/)
. (7)
2  ACCELERATION HARMONIC ESTIMATION  
AND SUPPRESSION APPROACH BASED ON ABC-CSS
2.1  The Standard ABC
The basic steps of the ABC algorithm are as follows: 
The colony size is N
p
, the employed bee size is N
e
, the 
number of nectar sources is N
p
 /2, the dimension of the 
nectar source is D, the number of iterations is K, the 
threshold of bee retention is Limit, and the population 
of employed bees is X = [X
1
, X
2
, …, X
Ne
]. Then the 
randomly distributed initial population of N
p
 solutions 
is expressed as
 XX XX
i
jj jj
 
minm ax min
(,)( ), rand 01 (8)
where i = {1, 2, …, N
p
} and j = {1, 2, …, D}, X
j
max
 and 
X
j
min
 are the lower and upper bound of the solution 
variable X
i
j
, respectively.
For the employed bees in step n, a new location is 
searched in the neighborhood of the current location 
vector, and the search formula is
 new_( ), XX XX
i
j
i
j
i
j
i
j
k
j
   (9)
where j = {1, 2, …, D} and k = {1, 2, …, N
e
}, k ≠ i, k and 
j are random numbers of [–1, 1]. 
According to the principle of optimal fitness 
selection, it is necessary to keep the optimal location 
of honey source and make the search direction of bee 
colony iterative in the direction of high honey source 
content. Then the probability distribution of new 
honey source search is represented as
 
PT
ff
ff
s
{( )}
()
()
X, new_Xn ew_X
newX X
newX X
ii i
ii
ii






1
0
()
()
_
_
 
, (10)
where X
i
 is the current time honey source position 
vector, new_X
i
 is the new honey source position 
vector.
The probability that different onlooker bees are 
recruited as corresponding employed bees is
 PT
f
fX
s
m
m
N
e
{( )}
()
()
, X
ii
i
X
X



1
 (11)
where T
s
 denotes stochastic mapping.
In the search process of nectar source, trail(i) is 
constantly used to record the search times of the same 
honey bee for the same nectar source location. When 
trail(i) > Limit and Eq. (10) is not satisfied, it means 
that the honey source in the neighborhood is generally 
low. If the honey source is searched here again, it will 
seriously affect the quality of honey source and search 
speed. So, it is necessary to re-specify the initial honey 
source location of such bees
 
Xn
XX Xi
Xn
i
i
()
(,)( )( )
() (
maxm in

 
1
01
min
rand trailL imit
trail i i)
.




Limit
 (12)
If the stop criterion is met, the calculation is 
stopped and the optimal fitness value is obtained, then 
the number of iterations is increased by 1. Otherwise 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 531-542
534 Wan, Z. – Yue, L. – Wang, Y. – Zhao, P.
the iteration continues. The flowchart of the ABC 
algorithm is shown in Fig. 1. 
2.2  The Improved ABC-CSS
In view of the lack of diversity, slow convergence 
speed and local optimum in the optimization process of 
ABC, an improved ABC based on CSS is introduced. 
In the improved method, the CSS is used to refine the 
search space of bee collecting and observing bees in 
the ABC algorithm, and a new neighborhood point of 
local optimal solution was generated in the iterative 
evolution, which accelerated the search of scout bees 
and made the bees find the optimal honey source at 
the fastest speed.
The chaotic search map can be expressed as
 yy y
nd nd nd 

1
1
,,,
() ,  (13)
where μ is the control parameter of chaotic state, 
n = 1, …, N
max
, d = 1, …, D, and y
n,d
 is the generated 
chaotic sequence.
According to the new neighborhood points of 
the local optimal solution, the chaotic variable is 
amplified by the carrier mode and applied to the single 
variable to be searched for honey source. The new 
variable is defined as
 yf Ry
nd id id nd ,
'
,,,
() ,   21 (14)
where y
'
n,d
 is the decision variable, f
i,d
 is the center of 
honey source, and R
i,d
 is the radius.
3  SIMULATION RESULTS
To validate the accuracy and convergence of the 
proposed ABC-CSS algorithm, a distorted waveform 
has been considered as a test signal, which includes 
six harmonics. In addition, several comparison 
algorithms are tested to validate the versatility of the 
developed method. The parameters are adjusted by 
trial-and-error method to ensure the good performance 
of the three controllers. The initialization parameters 
of ABC-CSS are N
p
 = 200, D = 100, N
e
 = 40, K = 2000, 
Limit = 100, μ = 4, R
i,d
 = 0.2. Fig. 2 shows the SA, 
GA, PSO, GSA, ABC and ABC-CSS, fitness against 
iteration times. It can be seen that the proposed ABC-
CSS obtained the smallest fitness value in minimum 
iteration times.
Fig. 2.  The fitting curves of different algorithms
Fig. 1.  The flowchart of the entire ABC algorithm

