


Advances in Technology Innovation, vol. 5, no. 2, 2020, pp. 84-97 

Free Vibration Frequency of Thick FGM Circular Cylindrical Shells  

with Simply Homogeneous Equation by Using TSDT 

Chih-Chiang Hong
*
 

Department of Mechanical Engineering, Hsiuping University of Science and Technology, Taichung, Taiwan 

Received 04 June 2019; received in revised form 20 July 2019; accepted 17 September 2019 

DOI: https://doi.org/10.46604/aiti.2020.4380 

Abstract 

The objective of this study is to provide the frequency solutions of free vibration in thick FGM circular 

cylindrical shells by mainly considering both shear correction coefficient and nonlinear coefficient term. This paper 

investigates the effects of third-order shear deformation theory (TSDT) and the varied shear correction coefficient on 

the free vibration of thick functionally graded material (FGM), the circular cylindrical shells with simply 

homogeneous equation under thermal environment. The approach of derivations are given as follows, the varied 

value of shear correction coefficient is included in the simple homogeneous equation. The nonlinear term of 

displacement field of TSDT is also included to derive the simply homogeneous equation, some reasonable 

simplifications in the elements of homogeneous matrix under free vibration of thick FGM circular cylindrical shells 

are assumed, thus, the natural frequency can be found. Three parameters effect on the frequency of thick FGM 

circular cylindrical shells are computed and investigated, they are nonlinear coefficient c1 term, environment 

temperature and power law index. There are some main conclusions obtained, generally the natural frequency results 

are in decreasing value with the mode shape numbers for the thicker circular cylindrical shells. The values of natural 

frequencies are also affected by the nonlinear coefficient term.  

 

Keywords: third-order shear deformation theory, TSDT, FGM, shells, vibration, frequency 

 

1. Introduction 

There are some free vibration frequency investigations with shear deformation effect and experimental studies in the 

functionally graded material (FGM) circular cylindrical shells. In 2019, Shahbaztabar et al. [1] presented the free vibration of 

FGM circular cylindrical shells, including the Pasternak foundation and stationary fluid by using the first order shear 

deformation theory (FSDT). Some effects on the result of the natural frequencies are investigated e.g. fluid depth ratio, elastic 

foundation, volume fraction exponent, geometrical parameters and boundary conditions. In 2019, Zippo et al. [2] used the 

experimental method and the model of finite element method (FEM) to study the linear and dynamic behavior of vibrations in 

the polymeric circular cylindrical shell with FGM equivalent thermal temperature properties. In 2018, Baltacıoğlu and Civalek 

[3] used the Love’s shell theory and FSDT of the displacements to obtain the numerical results for the circular cylindrical FGM 

with carbon nanotube reinforced (CNTR) panels. For the thick FGM shells, it is necessary to consider the nonlinear terms of 

displacement theories to obtain more accurate results of analyses, e.g. third-order shear deformation theory (TSDT), 

higher-order shear deformation theory and triangular function shear deformation theory. In 2018, Torabi and Ansari [4] 

presented a formulation of higher-order isoparametric supplement to study the free vibration of FGM shells, considering the 

structural effects of circular cylindrical, conical, spherical and toroid shells. In 2017, Baltacıoğlu and Civalek [5] used the 

extended FEM to obtain the frequency of the vibration of cracked FGM shells considering the structural effects of cylindrical 

                                                           
*
 
Corresponding author. E-mail address: cchong@mail.hust.edu.tw

 



Advances in Technology Innovation, vol. 5, no. 2, 2020, pp. 84-97 

 

85 

shell, conical shell and spherical on the vibrations. In 2017, Wang and Wu [6] presented the free vibration analysis of porous 

FGM cylindrical shells based on a sinusoidal shear deformation theory (SSDT). In 2014, Fazzolari and Carrera [7] presented 

the hierarchical trigonometric Ritz formulation (HTRF) and used in the free vibration analyses for the doubly curved FGM 

shells and sandwich shells with FGM core. In 2016, Fantuzzi et al. [8] presented free vibration analyses of cylindrical and 

spherical shells by using the FEM and the generalized differential quadrature (GDQ) methods. There are some new and 

improved TSDT used in the investigations of FGMs. In 2016, Bui et al. [9] presented the numerical results of deflection and 

frequency by using the TSDT and FEM for the static bending behaviors of FGM plates. A similar new TSDT in terms of five 

un-known variables was used in the eigenvalue equation to calculate the natural frequency. In 2017, Do et al. [10] presented the 

numerical results of deflection and stress by using the TSDT and FEM for the static buckling and bending behaviors of FGM 

plates. The same new TSDT in terms of five un-known variables was used in the bending equation and pre-buckling equation 

respectively to calculate the numerical solutions without considering the effect of shear correction factors. In 2018, Vu et al. 

