10848


FACTA UNIVERSITATIS  
Series: Mechanical Engineering  
https://doi.org/10.22190/FUME230428025F 

© 2020 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND 

Original scientific paper 

A CIRCULAR SECTOR VIBRATION SYSTEM IN A POROUS 

MEDIUM: A FRACTAL-FRACTIONAL MODEL AND HE’S 

FREQUENCY FORMULATION 

Guangqing Feng 

School of Mathematics and Information Science, Henan Polytechnic University,  

Jiaozuo, China 

Abstract. A circular sector is commonly used in a linkage mechanism, and its frequency 

property plays an important role in optimization of the linkage mechanism. Fast insight 

into its vibration property with simple calculation is very meaningful in scientific 

research. This paper studies the vibration of the circular sector in a porous medium (e.g. 

water), and a fractal-fractional oscillator is established using the two-scale fractal 

derivative. He’s frequency formula and Ma’s modification are used to elucidate the 

circular sector’s periodic property in a porous medium, the results show that the fractal 

dimension of the porous medium plays an important role in vibration attenuation.  

Key words: He’s frequency formula, Ma’s modification, Circular sector oscillator, 

Fractal variational theory, Two-scale transform 

1. INTRODUCTION 

A circular sector is commonly used in a linkage mechanism, it might vibrate in water 

or oil, and its vibration property has caught much attention in mechanical engineering [1]. 

There are many engineering applications in literature, such as the base of structures, the 

wipers of cars, smoothing filters and many other vibration systems [2-4].  

The traditional vibration theory cannot effectively elucidate the effect of the porous 

medium’s geometry on the vibration property, now the condition is changed, because of 

the new born fractal vibration theory [5]. The fractal vibration theory can make up for the 

deficiency of the traditional vibration theory. Let's take the vibration of a sector in the air 

as an example. In all previous vibration theories, the influence of air on vibration 

characteristics is considered as an air drag; however, the performance of thin air in a 

microgravity environment is different from that of near surface atmosphere. The vibration 

                                                                 
Received: April 28, 2023 / Accepted August 10, 2023 
Corresponding author: Guangqing Feng 

School of Mathematics and Information Science, Henan Polytechnic University, Shanyang District, 454011 

Jiaozuo, Henan, China 
E-mail: fgqing@hpu.edu.cn 



2 G. FENG  

model constructed by the fractal vibration theory can consider the influence of molecules’ 

size and distribution on vibration motion, the acceleration in the fractal space is a 

combination of the damping force and the inertia force in the traditional vibration system 

[6-10], so the vibration system in a porous medium(e.g. air, water ) can be modelled by a 

fractal-fractional model without considering the damping effect. 

In general, the fractal vibration follows a fractal variational theory, that is, the vibration 

system in a porous medium obeys the fractal energy conservation law, the fractal kinetic 

energy and the potential energy are changed during the vibrating process, but their total 

energy keeps unchanged.  Fractal nonlinear systems truly describe the dynamic problems 

of engineering science, and its academic research has greatly expanded the field of human 

cognition. Many phenomena can be explained by the fractal theory, which makes us realize 

that the world is totally discontinuous and nonlinear. The fractal nonlinear vibration can be 

closer to practical problems in both depth and breadth. It is very important to study the 

analytical or approximate solutions of nonlinear vibration equations that can provide deep 

insight into the essence of general properties of a fractal vibrating system. So, there are 

many analytical and numerical methods to find an approximate solution of a differential 

equation involving fractal-fractional derivatives, such as the variational iteration method 

[11-13], the homotopy perturbation [14,15], the Hamiltonian approach [16], and the Taylor 

series method [17].  

The most important property of a nonlinear system is the frequency-amplitude 

relationship, so how to estimate quickly its relationship is an urgent problem in practical 

applications. Many researchers devoted their efforts to studying fractal calculus that 

provides a powerful tool to characterizing the periodic behavior of a nonlinear oscillator 

[18-20]. Chun-Hui He suggested a fractal nano/microelectromechanical (N/MEMS) 

system [21], Ji-Huan He gave a tutorial review on fractal space and fractional calculus 

[22,23]. He, et al. studied the fractal Duffing oscillator with arbitrary conditions [24].  

Ji-Huan He suggested a frequency formula for a conservation nonlinear oscillator [25] 

and it was further improved to an extremely simple formula in Refs. [26,27], and it has 

been widely used to solve nonlinear oscillator problems, for examples, the N/MEMS 

oscillator [28], the attachment oscillator [29], the Tangent oscillator [30], the fractal 

undamped Duffing equation [31], the large amplitude vibration system [32] and the 

vibration system on a porous foundation [33]. Many modifications were appeared in 

literature to deal with more complex vibration systems [34-37]. 