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 531-542
535 Acceleration Harmonic Estimation and Suppression for Hydraulic Load Simulator Based on Artificial Bee Colony with Chaotic Search Strategy  
The detailed identification results are collected 
in Table 1. In order to ensure the fairness of the 
comparison, the experimental result is the average 
after 100 runs of each algorithm with different initial 
values. The harmonic parameters calculated by 
the proposed algorithm exhibit the best estimation 
accuracy, where the largest amplitude error is 0.3 %, 
which occurred at the 4
th
 harmonic estimation. 
Besides, the standard deviation results indicate that 
the ABS-CSS method is the best performance among 
the compared algorithms. The computation time of 
ABC-CSS is the smallest compared to the other five 
algorithms. To verify the performance of the proposed 
algorithm in solving local minima, Rosenbrock 
benchmark function is tested with six algorithms.  
As is shown in Fig. 3, the ABC-CSS has excellent 
performance in both convergence accuracy and 
convergence process.
Table 1.  Comparison results of harmonic identification based on different algorithms
Algorithms Parametres Fund 2
nd
3
rd
4
th
5
th
6
th
Time [s]
Actual
Frequency [Hz] 5 10 15 20 25 30
15.2 Amplitude [m/s
2
] 6 5 4 3 2 1
Phase [rad] -0.1 0.1 0.3 -0.3 0.7 -0.7
SA
Amplitude [m/s
2
] 5.876 4.897 3.921 2.935 1.957 0.981
13.4
Error [%] 2.067 2.067 1.975 2.167 2.150 1.900
Std [m/s
2
] 0.012 0.010 0.008 0.007 0.004 0.002
Phase [rad] -0.092 0.091 0.287 -0.287 0.687 -0.685
Error [%] 8.000. 9.000 4.330 4.300 1.857 2.143
Std [rad] 8E-4 9E-4 0.001 0.001 0.001 0.002
GA
Amplitude [m/s
2
] 5.898 4.905 3.945 2.942 1.961 0.985
12.6
Error [%] 1.700 1.900 1.375 1.933 1.950 1.500
Std [m/s
2
] 0.010 0.010 0.006 0.006 0.004 0.002
Phase [rad] -0.094 0.093 0.291 -0.289 0.689 -0.689
Error [%] 6.000 7.000 3.000 3.667 1.571 1.571
Std [rad] 6E-4 7E-4 9E-4 0.001 0.001 0.001
PSO
Amplitude [m/s
2
] 5.924 4.938 3.967 2.953 1.964 0.990
11.2
Error (%) 1.267 1.240 0.825 1.567 1.800 1.00
Std [m/s
2
] 0.008 0.006 0.003 0.005 0.004 0.001
Phase [rad] -0.095 0.095 0.293 -0.292 0.693 -0.692
Error [%] 5.000 5.000 2.333 2.667 1.000 1.142
Std [rad] 5E-4 5E-4 7E-4 8E-4 7E-4 8E-4
GSA
Amplitude [m/s
2
] 5.958 4.956 3.978 2.963 1.968 0.992
10.5
Error [%] 0.700 0.880 0.55 1.233 1.600 0.800
Std [m/s
2
] 0.004 0.004 0.002 0.004 0.003 8E-4
Phase [rad] -0.097 0.097 0.295 -0.295 0.694 -0.695
Error [%] 3.000 3.000 1.667 1.667 0.857 0.714
Std [m/s
2
] 3E-4 3E-4 5E-4 5E-4 6E-4 5E-4
ABC
Amplitude [m/s
2
] 5.984 4.987 3.997 2.967 1.987 0.995
9.7
Error [%] 0.267 0.260 0.075 1.100 0.650 0.500
Std [rad] 0.002 0.001 3E-4 0.003 0.001 5E-4
Phase [rad] -0.098 0.098 0.298 -0.297 0.695 -0.697
Error [%] 2.000 2.000 0.667 1.000 0.714 0.429
Std [m/s
2
] 2E-4 2E-4 2E-4 3E-4 5E-4 3E-4
ABC-CSS
Amplitude [m/s2] 5.999 4.992 4.001 2.991 2.003 0.999
6.5
Error [%] 0.017 0.160 0.025 0.300 0.150 0.100
Std [m/s
2
] 1E-4 8E-4 1E-4 9E-4 3E-4 1E-4
Phase [rad] -0.099 0.099 0.299 -0.298 0.699 -0.699
Error [%] 1.000 1.000 0.330 0.667 0.143 0.143
Std [rad] 1E-4 1E-4 1E-4 2E-4 1E-4 1E-4

