*Corr. Author’s Address: Guilin University of Electronic Technology, Jinji Road 1#, Guilin, China,pingguotree4426@163.com 181
Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 181-193 Received for review: 2023-10-09
© 2024 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2023-12-19
DOI:10.5545/sv-jme.2023.825 Original Scientific Paper Accepted for publication: 2023-12-23
Nonlinear Free Vibration Analysis of Functionally Graded Porous 
Conical Shells Reinforced with Graphene Nanoplatelets
Huang, X. – Wei, N. – Wang, C. – Zhang, X.
Xiaolin Huang – Nengguo Wei – Chengzhe Wang – Xuejing Zhang*
Guilin University of Electronic Technology, School of Architecture and Transportation Engineering, China
The nonlinear vibration analysis of functionally graded reinforced with graphene platelet (FG-GRC) porous truncated conical shells surrounded 
by the Winkler-Pasternak elastic foundation is presented in this paper. An improved model for evaluating the material properties of porous 
composites is proposed. Three types of porous distribution and three patterns of graphene nanoplatelets (GPLs) dispersion are estimated. 
Coupled with the effect of the Winkler-Pasternak elastic foundation, the nonlinear governing equations are developed by using the Hamilton 
principle. The Galerkin integrated technique is employed to obtain the linear and nonlinear frequencies of the shells. After the present method 
is validated, the effects of the pores, GPLs, the Winkler-Pasternak foundation, and the semi-vertex are investigated in detail. The results show 
that the linear frequency can be raised by increasing the values of the mass volume of the GPL and foundation parameters. In contrast, the 
ratio of nonlinear to linear frequency declines as the mass volume of the GPLs and foundation parameters rises. Furthermore, it is found 
that the minimum ratio of nonlinear to linear frequency can be obtained as the semi-vertex angle is about 55º, and the effect of porosity 
distribution on the linear and linear frequencies might be neglected.
Keywords: nonlinear vibration, truncated conical shell, graphene nanoplatelet, porous materials, elastic foundation
Highlights
• 	 A new model for estimating the material properties of FG-GRCs is presented.
• 	 The nonlinear vibrational equations for FG-GRCs conical shells are built.
• 	 The formulations for the linear and nonlinear frequency of FG-GRCs conical shells are presented.
0  INTRODUCTION
Because the nonlinear dynamic behaviour of 
structures inevitably appears in engineering 
applications, the nonlinear characteristics of the 
structures have attracted the attention of many 
researchers. For example, Lu et al. [1] and [2] and 
Hao et al. [3] developed a novel model to analyse 
the nonlinear vibration of isolation systems with a 
high-static-low-dynamic stiffness. They validated the 
analytical results by using direct time integration and 
experiments. The inherent vulnerability of nonlinear 
vehicle platoons was studied by Wang et al. [4]. They 
proposed a vibration-theoretic approach to compute 
the platoon’s resonance frequency. Zhou et al. [5] 
presented a high-order nonlinear friction model to 
investigate the nonlinear hysteresis characteristics of 
metal rubbers. They found that the nonlinear friction 
hysteresis dynamic model has a high prediction 
accuracy and signal-to-noise ratio. Yang and Kai [6] 
built the nonlinear coupled Schrodinger equation 
in fibre gratings, and the periodic solution was 
presented. For porous composite plates reinforced 
with graphene platelets, Huang et al. [7] presented 
a two-step perturbation technique to analyse the 
nonlinear vibration of the plates. In their studies, the 
effects of the pores, graphene platelets, and elastic 
foundations on the nonlinear to linear frequency ratio 
were discussed in detail.
Due to the advantages of withstanding severe 
high temperatures while maintaining structural 
integrity, functionally graded materials (FGMs) are 
widely used in engineering fields such as aerospace, 
nuclear, mechanical, and civil engineering. The 
nonlinear static and dynamic characteristics of FGM 
shell structures have gained much attention. Chan 
et al. [8] to [10] studied the nonlinear buckling and 
vibration of FGM truncated conical shells, and Duc et 
al. [11] and Vuong et al. [12] investigated the nonlinear 
stability of FGM toroidal shells. 
Applying the concept of FGMs, a new type of 
functionally graded nanocomposites, in which the 
carbonaceous nanofilters such as graphene platelets 
(GPLs) and carbon nanotubes (CNTs), are gradually 
distributed in the thickness direction of the polymeric 
matrix, has been developed. Because nanocomposites 
are one of the most promising materials in composite 
structures, research work has been devoted to 
examining the linear vibration of functionally graded 
truncated conical shells reinforced by GPLs and 
CNTs. Wang et al. [13] studied the free vibration of 
the composite conical shells reinforced with GPLs. 
They found that the fundamental frequency is greatly 
affected by the distribution patterns of GPLs. Afshari 
[14] and [15] examined the vibration characteristics 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 181-193
182 Huang, X. – Wei, N. – Wang, C. – Zhang, X.
of truncated conical shells reinforced with graphene 
nanoplatelets and discussed the influences of the 
boundary conditions and constant angular velocity 
on the natural frequencies. For the joined conical-
conical shells made of epoxy-enriched with graphene 
nanoplatelets, Damercheloo et al. [16] studied free 
vibration and examined the influences of the boundary 
conditions, the length-to-small radius ratio, and 
semi-vertex angle in two shell segments. Using the 
finite element method in conjunction with a higher-
order shear deformation theory, Singha et al. [17] 
investigated the free vibration behaviour of rotating 
pre-twisted sandwich conical shell panels with 
functionally graded graphene-reinforced composite 
face sheets and homogenous cores in a uniform 
thermal environment. In their studies, the influence 
of graphene distribution patterns on the fundamental 
frequencies is discussed with an emphasis on 
triggering parameters like pre-twist angle, cone length‐
to‐thickness ratio, core-to-face sheets thickness ratio, 
and dimensionless rotational speed. Adab et al. [18] 
and [19] presented a vibrational analysis on truncated 
conical sandwich microshells. For the vibration 
behaviour of the composite conical shell reinforced 
with CNTs, Yousef et al. [20] and [21] investigated 
the effect of CNTs agglomeration on the vibration 
characteristics of three-phase CNT/polymer/fibre 
laminated truncated conical shells. The results show 
that the subjoining of CNTs leads to a remarkable rise 
in the natural frequency. Employing the first-order 
shear deformation theory and the Eshelby-Mori-
Tanaka scheme along with the rule of mixture, Afshari 
and Amirabadi [22] conducted the free vibration 
analysis of rotating truncated conical shells reinforced 
with CNTs and the effects of different parameters on 
the forward and backward frequencies of the shells 
are investigated. Moreover, the aeroelastic stability 
of polymeric truncated conical shells reinforced with 
CNTs and under supersonic fluid flow was studied by 
Afshari et al. [23]. However, the published literature 
on the nonlinear vibrations of functionally graded 
graphene-reinforced composite (FG-GRC) plates and 
conical shells is limited. Using the 2-D differential 