The Hamiltonian-based frequency formula is a modification of He’s frequency formula 

[38]. The frequency formula starts with two arbitrary guesses for the frequency, and two 

residual integrals must be computed to estimate a more accurate frequency. This paper 

applies He’s frequency formula and Ma’s modification to study the fractal vibration 

systems and uses the fractal circular sector oscillator as an example. The results show that 

the two methods are very effective for fractal nonlinear oscillators. 

2. FRACTAL CIRCULAR SECTOR OSCILLATOR 

As the fractal variational theory is helpful in establishing a governing equation in a 

fractal space, it has become a significantly hot topic in both mathematics and mechanical 

engineering. Many fractal variational principles were appeared for the internal temperature 

response of a porous concrete [39], the fractal Benney-Lin equation [40], the fractal 



 A Circular Sector Vibration System in a Porous Medium: a Fractal-Fractional Model...  3 

shallow water wave [41], the fractal Bogoyavlenskii system [42], the fractal Schrodinger 

system [43], the fractal solitary wave [44] and the fractal economics [45].  

The variational formula of an oscillator in a fractal space is in the form 

 






duf

d

du
uJ   )}()(

2

1
{)(

2
 (1) 

where (du/d)2/2 is the fractal kinetic energy and f(u) is the potential energy.  

When f=-cosu, it is the well-known pendulum oscillator [46,47]; when f=u2/2+au4/4, it 

is the well-known Duffing oscillator [48]. When f=gln(3R2/2-2Rkcosu)/(2R), its variational 

formula is: 

 






duRkR

R

g

d

du
uJ   )}cos2

2

3
ln(

2
)(

2

1
{)(

22
 (2) 

It meets the following energy balance equation  

 HuRkR
R

g

d

du
 )cos2

2

3
ln(

2
)(

2

1 22



 (3) 

where H is the Hamiltonian constant.  

The fractal nonlinear oscillator can be obtained as follows 

 0

cos2
2

3

sin
)(

2







uRkR

ugk

d

du

d

d



 (4) 

It is the fractal circular sector oscillator [49], where u is the angular displacement, R is 

the semicircular radius, g is the gravity acceleration, R̅ is the height of mass center, 

R̅=2Rsin/(3),  indicates the semicircular angle, k is the dimensionless geometrical 

parameter, k= R̅/R=2sin/(3), as illustrated in Fig. 1. 

 

Fig. 1 Circular sector oscillator in water 



4 G. FENG  

By the two-scale transform [50], t=, the fractal circular sector oscillator of Eq. (4) can 

be converted to an ordinary form as follows 

 0)0(,)0(,0

cos2
2

3

sin

2





 uAu

uRkR

ugk
u  (5) 

where the derivative of the function u with respect to t.  

3. HE’S FREQUENCY FORMULA  

He’s frequency formula [26,27] is an effective tool to learn the frequency-amplitude 

relationship of fractal oscillators in a timely and efficient way.  

Consider the following nonlinear oscillator  

 0)0(,)0(,0)(  uAuuhu  (6) 

where h(u) is a nonlinear function, and it satisfies h(u)/u>0.  
He’s frequency formula is： 

 
Au

du

udh

5.0

2 )(



  (7) 

or 

 
NA

NAh )(2
  (8) 

where N is a constant, 0<N≤1. N is recommended as√3/2 for nonlinear oscillators in [26, 
27]. 

Convert Eq. (5) to the following approximate form   

 0
2
 uu   (9) 

where the square of the frequency can be calculated by Eq.(8): 

 

)]
2

3
cos(2

2

3
[

2

3

)
2

3
sin(

2

2

ARkRA

Agk



  (10) 

The approximate solution of Eq. (4) can be formed as follows 

 )cos()(


  Au   (11) 



 A Circular Sector Vibration System in a Porous Medium: a Fractal-Fractional Model...  5 

4. MA’S MODIFICATION 

Ma obtained a simplified form of Hamiltonian-based frequency formula as follows [38]: 

 
22

2

)1(5.0

)()(

AL

LAHAH




  (12) 

where dH(u)/du=h(u), and L is recommend as 2-0.5 [38], the simplified Hamiltonian-based 

frequency formula converts to 

 )]
2

()([
4

2

2 A
HAH

A
  (13) 

According to Eq.(5),  we have  

 )cos2
2

3
ln(

2
)(

2
uRkR

R

g
uH   (14) 

Ma’s modification leads to the following result  

 

)2/cos(2
2

3

)cos(2
2

3

ln
2

2

2

AkR

AkR

RA

g





  (15) 

The approximate solution of Eq. (4) can be formed as following 

 )cos()(

  Au  (16) 

The obtained approximate analytical solution must be compared with the numerical 

ones to demonstrate the validity of the above approaches. Therefore, Figs. 2-4 demonstrate 

the analytical solution from the simplified Hamiltonian-based frequency method (SHF), 

together with the numerical solution (NS) and the solution from He’s frequency method 

(HF) for different values of A, R and α. The approximate periodic solutions agree well with 

exact numerical results.  