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 531-542
536 Wan, Z. – Yue, L. – Wang, Y. – Zhao, P.
Fig. 3.  The fitting curves of Rosenbrock function
The inherent nonlinear factors of the servo system 
cause serious harmonic distortion of the load simulator 
sinusoidal acceleration response. The adaptive notch 
filter can suppress the harmonic of the acceleration 
response signal in real time, so as to significantly 
improve the reliability of the load simulator’s 
sinusoidal acceleration test. The flowchart of ABC-
CSS algorithm for acceleration harmonic estimation 
and suppression is shown in Fig. 4, where r and a are 
reference signal and response signal, respectively. 
The amplitude and phase information of harmonics 
are estimated by the ABC-CSS algorithm based on 
the acceleration response. The adaptive notch filter 
generates a signal with the frequency of harmonic that 
should be eliminated from the acceleration response, 
where the centre frequency is determined by reference 
acceleration signal.
Fig. 5 shows the estimation amplitude and phase 
of each harmonic. It is noted that the amplitude and 
phase reach the given parametres after 0.4 s, which 
shows the fast convergence of the proposed algorithm. 
In addition, 5 dB, 10 dB, 15 dB and 20 dB gaussian 
noise are conducted to validate the estimation 
accuracy and robustness of the proposed algorithm. 
The harmonic amplitude and phase results under 
different SNR (signal-to-noise ratios) are shown 
in Fig. 6. These results demonstrate that the ABC-
Fig. 4.  Flowchart of ABC–CSS algorithm for acceleration harmonic estimation and elimination
Fig. 5.  The estimation amplitude and phase of each harmonic; a) amplitude, and b) phase

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 531-542
537 Acceleration Harmonic Estimation and Suppression for Hydraulic Load Simulator Based on Artificial Bee Colony with Chaotic Search Strategy  
CSS has good anti-interference performance in the 
estimation of each harmonic component containing 
noise component. Fig. 7 gives the time domain and 
frequency domain response results before and after 
harmonic suppression. The results show that the 
distortion of the sinusoidal acceleration signal is 
obviously improved after the harmonic suppression, 
and the response signal can reproduce the given 
reference signal well.
4  EXPERIMENTAL RESULTS
An experimental system of the hydraulic load simulator 
is shown in Fig. 8, which includes controller module, 
servo valve, sensor system and hydraulic system. The 
proposed algorithm is coded by MATLAB/Simulink 
and compiled by Microsoft Visual Studio.NET. The 
hydraulic valve drives the load simulator movement, 
where the control signal is processed by control 
Fig. 6.  The estimation harmonic amplitude and phase under noise; a) amplitude, and b) phase
Fig. 7.  Time domain and frequency domain response before and after harmonic suppression; a) the time domain before elimination,  
b) the frequency domain before elimination, c) the time domain after elimination, and d) the frequency domain after elimination