quadrature method, arc-length continuation technique, 
and harmonic balance technique, Jamalabadi et al. 
[24] studied the nonlinear vibration of FG-GRC 
truncated conical shells and investigated the effects 
of the mass volume and distribution of the GPLs, 
semi-vertex, and foundation parameters on the ratio 
of linear to nonlinear frequency. They found that 
the rising valve of the GPL mass volume can raise 
the linear frequency. In contrast, the ratio of the 
nonlinear to linear frequency declines. Using the 
variational differential quadrature and finite element 
method, Ansari et al. [25] investigated the nonlinear 
vibration characteristics of FG-GRC conical panels 
with arbitrary-shaped cutouts. The results show that 
the natural frequency rises as the weight fraction of 
the GPLs increases. Yang et al. [26] employed the 
Galerkin and harmonic balance methods to obtain 
the nonlinear frequencies of FG-GRC conical shells 
and studied the periodic and chaotic motions. To 
investigate the nonlinear dynamic characteristics of 
FG-GRC conical shells, Wang et al. [27] employed 
the Galerkin method and fourth-order Runge-Kutta 
technique to obtain the frequency response. They 
found that both the mass volume and distribution of 
GPLs significantly affect the resonance response of 
the shells. Ding and She [28] obtained the nonlinear 
dynamic response of FG-GRC truncated conical 
shells and discussed the effects of the mass fraction 
of the GPLs, geometrical parameters, and the position 
of the external load on the response. Additionally, 
Bidgoli and Arefi [29] presented the nonlinear 
vibration analysis of sandwich plates with graphene 
nanoplatelet-reinforced face sheets. They found that 
the nonlinear to linear frequency ratio can be affected 
by the weight fraction and geometric parameters of 
graphene nanoplatelets.
In the literature above, the conical shells were 
regarded as the perfect structures without pores. 
However, internal pores may appear inside composite 
materials [30]. Thus, it is necessary to investigate the 
effect of internal pores on the dynamic behaviour 
of porous structures. Some researchers studied the 
vibration characteristics of porous isotropic [31] and 
[32], sandwich [33] and [34]. For the functionally 
graded porous (FGP) truncated conical panels with 
piezoelectric actuators in thermal environments, 
Chan et al. [35] investigated the nonlinear dynamic 
response and free vibration. In their studies, the effect 
of the porosity distribution on the natural frequency 
and the deflection amplitudes were discussed. 
Considering the two types of porosity distribution, 
the buckling and vibration of sigmoid functionally 
graded material shells were studied by Huang et 
al. [36]. The results show both the porosity volume 
fraction and distribution have significant effects 
on the buckling pressures and lowest frequency. 
Nevertheless, little work has been done for the porous 
truncated conical shells reinforced by nano filters [37]. 
Bahhadini et al. [38] and Yan et al. [39] studied the 
linear vibration of porous FG-GRC nanocomposite 
sandwich conical shells and found that the natural 
frequencies were increased with the rising material 
length-scale parameter and porosity coefficient. The 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 181-193
183 Nonlinear Free Vibration Analysis of Functionally Graded Porous Conical Shells Reinforced with Graphene Nanoplatelets
vibration behaviour of porous sandwich truncated 
conical shells with FG face sheets and a saturated 
FGP core was studied by Rahmani et al. [40] and 
Esfahani et al. [41]. The effect of internal pores on 
the natural frequency was also discussed in detail. 
Applying the isogeometric analysis, Le et al. [42] 
obtained the three-dimensional solution of the free 
vibration and buckling for functionally graded porous-
cellular conical shells. Sobhani et al. [43] studied 
the vibrational behaviour of porous nanocomposite 
joined hemispherical-cylindrical-conical shells and 
examined the effects of the porosity distributions 
and geometric properties. The results revealed that 
the natural frequency inclines with the increase of 
the porosity factor. Additionally, the free vibration of 
FG-joined conical-cylindrical shells reinforced with 
graphene nanoplatelets was investigated by Kiarasi 
et al. [44]. It was found that the volume fraction and 
distributions of the internal pores have an insignificant 
impact on the natural frequencies of FG-GRC porous 
conical shells.
Unlike the Winkler elastic foundation model, 
the Winkler-Pasternak model incorporates the shear 
interaction between structures and foundations and has 
been widely used to examine the dynamic behaviour 
of FGM-truncated conical shells [45] to [47]. However, 
the model was used less to investigate the vibration of 
FG-GRC conical shells. Safarpour et al. [48] studied 
the free vibration of perfect and imperfect FG-GRC 
conical shells resting on Pasternak foundations. Their 
results show that the effect of the elastic foundation 
becomes more dominant as the foundation parameters 
increase. Eyvazian et al. [49] computed the natural 
frequencies and the corresponding mode shapes of 
FG-GRC conical panels and discussed the influences 
of boundary conditions, GPLs volume fraction, and 
foundation parameters. Furthermore, the frequency 
responses of rotating two-directional FG-GPLs 
conical shells on elastic foundations were presented by 
Amirabadi et al. [50]. They found that the increasing 
value of the GPL mass fraction can raise the natural 
frequency. The pattern of GPLs scattering near the 
inner surface has a higher effect on the frequency. 
As reviewed above, the published literature on the 
free vibration of FG-GRC porous truncated conical 
shells remains limited. Most of the open literature 
focused on the case of the linear problem. The model 
used for evaluating the material properties of porous 
nanocomposites was based on the three assumptions 
about Young’s elastic modulus, shear modulus, and 
mass density, which is too complicated. Hence, the 
present work attempts to solve these problems, that 
is, to propose an improved model for estimating the 
material properties and present the analytical solution 
for the nonlinear free vibration of FG-GRC porous 
truncated shells. Also, the effects of the internal 
pores, GPL, and elastic foundation on the linear and 
nonlinear frequencies are investigated.
1  A POROUS FG-GPLS TRUNCATED CONICAL SHELL
As shown in Fig. 1, a porous FG-GPLs truncated 
conical shell surrounded by Winkler-Pasternak elastic 
medium is considered. The curvilinear coordinate 
system (x, θ, z) is located on the middle surface of the 
shell. s
1
 and s
2
 are the distances from the vertex to the 
small and large ends, respectively. L, h and α denote 
the length, thickness, and semi-vertex of the shell.
Fig. 1.  A FG-GPLs truncated conical shell surrounded by a Winkler-
Pasternak elastic foundation
Three types of porous distribution in the thickness 
direction, denoted by P-1, P-2, and P-3, are taken into 
account, as depicted in Fig. 2. 
It is assumed all the internal pores are tiny and 
the total volume fraction of the pores is small. Unlike 
other models for evaluating the material properties 
[30] and [39], in which Young’s modulus, shear 
modulus, and mass density need to be assumed, the 
present model is based on the following assumption 
about the volume fraction V
P
(z) of the internal pores:
 