The value of α changes from /6 to /3, the change amount is =/6, and the period 

increment is T=4.5 in Fig. 2. 

The value of α changes from /2 to 2/3, the change amount is =/6, and the period 

increment is T=14 in Fig. 3. The larger the value of α, the more obvious the period change. 
It concludes that by increasing the semicircular angle α with constant semicircular radius 

R, the period of the oscillation will increase considerably. 

In Fig. 4, the value of the amplitude A determines the maximum value of u. By 

comparing Figs. 3 and 4, it is found that by reducing the semicircular radius R, the period 

of the oscillation will decrease. 



6 G. FENG  

 

Fig. 2 Comparisons of the exact solutions (NS) with the approximate solutions based on 

Eqs. (11) (HF) and (16) (SHF) for α ranging from /6 to /3 

  

Fig. 3 Comparisons of the exact solutions (NS) with the approximate solutions based on 

Eqs. (11) (HF) and (16) (SHF) for α ranging from /2 to 2/3 

  

Fig. 4 Comparisons of the exact solutions (NS) with the approximate solutions based on 

Eqs. (11) (HF) and (16) (SHF) for amplitude A ranging from /8 to /6 

0 10 20 30 40 50 60 70 80
-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

t

u
R = 15 , A =  / 8 ,  =  / 6

N S

H F

S H F

0 10 20 30 40 50 60 70 80
-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

t

u

R = 15 , A =  / 8 ,  =  / 3

N S

H F

S H F

0 10 20 30 40 50 60 70 80
-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

t

u

R = 15 , A =  / 8 ,  =  / 2

N S

H F

S H F

0 10 20 30 40 50 60 70 80
-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

t

u

R= 15 , A =  / 8 ,  = 2  / 3

N S

H F

S H F

0 10 20 30 40 50 60 70 80
-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

t

u

R = 10 , A =  / 8 ,  =  / 4

R=10, A=/6, =/4R=10, A=/6, =/4

N S

H F

S H F

0 10 20 30 40 50 60 70 80

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

t

u

R = 10 , A =  / 6,  =  / 4

N S

H F

S H F



 A Circular Sector Vibration System in a Porous Medium: a Fractal-Fractional Model...  7 

Based on Eqs. (10) and (15), results are extracted from the analytical solution and are 

shown in Fig. 5. It visually shows the relationship between A, α and the frequency when 

the value of R is fixed. An increase in the semicircle radius R or semicircle α results in a 
decrease in frequency, which implies an increase in the oscillation period. 

Five sequences of the fractal parameter θ are considered for Eq. (4) with the fixed 

parameters R=10, A=π/6, α=π/4 and shown together in Fig. 6. Numerical simulations 

indicate that the oscillation frequency becomes faster and the vibration attenuation occurs 

greater for increasing the values of the fractal exponent θ.  

  

   Fig. 5 Sensitivity analysis of frequency (0<A<π/4, 0<α<π/2) with R = 10 (left) and R = 

15 (right)  

  

Fig. 6 Eq. (4) at different  values for R=10, A=π/6, α=π/4 

0
0.2

0.4
0.6

0.8

0
0.4

0.8
1.2

1.6
0.16

0.17

0.18

0.19

0.2

0.21

0.22

Aα

fr
e

q
u

e
n

c
y

0.17

0.18

0.19

0.2

0.21

0
0.2

0.4
0.6

0
0.4

0.8
1.2

1.6
0.1

0.11

0.12

0.13

0.14

0.15

Aα

fr
e

q
u

e
n

cy

0.11

0.115

0.12

0.125

0.13

0.135

0 5 10 15 20 25 30 35 40

-0.4

-0.2

0

0.2

0.4

0.6



u

θ=0.7

θ=0.9

θ=1

θ=1.1

θ=1.3



8 G. FENG  

5. CONCLUSION 

The most important property of a fractal nonlinear system is the relationship between 

frequency and amplitude, which plays a crucial role in construction engineering. This paper 

presents He’s frequency formula and Ma’s modification of Hamiltonian-based frequency 

formula and discusses how to quickly calculate the approximate frequency of the fractal 

circular sector in a porous medium. The result shows that two methods are accurate tools 

to quickly calculating the periodic properties of fractal nonlinear oscillators, and that the 

fractal dimension of a porous medium plays an important role in vibration attenuation. Both 

the frequency formulations will play an important role in the fractal nonlinear vibration 

theory and open up numerous opportunities to elucidate the effect of the porous medium’s 

geometry on the vibration property, and also probably pave a new avenue for optimizing 

the circular sector in a nanofluid [51] where the nanoparticles’ size and distribution can be 

effectively modelled by the fractal-fractional model. 

 

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