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 531-542
538 Wan, Z. – Yue, L. – Wang, Y. – Zhao, P.
amplitude is higher than that of the phase. Specially, 
the fundamental amplitude is the largest, and the other 
harmonic amplitudes are relatively smaller. It can be 
seen from the identification amplitude that the fifth 
harmonic component in the acceleration response 
is large, so the influence of the fifth harmonic on 
the waveform distortion should be suppressed 
first. The sinusoidal acceleration response after 
adopting the adaptive harmonic suppression strategy 
is shown in Fig. 11. It can be seen from the time-
domain response that the waveform distortion of 
the acceleration response signal is greatly improved 
after suppressing the fifth harmonic. Compared 
with the frequency domain response before and 
after harmonic suppression, it can be seen that the 
amplitude decreases by 30 dB after the fifth harmonic 
suppression. The THD analysis results before and 
after harmonic suppression are collected in Table 
2, in which it can be seen that the amplitude of the 
fifth harmonic drops sharply from 0.364 m/s
2
 before 
harmonic suppression to 0.006 m/s
2
 after harmonic 
suppression, and the THD value drops from 14.17 % 
to 5.23 %.
module. The position signal and pressure signal of the 
load simulator are measured by corresponding sensors 
and then transmitted to the controller and finally fed 
back to the computer. The hydraulic valve converts 
the hydraulic energy of the hydraulic system into the 
working force of the hydraulic cylinder according to 
the flow rate of the oil source, so as to drive the load 
simulator to produce the desired movement. It should 
be noted that the mechanical and hydraulic parameters 
are changed in real time during the experiment.
The time domain response and frequency 
domain response of low-frequency acceleration 
signal y(t) = 4sin(2 πt) [m/s
2
] are shown in Fig. 9. 
It can be seen from the time domain response that 
the distortion of the acceleration response signal is 
relatively serious, and the frequency domain response 
indicates the acceleration signal is contaminated by 
higher order harmonic components. The amplitude 
and phase estimated by the proposed method of each 
harmonic is shown in Fig. 10. It can be seen that the 
convergence speed of the identification result is very 
fast, the amplitude and phase converge to the stable 
state within 2 s, and the identification precision of the 
Fig. 8.  Hydraulic load simulator system
Fig. 9.  Time domain and frequency domain response; a) time domain, and b) frequency domain 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 531-542
539 Acceleration Harmonic Estimation and Suppression for Hydraulic Load Simulator Based on Artificial Bee Colony with Chaotic Search Strategy  
Table 2.  THD analysis results of y(t) = 4sin(2 πt)
Harmonic Before suppression After suppression 
Fund 3.642 3.984
2
nd
0.185 0.098
3
rd
0.124 0.101
4
th
0.245 0.112
5
th
0.364 0.006
6
th
0.172 0.105
THD [%] 14.17 5.23
Fig. 12 shows the time and frequency 
domain responses of the sinusoidal input signal of 
y(t)=6sin(8 πt) [m/s
2
]. It can be seen from the time-
domain response that there are amplitude attenuation 
and phase lag in the response signal to some extent, 
and especially the distortion at the crest and trough 
is serious. The frequency domain response shows 
that the amplitude of the 5
th
 harmonic is the largest; 
the 3
rd
 harmonic is the smallest. Thus, suppressing 
the 5
th
 harmonic is helpful to improve the waveform 
reproduction accuracy of the desired sinusoidal input 
signal. The identified harmonic amplitude and phase 
are shown in Fig. 13. It can be seen from the figure 
that the identification speed of amplitude and phase 
is very fast, and it only takes 0.4 s to accurately 
identify the amplitude and phase information of each 
harmonic.
The time domain and frequency domain 
responses of the sine acceleration after the suppression 
of the 5
th
 harmonic are shown in Fig. 14. From the 
time domain diagram, it can be seen that the response 
signal after the suppression of the 5
th
 harmonic fits the 
input signal well. It can be seen from the frequency 
domain response diagram that the higher harmonics 
all decrease to some extent, which indicates that the 
harmonic suppression strategy based on adaptive 
notch filter can weaken the 5
th
 harmonic in the 
response signal well. The THD analysis results before 
and after harmonic suppression are shown in Table 3. 
It can be seen that THD decreases by nearly 13.3 % 
after the fifth harmonic suppression, which indicates 
that the waveform distortion is greatly improved. The 
acceleration response after harmonic suppression can 
reproduce the given time domain reference signal 
well and accurately simulate the expected sinusoidal 
motion.
Fig. 10.  Estimated amplitude and phase; a) amplitude, and b) phase
Fig. 11.  Acceleration response after harmonic suppression; a) time domain, and b) frequency domain

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 531-542
540 Wan, Z. – Yue, L. – Wang, Y. – Zhao, P.
Table 3.  THD analysis results of y(t) = 5sin(8 πt)
Harmonic Before suppression After suppression
Fund 5.321 5.797
2
nd
0.313 0.212
3
rd
0.245 0.131
4
th
0.407 0.198
5
th
0.811 0.021
6
th
0.271 0.124
THD [%] 19.30 5.91
5  CONCLUSIONS
This paper developed an ABC-CSS method to 
accurately estimate the amplitude and phases of 
acceleration response  contaminated with nonlinear 
factors and disturbances. In this hybrid algorithm, 
the chaotic search strategy is introduced to map the 
value of chaos variable to the value range of the 
optimization variable by the carrier mode. The new 
Fig. 12.  Time and frequency domain response; a) time domain e, and b) frequency domain
Fig. 13.  Estimated amplitude and phase; a) amplitude, and b) phase
Fig. 14.  Acceleration response after harmonic suppression; a) time domain, and b) frequency domain

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 531-542
541 Acceleration Harmonic Estimation and Suppression for Hydraulic Load Simulator Based on Artificial Bee Colony with Chaotic Search Strategy  
neighborhood points of the local optimal solution are 
generated, which speeds up the global search ability 
and local search ability, enhances the diversity of 
bee colonies, and improves the convergence speed 
of the algorithm. Simulation and experiments were 
conducted to verify the effectiveness of the harmonic 
information estimation including amplitude and 
phase. The harmonic suppression strategy based on 
an adaptive notch filter neither needs to identify the 
model of the system, norto be targeted at specific kinds 
of non-linear factors, so it has strong universality and 
practical application. In addition, simple structure 
and less computation can ensure the real-time on-line 
harmonic suppression.
6  ACKNOWLEDGEMENTS
This research was funded by the Henan Key Laboratory 
of Superhard Abrasives and Grinding Equipment, 
Henan University of Technology (JDKFJJ2023005), 
the Key Science and Technology Program of Henan 
Province (242102221001, 232102220085).
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