Vz e
z
h
Vz e
z
h
V
P
P
P
 







 







0
0
1
24
2
cos, ()
cos, ()


P
P
z ze  
0
3 ,( ) P (1)

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 181-193
184 Huang, X. – Wei, N. – Wang, C. – Zhang, X.
here, e
0
 denotes the porosity coefficient. Young’s 
elastic modulus E(z) and mass density ρ(z) for various 
porosity distributions can be expressed as [39].
 
Ez EV z
z
e
e
Vz
P
m
P
() (),
() (),
 







1
1
0
1
1  (2)
where E
1
 and ρ
1
 denote the maximum values of 
Young’s modulus and mass density, and e
m
 is the mass 
coefficients. Furthermore, the coefficients e
0
, e
m
 and 
the Poisson ratio μ(z) are calculated as follows [39]:
 
e
eV z
Vz
zp pp
P
P
m


 
1 121 11
0 221 0 342 12 1
0
23
1
2
.[ () ]
()
,
() .( ..
.
  
 
10
1 12111
23
.) ,
.( () ),
.
pV z
P
 (3)
in which μ
1
 is the Poisson ratio of the composite 
without pores, and p is the coefficient of the Poisson 
ratio.
a) 
b) 
c) 
Fig. 2.  Three types of porosity distribution:  
a) symmetric distribution (P-1), b) asymmetric distribution (P-2), 
and c) even distribution (P-3)
a)                        b)                           c)
Fig. 3.  Three patterns of GPLs dispersion:  
a) symmetric distribution (G-1), b) asymmetric distribution (G-2), 
and c) even distribution (G-3)
It is noted that the present model in Eqs. (1) to (3) 
can be used not only for FG-GRCs but also for other 
composites in which the foam metal is the matrix. 
In the present study, three patterns of GPL 
dispersion, denoted by G-1, G-2, and G-3, are 
considered, as shown in Fig. 3. The volume fractions 
V
GPL
(z) for the three patterns are expressed as follows 
[43]:
 
Vzf
z
h
Vzf
z
h
i
i
GPL
GPL
G  













  
1
2
11
1
2
cos, ()
cos

  
4
2
3
3













 
,( )
,( )
G
G
GPL
Vzf
i
 (4)
where f
i1
, f
i2
 and f
i3
 are the maximum values of various 
GPL distributions, calculated by [39]:
W
WW
e
e
Vzdz
V
P
h
h
GPL
GPLG PL mG PL
m
GPL
  
 









 /
/
/
1
1
0
2
2
z z
e
e
Vzdz
P
h
  








1
0
2
2
m
h/
/
. (5)
In Eqs. (2) and (3), E
1
, ρ
1
, and μ
1
 can be 
determined by using the Halpin-Tsai micromechanics 
model and the rule of the mixture as follows [39]:
   E
V
V
E
V
V
1
3
8
1
1
3
8
1
1






















aaGPL
aG PL
m
bbGPL
bG PL
E E
m
, (6)
  
1
1 zV zV z      


 GPLG PL GPLm
, (7)
  
1
1 zV zV z      


 GPLG PL GPLm
, (8)
in which the geometric parameters of GPL ξ
a
 , ξ
b, 
η
a
 
and η
b
 can be stated as
 




a
GPL
GPL
b
GPL
GPL
a
GPLm
GPLma
GPL





22
1
a
h
b
h
EE
EE
EE
b
,,
/
/
,
/
m m
GPLmb


1
EE /
,

 (9)

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 181-193
185 Nonlinear Free Vibration Analysis of Functionally Graded Porous Conical Shells Reinforced with Graphene Nanoplatelets
here, E
GPL
, ρ
GPL
 and μ
GPL
 denote the Young’s 
modulus, mass density and Poisson ratio of GPL, and 
E
m
, ρ
m
 
and μ
m
 are the corresponding values of the 
matrix. a
GPL
, b
GPL
 and h
GPL
 are the length, width, and 
thickness of the GPL.
2  FORMULATIONS
2.1  Governing Equations
It is assumed that the shell is thin and has a large 
deformation. According to the classic shell theory 
with the geometrical nonlinearity, the normal strains 
( ε
1
, ε
2
) and shear strain γ
12
 can be expressed as [51]
   
12
  
1
0
12
0
21 21 2
0
12
2 zz z ,, , (10)
in which the strains 

1
0
2
0
12
0
,,
 , curvatures ( κ
1
, κ
2
) 
and twist κ
12
 in the middle surface are defined by [51]





1
0
2
2
0
22
1
2
11
2















 

u
x
u
x
x
vu
x
w
x x
,
sin
cot
sin
w w
x
uv
x
v
xx
w
x
w










 












2
12
0
1
11
,
sin
cot
sin
,
2 2
2 2 22
2
2
12
2
2
11
11
w
xx
w
x
w
x
x
w
x x
w













,
sin
,
sin



  
, (11)
where u, v and w are the displacements along the 
directions of x, θ and z axes, respectively.
According to Hooke’s law, the normal stresses σ
1
  and σ
2
, and the shear stress σ
12
 can be stated as [51]
 






1
2
12
11 12
12 22
66
1
2
12
0
0
00

























QQ
QQ
Q  





. (12)
Here, Q
ij
 (i, j = 1, 2, 6) are the reduced stiffness 
coefficients, defined by [51]
 
QQ
Ez
z
Q
zEz
z
Q
Ez
z
11 22 2 12 2
66
11
21





 
()
()
,
() ()
()
,
()
()
.




 (13)
The membrane forces (N
1
, N
2
, N
12
) and moments 
(M
1
, M
2
, M
12
)  are expressed as [55]
 
NNNM MM
zd z
h
h
1
,, ,, ,
,, ,.
/
/
21 21 21 2
1212
2
2
1
  



  


 (14)
Substituting Eq. (12) into (14), the following 
equations can be obtained [51]:
 
NA ABB
NA ABB
11 11
0
12 2
0
11 11 22
21 21
0
22 2
0
12 12 22




,
,
N NA B
MB BDD
MB
12 66 12
0
66 12
11 11
0
12 2
0
11 11 22
21 21
2 





,
,
0 0
22 2
0
12 12 22
12 66 12
0
66 12
2


BDD
MB D


,
. (15)
The stiffness constants A
ij
, B
ij
, and D
ij
, can be 
calculated by [51]
 ABDQ zz dz
ij ij ij
h
h
ij
,, ,, .
/
/





1
2
2
2
 (16)
The interaction force between the Winkler-
Pasternak medium and shell is assumed to be 
Fk wk w  
wp
2
, in which k
w
 and k
p
 are the 
parameters of Winkler and Pasternak foundation, 
and ∇
2
 is the Laplace operator. Using the Hamilton 
principle, the nonlinear dynamic Equilibrium 
equations of the conical shell surrounded by the 
Winkler-Pasternak medium can be derived as [51]:
 
x
N
x
N
NN
u
t
N
x
N
x
N
t














11 2
12
2
2
21 2
12
1
1
2
sin
,
sin



 
 
t
v
t
x
M
x
M
x
M
xx
M




















2
2
2
1
2
2
1
2
12 12
2
21
1
,
sin
x x
w M
x
N
xN
w
x
N
w
x
sin
cot
sin
s
,
2
2
2
2
2
11 2
1
1






















i in sin
,
 

xN
w
x
N
w
11 2
1 










       
 


xk wx kw x
w
t
pt w
2
2
2
 ,
 (17)
where the mass inertia is 
t
h
h
zd z 


()
.
.
05
05
.
Substituting Eqs. (11) and (15) into (17), the 
nonlinear vibrational equations of the shell can be 
derived as follows [51]:
 Lu Lv Lw Lw
u
t
t 11 12 13 14
2
2
() () () () , 


 (18)
 Lu Lv LwLw
v
t
t 21 22 23 24
2
2
() () () () , 


 (19)

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 181-193
186 Huang, X. – Wei, N. – Wang, C. – Zhang, X.
Lu Lv Lw Lu vw x
w
t
t 31 32 33 34
2
2
() () () (,,) , 


 (20)
where the linear operators L
ij
 (i, j = 1, 2, 3) have been 
given by Duc et al. [51]. The nonlinear operators L
i4
 
(i, j = 1, 2, 3) are listed in Appendix. According to 
Volmir’s assumption [26], the inertia forces  

t
u
t


2
2
 and 
t
v
t


2
2
 can be neglected.
2.2  The Solution of the Governing Equations
In this study, simply supported boundaries are 
considered. The boundary conditions are written as
  vw NM Mx ss = = = = = = 00 0
11 12 12
,, ,, . at (21)
In the present case, the asymmetric solution 
is taken into account. The solution satisfying the 
boundary conditions is assumed to be [51]
 
uut
mxs
L
n
vvt
mxs
L
n
mn
mn

 







() cos
()
sin,
() sin
()
cos
 
 
1
1
2
2 2
2
1







 





,
() sin
()
sin, wwt
mxs
L
n
mn
 
 (22)
where m and n are the numbers of half-waves along 
the generator and parallel circle, respectively. 
In Eq. (22), if the terms of sin( nθ/2) and cos( nθ/2) 
are eliminated, the asymmetric solution is transformed 
into the corresponding symmetric solution. 
For the sake of convenience in integration, 
multiplying Eqs. (18) and (19) by x and Eq. (20) by 
x
2
, then applying the Galerkin method for the resulting 
equations, Eq. (18) to (20) can be developed as 
follows [51]:


1
0
2
1
2
0
2
1
2
1
2
2
0


 


 
 


s
s
s
s
mxs
L
n
xd xd
m
cos
()
sins in ,
(
sin
x xs
L
n
xd xd
mxs
L
n
x
s
s



 
1
3
0
2
1
2
0
2 1
2
)
coss in ,
()
sins in


 

 sin  

 


dxd
w
t
mxs
L
n
xd xd
t
s
s




 
2
2
0
2
1 3
1
2
2
sin
()
sins in ,(23)
in which
 


11 11 21 31 4
22 12 22 3
  
 
xL uL vL wL w
xL uL vL w
() () () () ,
() () ()   
  
Lw
xLuL vL wLw
24
3
2
31 32 33 34
() ,
() () () () .  (24)
Substituting Eqs. (22) and (24) into Eq. (23), 
the following ordinary differential equations can be 
derived:
 a
dw
dt
aw aw aw
mn
mn mn mn 1
2
2 23
2
4
3
0  , (25)
where a
i
 (i, j = 1, 2, 3, 4) are integral coefficients.
According to Eq. (25), the linear frequency 
is obtained as 
L
aa 
21
/ , and the nonlinear 
frequency can be derived as [52]: 
 
NL L
aa a
a
A 

1
91 0
12
42 3
2
2
2
2
. (26)
Here, the dimensionless vibrational amplitude A 
is w
max
 / h, in which w
max
 is the maximum dynamic 
deflection.
3  RESULTS AND DISCUSSION
3.1  Comparison Studies
In this subsection, two examples are given to validate 
the accuracy of the present method.
Example 1. In Fig. 4, the curves of the 
dimensionless linear frequency λ versus the porosity 
coefficient e
0
 for a porous FG-GRC truncated 
conical shell are depicted. The material properties 
of the matrix are E
m
 = 130 GPa, ρ
m
 = 8960 kg/m³ and  
μ
m
 = 0.34. The pattern of GPL dispersion is G – 1. The 
material and geometrical parameters are E
GPL
 = 1.01 
TPa, ρ
GPL
 = 1062.5 kg/m³, μ
GPL
 = 0.86, a
GPL
 =  2.5 µm, 
b
GPL
 = 1.5 µm, and h
GPL
 = 1.5 nm. The geometrical 
parameters of the shell are R
2
 / h = 200 cos α, L = s
1
 
and α  =  10°. The dimensionless linear frequency is 
λ = ω[ ρ
t
h
2
L
2
 / D
1
]
0.5
. The figure reveals that the present 
results are in good agreement with those given by 
Bahaadini et al. [38]. It is noted the natural frequency 
decreases with the rising porous coefficient e
0
. 
Although both the stiffness and mass density decline 
with the increase of the coefficient, the decreasing 
speeds are different. As the decreasing speed of 
the mass density is higher than that of stiffness, the 
natural frequency reduces. In the opposite case, the 
natural frequency increases.
Example 2. The fundamental linear and nonlinear 
frequencies for an isotropic truncated conical shell 
surrounded by a Winkler-Pasternak foundation are 
calculated and listed in Table 1. The geometrical 
parameters of the shell are R
1
 / h = 300, L = 2R
1
, γ  =  30°. 
The vibrational amplitude is A = 3. The dimensionless 
frequencies are

LL
   RE
2
2
1 () / , 
NL NL
   RE
2
2
1 () / . 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 181-193
187 Nonlinear Free Vibration Analysis of Functionally Graded Porous Conical Shells Reinforced with Graphene Nanoplatelets
It is seen that the present results agree well with those 
given by Najafov and Sofiyev [53].
Fig. 4.  Comparison of the fundamental frequency for a porous FG 
truncated conical shell
3.2  Parametric Studies
After the present method is validated, the effects of 
GPL, pore, and semi-vertex angle on the dimensionless 
linear and nonlinear frequencies for porous FG-GRC 
truncated conical shells surrounded by Winkler-
Pasternak elastic foundations are investigated in this 
subsection. The material parameters of the GPL are 
the same as those given in Example 1. The material 
properties of the matrix are E
m
 = 3.0 GPa, ρ
m
 = 1200 
kg/m³ and ν
m
 = 0.34 and the geometrical parameters of 
the shell are R
1
 / h = 100, L = R
1
 and α  =  30°. The 
dimensionless linear frequency  
LL mm
 RE
2
/ . 
Unless specially stated, the type of the porosity 
distribution is P–1, and the pattern of GPL dispersion 
is G–1.
Table 1. Dimensionless linear and nonlinear fundamental 
frequencies for an isotropic truncated conical shell rested on an 
elastic foundation
k
w
 [N/m³] k
p 
[N/m]
Ω
L
Ω
NL
Ref. [53] Present Ref. [53] Present
10
5
0 0.101 0.102 0.119 0.120
2.5×10
4
0.106 0.106 0.123 0.125
5.0×10
4
0.110 0.111 0.126 0.128
7.5×10
4
0.115 0.113 0.129 0.130
5.0×10
4
0 0.119 0.117 0.132 0.134
2.5×10
4
0.123 0.123 0.138 0.140
5.0×10
4
0.131 0.129 0.140 0.141
7.5×10
4
0.134 0.131 0.143 0.145
The variation of the linear frequency ϖ
L
 
with the generatrix half-wave number m and 
circumferential half-wave number n is shown in Fig. 
5. As demonstrated by other literature [37] to [39], the 
number m has a different impact on the frequency with 
that of the genetrix number n. The frequency is raised 
with the rising value of m. If n < 13, the frequency 
is monotonously decreased. However, if n > 13, the 
frequency is increased. This is because the mass 
density declines more significantly than the stiffness 
when n < 13. In contrast, the mass density declines 
more slowly than the stiffness when n > 13. Hence, the 
fundamental frequency can be obtained when m ≈ 1, 
n ≈ 13. The following linear and nonlinear frequencies 
are calculated at the vibration mode (1,13).
a) 
b) 
Fig. 5.  Variation of the linear frequency with a) the generatrix half-
wave number m and b) the circumferential half-wave number n
Tables 2 to 4 list the dimensionless linear 
frequencies ϖ
L
 of the shell with the different 
parameters of pores, GPL, and elastic foundations. 
Because the Winkler-Pasternak foundation can 
raise the effective stiffness of the conical shell, the 
frequency inclines with the increases of foundation 
parameters k
w
 and k
p
. Furthermore, the effect of this is 
more significant than that of the parameter because it 
can enhance the effective stiffness more significantly 
k
w
. The three tables show that the frequency increases 
with the increasing mass fraction of the GPLs, which 
is due to the fact that the elastic modulus of the GPL is 
higher than that of the matrix. Also, the tables reveal 
that the effect of the porosity coefficient e
0
 is related 
to the parameters of the elastic foundation. If the 
values of k
w
 and k
p
 are not zero, the frequency reduces 
with the increasing value of the porosity coefficient
 
e
0
. In contrast, if the values of k
w
 and k
p
 are zero 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 181-193
188 Huang, X. – Wei, N. – Wang, C. – Zhang, X.
Table 2. Dimensionless linear frequencies ϖ
L
 for a porous FG-GRC conical shell distributed with G-1 GPLs and rested on an elastic foundation
k
w
   
10
6
 [N/m³]
k
p
  
10
4
 [N/m]
e
0
 = 0.0
P-1 P-2 P-3
e
0
 = 0.2 e
0
 = 0.4 e
0
 = 0.2 e
0
 = 0.4 e
0
 = 0.2 e
0
 = 0.4
W
GPL
 = 0.0 %
0
0 0.192 0.188 0.184 0.185 0.178 0.196 0.192
2.5 0.337 0.342 0.350 0.341 0.347 0.337 0.338
5.0 0.436 0.446 0.460 0.445 0.457 0.436 0.437
1.0
0 0.413 0.421 0.434 0.421 0.431 0.413 0.414
2.5 0.497 0.510 0.526 0.509 0.524 0.497 0.499
5.0 0.568 0.584 0.605 0.584 0.604 0.568 0.569
W
GPL
 = 0.1 %
0
0 0.231 0.226 0.221 0.223 0.213 0.231 0.230
2.5 0.360 0.365 0.372 0.363 0.366 0.360 0.361
5.0 0.454 0.463 0.476 0.462 0.472 0.454 0.455
1.0
0 0.432 0.440 0.452 0.439 0.474 0.432 0.434
2.5 0.513 0.525 0.541 0.524 0.537 0.513 0.515
5.0 0.583 0.598 0.617 0.597 0.614 0.583 0.586
W
GPL
 = 0.3 %
0
0 0.294 0.289 0.283 0.284 0.271 0.294 0.293
2.5 0.403 0.406 0.411 0.403 0.403 0.403 0.404
5.0 0.489 0.497 0.508 0.494 0.501 0.489 0.490
1.0
0 0.469 0.475 0.485 0.472 0.477 0.469 0.471
2.5 0.544 0.555 0.569 0.552 0.563 0.544 0.546
5.0 0.610 0.627 0.642 0.622 0.637 0.610 0.613
Table 3.  Dimensionless linear frequencies ϖ
L
 of a porous FG-GRC conical shell distributed with G-2 GPLs and rested on an elastic foundation
k
w
   
10
6
 [N/m³]
k
p
  
10
4
 [N/m]
e
0
 = 0.0
P-1 P-2 P-3
e
0
 = 0.2 e
0
 = 0.4 e
0
 = 0.2 e
0
 = 0.4 e
0
 = 0.2 e
0
 = 0.4
W
GPL
 = 0.1 %
0
0 0.220 0.215 0.210 0.212 0.202 0.220 0.219
2.5 0.353 0.358 0.364 0.356 0.360 0.353 0.354
5.0 0.484 0.458 0.471 0.456 0.467 0.448 0.449
1.0
0 0.426 0.434 0.446 0.433 0.442 0.426 0.426
2.5 0.508 0.520 0.536 0.519 0.533 0.508 0.510
5.0 0.578 0.594 0.613 0.592 0.611 0.578 0.579
W
GPL
 = 0.3 %
0
0 0.259 0.253 0.247 0.250 0.238 0.259 0.258
2.5 0.379 0.382 0.387 0.380 0.382 0.379 0.380
5.0 0.469 0.477 0.488 0.475 0.484 0.469 0.471
1.0
0 0.480 0.455 0.465 0.453 0.460 0.448 0.449
2.5 0.526 0.537 0.552 0.536 0.548 0.526 0.528
5.0 0.595 0.609 0.627 0.607 0.624 0.595 0.597
Table 4. Dimensionless linear frequencies ϖ
L
 of a porous FG-GPLs conical shell distributed with G-3 GPLs and rested on an elastic foundation
k
w
   
10
6
 [N/m³]
k
p
  
10
4
 [N/m]
e
0
 = 0.0
P-1 P-2 P-3
e
0
 = 0.2 e
0
 = 0.4 e
0
 = 0.2 e
0
 = 0.4 e
0
 = 0.2 e
0
 = 0.4
W
GPL
 = 0.1 %
0
0 0.222 0.218 0.214 0.214 0.205 0.224 0.220
2.5 0.369 0.374 0.382 0.372 0.377 0.368 0.370
5.0 0.471 0.482 0.496 0.481 0.403 0.470 0.473
1.0
0 0.427 0.436 0.448 0.434 0.444 0.426 0.429
2.5 0.519 0.532 0.548 0.530 0.545 0.518 0.521
5.0 0.569 0.612 0.633 0.611 0.630 0.595 0.599
W
GPL
 = 0.3 %
0
0 0.268 0.263 0.258 0.259 0.247 0.269 0.267
2.5 0.398 0.402 0.409 0.399 0.402 0.397 0.399
5.0 0.495 0.504 0.517 0.502 0.512 0.494 0.496
1.0
0 0.453 0.460 0.471 0.458 0.465 0.452 0.454
2.5 0.540 0.551 0.567 0.550 0.562 0.538 0.540
5.0 0.615 0.630 0.650 0.628 0.645 0.613 0.616

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 181-193
189 Nonlinear Free Vibration Analysis of Functionally Graded Porous Conical Shells Reinforced with Graphene Nanoplatelets
(k
w
 = 0, k
p
 = 0), the frequency rises. Additionally, it 
can be observed that the effects of the pattern of GPL 
dispersion and the type of porosity distribution on the 
linear frequency can be neglected.
The influences of the type of GPL dispersion and 
mass fraction of the GPL on the ratio of nonlinear to 
linear frequency ϖϖ
NL L
/ are shown in Fig. 6. Among 
the three patterns of GPL dispersion, the ratio for G–2 
is slightly larger than those for G–3 and G–1. That is 
because the nonlinear frequency for G–2 increases 
faster than that for G–3 and G–1. Moreover, the mass 
fraction of the GPLs W
GPL
 rises with the increase of 
the ratio. For instance, when W
GPL
 changes from 0 % 
to 0.5 %, the ratio increases by about 9 % at A = 5.
a) 
b) 
Fig. 6.  Influence of the GPLs on the frequency ratio; a) the pattern 
of GPL dispersion, and b) the mass fraction of the GPLs
The influences of the porosity distribution and 
porosity coefficient on the frequency ratio ϖϖ
NL L
/ 
are shown in Fig. 7. It is found that the porosity 
distribution has an insignificant effect on the ratio. 
When A = 5, the ratio for P–1 is only higher by 1.5 % 
than that for P–3. Also, it is found that the ratio can be 
raised by increasing the porosity coefficient. 
Fig. 8 shows the effects of parameters k
w
 and k
p
 
on the frequency ratio ϖϖ
NL L
/ . It is seen that the 
effects are very significant. For example, when the 
parameter k
p
 changes from 0 kN/m to 75 kN/m, the 
ratio declines by 98.9 %.
a) 
b) 
Fig. 7.  Influence of pores on the frequency ratio; a) porosity 
distribution, and b) porosity coefficient
a) 
b) 
Fig. 8.  Influence of foundation parameters on the frequency ratio; 
a) Winkler parameter, and b) Pasternak parameter
Finally, the effects of the semi-vertex angle α
 
on the linear frequency ϖ
L
 and the ratio of linear to 
nonlinear frequency ϖϖ
NL L
/ are shown in Fig. 9. It 
is shown that the linear frequency declines when the 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 181-193
190 Huang, X. – Wei, N. – Wang, C. – Zhang, X.
semi-vertex angle increases. This is because the 
effective stiffness is raised with the increase of the 
semi-vertex angle. The figure also shows that the ratio 
reduces as the angle increases from 15º to 55º. In 
contrast, it is increased as the angle changes from 55º 
to 75º. Thus, the minimum value of frequency ratios 
can be obtained at α  ≈ 55º.
a) 
b) 
Fig. 9.  Influence of the semi-vertex angle on  
a) the linear frequency and b) the frequency ratio
4  CONCLUSIONS 
In this study, an improved method for evaluating the 
material properties of porous FG-GRCs was proposed. 
An analytical method to investigate the nonlinear 
vibration of composite such as FGM, FG-GRC, and 
FG-CNT porous truncated conical shells was 
presented. The effects of the pore, GPL, elastic 
foundation, and semi-vertex angle on the linear 
frequency and the ratio of nonlinear to linear 
frequency were discussed. Compared with other 
methods for investigating the nonlinear vibration 
behaviour of FG-GRC porous conical shells, the 
present method is simpler and briefer. However, it is 
not more accurate because the inertia forces 
t
u
t


2
2
 
and 
t
v
t


2
2
 are neglected. Moreover, some interesting 
conclusions can be drawn from the parametric studies 
as follows:
1. As the values of the mass fraction of the GPLs 
is raised. In contrast, the frequency ratio reduces. 
Among the three patterns of GPLs dispersion, the 
frequency ratio for G-1 is the lowest, and that for 
G-2 is the highest.
2. The natural frequency is not monotonously varied 
with the rise of the porous coefficient 
0
e . If the 
mass density decreases faster than the stiffness, 
the natural frequency increases. In contrast, the 
natural frequency declines.
3. The frequency ratio can be decreased by 
increasing the porosity coefficient. However, the 
effect of porosity distribution on the frequency 
ratio might be negligible.
4. The linear frequency reduces if the semi-vertex 
angle increases. The minimum value of the 
frequency ratio can be obtained as the semi-
vertex angle is approximately 55º.
5. Both the Winkler and Pasternak foundation 
parameters can significantly raise the natural 
frequency. However, they lower the ratio of 
nonlinear to linear frequency.
5  ACKNOWLEDGEMENTS
The authors thank for the financial support of 
the Natural Science Foundation of Guangxi 
[No.2021GXNSFAA220087] and the National 
Natural Science Foundation of China [No.12162010].
6  REFERENCES
[1] Lu, Z.Q., Yang, T.J., Brennan, M.J., Liu, Z.G., Chen, L.Q. (2017). 
Experimental investigation of a two-stage nonlinear vibration 
isolation system with high-static-low-dynamic stiffness. 
Journal of Applied Mechanics, vol. 84, no. 2, art. ID 021001, 
DOI:10.1115/1.4034989.
[2] Lu, Z.Q., Gu, D.H., Ding, H., Lacarbonara W., Chen, L.Q. (2020). 
Nonlinear vibration isolation via a circular ring. Mechanical 
Systems and Signal Processing, vol. 136, art. ID 106490, 
DOI:10.1016/j.ymssp.2019.106490.
[3] Hao, R.B., Lu, Z.Q., Ding, H., Chen, L.Q. (2022). A nonlinear 
vibration isolator supported on a flexible plate: analysis and 
experiment. Nonlinear Dynamics, vol. 108, p. 941-958, 
DOI:10.1007/s11071-022-07243-7.
[4] Wang, P.C., Wu, X.K., He, X.Z. (2023). Vibration-theoretic 
approach to vulnerability analysis of Nonlinear vehicle 
platoons. IEEE Transactions on Intelligent Transportation 
Systems, vol. 24, no. 10, p. 11334-11344, DOI:10.1109/
TITS.2023.3278574.
[5] Zhou, C.H., Ren, Z.Y., Lin, Y.X., Huang, Z.H., Shi, L.W., Yang, Y., 
Mo, J.L. (2023). Hysteresis dynamic model of metal rubber 
based on higher-order nonlinear friction (HNF). Mechanical 
Systems and Signal Processing, vol. 189, art. ID 110117, 
DOI:10.1016/j.ymssp.2023.110117.
[6] Yang, Y., Kai, Y. (2023). Dynamical properties, modulation 
instability analysis and chaotic behaviors to the nonlinear 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 181-193
191 Nonlinear Free Vibration Analysis of Functionally Graded Porous Conical Shells Reinforced with Graphene Nanoplatelets
coupled Schrödinger equation in fiber Bragg gratings. 
Modern Physics Letters B, vol. 38, no. 6, art. ID 2350239, 
DOI:10.1142/S0217984923502391.
[7] Huang, X.L., Wang, C.Z., Wang, J.H., Wei, N.G. (2022). 
Nonlinear vibration analysis of functionally graded porous 
plates reinforced by graphene platelets on nonlinear elastic 
foundations. Strojniški vestnik - Journal of Mechanical 
Engineering, vol. 68, no. 9, p. 571-582, DOI:10.5545/sv-
jme.2022.274.
[8] Chan, D.Q., Long, V.D., Duc, N.D. (2019). Nonlinear buckling 
and post-buckling of FGM shear-deformable truncated conical 
shells reinforced by FGM stiffeners. Mechanics of Composite 
Materials, vol. 54, p. 745-764, DOI:10.1007/s11029-019-
9780-x.
[9] Chan, D.Q., Quan, T.Q., Kim, S.E., Duc, N.D. (2019). Nonlinear 
dynamic response and vibration of shear deformable 
piezoelectric functionally graded truncated conical panel 
in thermal environments. European Journal of Mechanics 
/ A Solids, vol. 77, art. ID 103795, DOI:10.1016/j.
euromechsol.2019.103795.
[10] Chan, D.Q., Anh, V.T.T., Duc, N.D. (2019). Vibration and 
nonlinear dynamic response of eccentrically stiffened 
functionally graded composite truncated conical shells 
surrounded by an elastic medium in thermal environments. 
Acta Mechanics, vol. 230, p. 157-178, DOI:10.1007/s00707-
018-2282-4.
[11] Duc, N.D., Anh, V.T.T., Cong, P.H. (2014). Nonlinear 
axisymmetric response of FGM shallow spherical shells on 
elastic foundations under uniform external pressure and 
temperature. European Journal of Mechanics A/Solids, vol. 
45, p. 80-89, DOI:10.1016/j.euromechsol.2013.11.008.
[12] Vuong, P.M., Duc, N.D. (2020). Nonlinear static and dynamic 
stability of functionally graded toroidal shell segments under 
axial compression. Thin-Walled Structures, vol. 155, art. ID 
106973, DOI:10.1016/j.tws.2020.106973.
[13] Wang, Y.A., Sheng, Y.P., Jiang, P.C. (2019). Free vibration 
of graphene reinforced composite truncated conical shell. 
International Journal of Mechanics Research, vol. 8, p. 101-
108, DOI:10.12677/ijm.2019.82012. (in Chinese)
[14] Afshari, H. (2022). Free vibration analysis of GNP-reinforced 
truncated conical shells with different boundary conditions. 
Australian Journal of Mechanical Engineering, vol. 20, no. 5, 
p. 1363-1378, DOI:10.1080/14484846.2020.1797340.
[15] Afshari, H. (2020). Effect of graphene nanoplatelet 
reinforcements on the dynamics of rotating truncated conical 
shells. Journal of the Brazilian Society of Mechanical Sciences 
and Engineering, vol. 42, art. ID 519, DOI:10.1007/s40430-
020-02599-6.
[16] Damercheloo, R.A., Khorshidvand, R.A., Khorsandijou, M.S., 
Jabbari, M. (2021). Free vibrational characteristics of GNP-
reinforced joined conical-conical shells with different boundary 
conditions. Thin-Walled Structures, vol. 169, art. ID 108287, 
DOI:10.1016/j.tws.2021.108287.
[17] Singha, D.T., Rout, M., Bandyopadhyay, T., Karmakar, A. 
(2021). Free vibration of rotating pretwisted FG-GRC sandwich 
conical shells in thermal environment using HSDT. Composite 
Structures, vol. 257, art. ID 113144, DOI:10.1016/j.
compstruct.2020.113144.
[18] Adab, N., Arefi, M., Amabili, M. (2022). A comprehensive 
vibration analysis of rotating truncated sandwich conical 
microshells including porous core and GPL-reinforced face-
sheets. Composite Structures, vol. 279, art. ID 114761, 
DOI:10.1016/j.compstruct.2021.114761.
[19] Adab, N., Arefi, M. (2023). Vibrational behavior of truncated 
conical porous GPL-reinforced sandwich micro/nano-
shells. Engineering with Computers, vol. 39, p. 419-443, 
DOI:10.1007/s00366-021-01580-8.
[20] Yousefi, A.H., Memarzadeh, P., Afshari, H., Hosseini, J.S. 
(2020). Agglomeration effects on free vibration characteristics 
of three-phase CNT/polymer/fiber laminated truncated 
conical shells. Thin-Walled Structures, vol. 157, art. ID 
107077, DOI:10.1016/j.tws.2020.107077.
[21] Yousefi, A.H., Memarzadeh, P., Afshari, H., Hosseini, J.S. 
(2023). Optimization of CNT/polymer/fiber laminated 
truncated conical panels for maximum fundamental frequency 
and minimum cost. Mechanics Based Design of Structures 
and Machines, vol. 51, no. 5, p. 3922-3944, DOI:10.1080/15
397734.2021.1945932.
[22] Afshari, H., Amirabadi, H. (2022). Vibration characteristics of 
rotating truncated conical shells reinforced with agglomerated 
carbon nanotubes. Journal of Vibration and Control, vol. 28, 
no. 15-16, p. 1894-1914, DOI:10.1177/10775463211000499.
[23] Afshari, H., Ariaseresht, Y., Koloor, S.S.R., Amirabadi, H., 
Amirabadi, H., Bidgoli, M.O. (2022). Supersonic flutter 
behavior of a polymeric truncated conical shell reinforced with 
agglomerated CNTs. Waves of Random Complex Media, p. 
1-25, DOI:10.1080/17455030.2022.2082581.
[24] Jamalabadi, M.Y.A., Borji, P., Habibi, M., Pelalak, R. (2021). 
Nonlinear vibration analysis of functionally graded GPL-
GRC conical panels resting on elastic medium. Thin-Walled 
Structures, vol. 160, art. ID 107370, DOI:10.1016/j.
tws.2020.107370.
[25] Ansari, R., Hassani, R., Hasrati, E., Rouhi, H. (2022). Studying 
nonlinear vibrations of composite conical panels with arbitrary-
shaped cutout reinforced with graphene nanoplatelets 
based on higher-order shear deformation theory. Journal 
of Vibration and Control, vol. 28, no. 21-22, p. 3019-3041, 
DOI:10.1177/10775463211024847.
[26] Yang, S.W., Hao, Y.X., Zhang, W., Yang, L., Liu, L.T. (2021). 
Nonlinear vibration of functionally graded graphene platelet 
reinforced composite truncated conical shell using first-
order shear deformation theory. Applied Mathematics 
and Mechanics (English Edition), vol. 42, p. 981-998, 
DOI:10.1007/s10483-021-2747-9.
[27] Wang, A.W., Pang, Y.Q., Zhang, W., Jiang, P.C. (2019). 
Nonlinear dynamic analysis of functionally graded graphene 
reinforced composite truncated conical shells. International 
Journal of Bifurcation and Chaos, vol. 29, no. 11, p. 12244-
12252, DOI:10.1142/S0218127419501487.
[28] Ding, H.X., She, G.L. (2023). Nonlinear primary resonance 
behavior of graphene platelet reinforced metal foams conical 
shells under axial motion. Nonlinear Dynamics, vol. 111, p. 
13723-13752, DOI:10.1007/s11071-023-08564-x.
[29] Bidgoli, E.M.R., Arefi, M. (2023). Nonlinear vibration analysis 
of sandwich plates with honeycomb core and graphene 
nanoplatelet-reinforced face-sheets. Archives of Civil and 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 181-193
192 Huang, X. – Wei, N. – Wang, C. – Zhang, X.
Mechanical Engineering, vol. 23, art. ID. 56, DOI:10.1007/
s43452-022-00589-0.
[30] Wu, H.L., Yang, J., Kitipornchai, S. (2020). Mechanical analysis 
of functionally graded porous structures: A review. International 
Journal of Structural Stability and Dynamics, vol. 20, no. 13, 
art. ID 2041015, DOI:10.1142/S0219455420410151.
[31] Li, H., Hao, Y.X., Zhang, W., Liu, L.T., Yang, S.W., Wang, D.M. 
(2021). Vibration analysis of porous metal foam truncated 
conical shells with general boundary conditions using 
GDQ. Composite Structures, vol. 269, art. ID 114036, 
DOI:10.1016/j.compstruct.2021.114036.
[32] Amirabadi, H., Afshari, H., Afjaei, M.A., Sarafeaz, M. (2022). 
Effect of variable thickness on the aeroelastic stability 
boundaries of truncated conical shells. Waves of Random 
Complex Media, DOI:10.1080/17455030.2022.2157517.
[33] Mohammadrezazadeh, S., Jafari, A.A. (2022). The study of the 
nonlinear vibration of S-S and C-C laminated composite conical 
shells on elastic foundations through an approximate method. 
Proceedings of the Institution of Mechanical Engineers, Part 
C: Journal of Mechanical Engineering Science, vol. 236, no. 3, 
p. 1377-1390, DOI:10.1177/09544062211021439.
[34] Hao, X.Y., Li, H., Zhang, W., Ge, X.S., Yang, W.S., Cao, Y.T. 
(2022). Active vibration control of smart porous conical shell 
with elastic boundary under impact loadings using GDQM 
and IQM. Thin-Walled Structures, vol. 175, art. ID 109232, 
DOI:10.1016/j.tws.2022.109232.
[35] Chan, D.Q., Thanh, N.T., Khoa, N.K., Duc, N.D. (2020). 
Nonlinear dynamic analysis of piezoelectric functionally graded 
porous truncated conical panel in thermal environments. Thin-
Walled Structures, vol. 154, art. ID 106837, DOI:10.1016/j.
tws.2020.106837.
[36] Huang, X.L., Wang, J.H. Wei, N.G., Wang, C.Z., Bin, M. (2023). 
Buckling and vibration of porous sigmoid functionally graded 
conical shells. Journal of Theoretical and Applied Mechanics, 
vol. 61, p. 559-570, DOI:10.15632/jtam-pl/168072.
[37] Kiarasi, F., Babaei, B., Sarvi, P., Asemi, K., Hosseini, M., Bidgol, 
M.O. (2021). A review on functionally graded porous structures 
reinforced by graphene platelets. Journal of Computational 
Applied Mechanics, vol. 52, no. 4, p. 731-750, DOI:10.22059/
jcamech.2021.335739.675.
[38] Bahhadini, R., Saidi, A.R., Arabjamaloei, Z., Chanbari-
Nejad-Parizi, A. (2019). Vibration analysis of functionally 
graded graphene reinforced porous nanocomposite shells. 
International Journal of Applied Mechanics, vol. 11, no. 7, art. 
ID 1950068, DOI:10.1142/S1758825119500686.
[39] Yan, K., Zhang, Y., Cai, H., Tahouneh, V. (2020). Vibrational 
characteristic of FG porous conical shells using Donnell’s shell 
theory. Steel and Composite Structures, vol. 35, no. 2, p. 249-
260, DOI:10.12989/scs.2020.35.2.249.
[40] Rahmani, M., Mohammadi, Y., Kakavand, F. (2019). 
Vibration analysis of sandwich truncated conical shells with 
porous FG face sheets in various thermal surroundings. 
Steel and Composite Structures, vol. 32, no. 2, p. 239-252, 
DOI:10.12989/scs.2019.32.2.239.
[41] Esfahani, M.N., Hashemian, M., Aghadavoudi, F. (2022). 
The vibration study of a sandwich conical shell with a 
saturated FGP core. Scientific Reports, vol. 12, art. ID 4950, 
DOI:10.1038/S41598-022-09043-W.
[42] Cuong-Le T., Nguyen, K.D., Nguyen-Trong, N., Khatir, 
S., Nguyen-Xuan, H., Abdel-Wahab, M. (2021). A three-
dimensional solution for free vibration and buckling of annular 
plate, conical, cylinder and cylindrical shell of FG porous-
cellular materials using IGA. Composite Structures, vol. 259, 
art. ID 113216, DOI:10.1016/j.compstruct.2020.113216.
[43] Sobhani, E., Arbabian, A., Civalek, Ö., Avcar, M. (2022). The 
free vibration analysis of hybrid porous nanocomposite joined 
hemispherical-cylindrical-conical shells. Engineering with 
Computers, vol. 38, p. 3125-3152, DOI:10.1007/s00366-021-
01453-0.
[44] Kiarasi, F., Babaei, M., Mollaei, S., Mohammadi, M., Asemi, K. 
(2021). Free vibration analysis of porous FG joined truncated 
conical-cylindrical shell reinforced by graphene nanoplatelets. 
Advances in Nano Research, vol. 11, no. 4, p. 361-380, 
DOI:10.12989/anr.2021.11.4.361.
[45] Sofieyev, A.H., Kuruoglu, N. (2011). Natural frequency 
of laminated orthotropic shells with different boundary 
conditions and resting on the Pasternak type elastic 
foundation. Composites Part B: Engineering, vol. 42, no. 6, p. 
1562-1570, DOI:10.1016/j.compositesb.2011.04.015.
[46] Fu, T., Wu, X., Xiao, Z.M., Chen, A.B., Li, B. (2021). Analysis 
of vibration characteristics of FGM sandwich joined conical-
conical shells surrounded by elastic foundations. Thin-
Walled Structures, vol. 165, art. ID 107979, DOI:10.1016/j.
tws.2021.107979.
[47] Deniz, A., Zerin, Z., Karaca, Z. (2016). Winkler-Pasternak 
foundation effect on the frequency parameter of FGM 
truncated conical shells in the framework of shear deformation 
theory. Composites Part B: Engineering, vol. 104, p. 57-70, 
DOI:10.1016/j.compositesb.2016.08.006.
[48] Safarpour, M., Rahimi, A.R., Alibeigloo, A. (2020). Static and 
free vibration analysis of graphene nanoplatelets reinforced 
composite truncated conical shell, cylindrical shell, and 
annular plate using theory of elasticity and DQM. Mechanics 
Based Design of Structures and Machines, vol. 48, no. 4, p. 
496-524, DOI:10.1080/15397734.2019.1646137.
[49] Eyvazian, A., Musharavati, F., Tarlochan, F., Pasharavesh, A., 
Rajak, D.K., Husain, M.B., Tran, T.N. (2020). Free vibration 
of FG-GPLRC conical panel on elastic foundation. Structural 
Engineering and Mechanics, vol. 75, no. 1, p. 1-18, 
DOI:10.12989/sem.2020.75.1.001.
[50] Amirabadi, H., Farhatnia, F., Civalek, O. (2021). Frequency 
response of rotating two-directional functionally graded GPL-
reinforced conical shells on elastic foundation. Journal of the 
Brazilian Society of Mechanical Sciences and Engineering, 
vol. 43, art. ID 349, DOI:10.1007/s40430-021-03058-6.
[51] Duc, N.D., Cong, P.H., Tuan, N.D., Tran, P., Thanh, N.V. (2017). 
Thermal and mechanical stability of functionally graded 
carbon nanotubes (FG CNT)-reinforced composite truncated 
conical shells surrounded by the elastic foundations. Thin-
Walled Structures, vol. 115, p. 300-310, DOI:10.1016/j.
tws.2017.02.016.
[52] Huang, X.L., Shen, H.S. (2004). Nonlinear vibration 
and dynamic response of functionally graded plates in 
thermal environments. International Journal of Solids and 
Structures, vol. 41, no. 9-10, p. 2403-2427, DOI:10.1016/j.
ijsolstr.2003.11.012.

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 181-193
193 Nonlinear Free Vibration Analysis of Functionally Graded Porous Conical Shells Reinforced with Graphene Nanoplatelets
7  APPENDIX
Lw Ax
w
x
w
x
AA
x
wA A
x
14 11
2
2
12 22
22
2
12 66
2
()
sins
















 i in sin
2
2
66 2
2
2 11 22
11
2   









 







ww
x
A
x
w
x
w
AA
w
x
 
2
,
 
Lw AA
w
x
w
x
A
x
ww
A
24 12 66
2
22 23
2
2 66
11
  










sins in  
1 11
66
2
2
x
w
x
w
A
w
x
w
sins in
,
 









 
Lu vw BB
w
x
w
x
BB B
x
w
x
34 11 12
2
2 12 22 66 22
32
1
,,
sin
  




  

 
 




















2
2 12 22
2
12
2
2
66
1
2
2
w
AA
w
x
Aw
w
x
B

 cotc ot
  




 




 B
x
w
x
w
BB
x
ww
x
12 2
2
2
2
2 66 12 22
2
1
24
1
sins in   
B B
x
ww
x
BB BA x
x
12 2
3
2
12 22 66 22 3
1
4
1
2
1
sin
cot
si
 





 






n n
cot
sins in
2
2
22 22
2
2 22 34
3
11


 
















w
A
x
w
w
B
x
ww
 
  
3
66 23
3
2 12 22
2
1
2
11
















 B
x
ww
x
A
x
w
x
w
A
sins in
2 2 2
2
11
2
2
2
12
11
2
1
2
1
x
w
x
ww
x
Ax
w
x
w
x
A
x
sin
si
 


















n ns in sin
2
2
2
2
66 22
2
66 2
11
 






















w
x
w
A
x
w
x
w
A
x    




















2
2
2
66 2
2
66 2
4
1
1
w
x
w
A
x
ww
x
w
x
A
x
sin
sin
2 2
2
2
66 23
2
2
66 3
11 ww
x
A
x
w
x
w
A
x 




















   sins in
 

















w
x
ww
A
x
ww
x
A
x
 
2
2 12 2
2
2
2
22 34
1
2
1
1
2
1
sin
sin   























2
2
2
11 11
2
2 12
1 ww
A
w
x
u
x
Ax
w
x
u
x
A
w
sin x x
v
x
A
w
x
u
x
Ax
w
x
u
x
A
w
x
v
A


















2
12 11
2
2
12
2
2 1
1

 sin
2 2
2
2 66 66 2
2
66 2
11 1
u
w
x
A
wu
A
x
wu
x
A
x













sins in sin     
 














w
vA
x
wv
x
A
wv
x
A
x
w
66
66
2
2 66 2
2
1
11
sin
sins in  











x
u
A
x
v
w
x
A
w
x
v
x
A
x    
66
2
66
2
66 2
11 1
sins in sin
 














w
x
u
A
x
w
x
v
A
w
x
v
x
A
x
2
2
66 66
2
66
11 1

  sins in sin
2 2
2
66
2
66
2
66
11
1
   













w
x
u
A
x
w
x
vA
w
x
v
x
A
x
sins in
2 23
2
2 66 22 66 2
2
11
sins in sin   














wu
A
x
wv
A
x
v
x
w
  
 












A
x
wu
x
A
x
wv
A
x
12 2
2
2
22 23
2
2 22 22
1
11
sin
sins in
2 2
2
w
u


         








A
x
wv
A
x
w
u
22 23
2
2 22 22
2
2
11
sins in
.
  
[53] Najafov, A.M., Sofiyev, A.H. (2013). The non-linear dynamics 
of FGM truncated conical shells surrounded by an elastic 
medium. International Journal of Mechanical Sciences, vol. 
66, p. 33-44, DOI:10.1016/j.ijmecsci.2012.10.006.