[11] presented the numerical results of deflection and frequency by using the TSDT and meshfree method for the static bending, 

free vibration and buckling behaviors of FGM plates. A similar refined TSDT in terms of four un-known variables was used in 

the equations to calculate the numerical solutions.  

 

Fig. 1 Two-material thick FGM circular cylindrical shells 

There are some importance and relevance of studied topics in the dynamics of cylindrical shells composed of FGMs. In 

2012, Zhang et al. [12] presented the nonlinear dynamics of clamped-clamped FGM circular cylindrical shells under an 

external excitation and uniform temperature change. The similar equations were used and based on the FSDT and von-Karman 

nonlinear strains-displacement relation to obtain the numerical response of displacements. In 2016, Dai et al. [13] presented the 

reviews of coupled mechanics on the FGM cylindrical structures during years 2000-2015. Some of the existing mechanical 

theories and hypotheses were assumed and would be improved in the future for obtaining the more accuracy of the results. In 

2008, Ansari and Darvizeh [14] presented a general analytical approach in arbitrary boundary conditions of FGM circular 

cylindrical shells. The FSDT of displacements was used to derive the homogeneous linear system and obtained the natural 

frequency under different boundary conditions. The author has some GDQ computational experiences in the composited FGM 

circular cylindrical shells. In 2017, Hong [15] used the approach of FSDT model and the varied shear correction factor to 

present the numerical GDQ results of thermal vibration and flutter of a supersonic air flowed over thick FGM circular 

cylindrical shells. In 2017, Hong [16] used the approach of Love’s theory for thin multilayered shells to present the numerical 

GDQ results of displacement and stresses of thin FGM laminated magnetostrictive shells with the value effects of velocity 

feedback and control gain subjected to thermal vibration. It is interesting to investigate the natural frequency in the TSDT 

approach of thick FGM circular cylindrical shells under free vibration with simply homogeneous equation and four edges in 

simply supported boundary conditions. The value effects of three parametric: nonlinear coefficient 1C  term, environment 

temperature and power law index on the natural frequency of thick FGM circular cylindrical shells are investigated. The main 

contribution and novelty of paper is to provide and investigate the analytic solutions of natural frequencies in the free vibration 

,   
x,u  

z ,w  

R  

T  z  

R  

*
h  

L  

x  

FGM material 1 
FGM material 2 

1h  

2h  



Advances in Technology Innovation, vol. 5, no. 2, 2020, pp. 84-97 86 

of thick FGM circular cylindrical shells by considering the varied effects of shear correction coefficient and nonlinear terms of 

TSDT. The motivation of the paper is listed as follows. To obtain the natural frequency values of FGM shells considering the 

nonlinear coefficient terms of TSDT within the simply homogeneous matrix equation. Also by considering the calculated 

values of shear correction coefficient usually varied with thickness, power law index and environment temperature. 

2. Formulation 

For a two-material thick FGM circular cylindrical shells in thermal environment with thickness 1h  of FGM material 1 and 

thickness 2h  of FGM material 2. Fig. 1 shows a colorful figure that can illustrate the FGM material for the approach of the 

study. The material properties of power-law function of FGM circular cylindrical shells are considered with a Young’s 

modulus fgmE  of FGM in the standard variation form of power law index nR , the others are assumed in the simple average 

form [17]. The properties iP  of individual constituent material of FGMs are functions of environment temperature T in the 

following form [18], 

1 2 3
0 1 1 2 3( 1 )iP P P T PT P T P T




      (1) 

where 0 1 1 2, , ,P P P P  and 3P  are the temperature coefficients. 

The time dependent of nonlinear displacements u , v  and w  of thick FGM circular cylindrical shells are assumed in the 

nonlinear coefficient 1C  term of TSDT equations [19] as follows, 

3
0 1

3
0 1

( , , ) ( , , ) ( )

( , , ) ( , , ) ( )

( , , )

x x
w

u u x t z x t c z
x

w
v v x t z x t c z

R

w w x t

 

   

 


 




   




   





 
(2) 

where 0u  and 0v  are tangential displacements in the in-surface coordinates x  and   axes direction, respectively, w  is 

transverse displacement in the out of surface coordinates z  axis direction of the middle-plane of circular cylindrical shells, x  

and   are the shear rotations, R  is the middle-surface radius of FGM shell, t  is time. Coefficient for 
*2

1 4 / (3 )c h  is given 

as in TSDT approach, in which *h  is the total thickness of circular cylindrical shells. The linear time dependent of 

displacements also can be obtained by letting 1 0c   in the equations. The nonlinear coefficient 1c  term of displacement fields 

of TSDT [19] is used in the thick FGM circular cylindrical shells to investigate the nonlinear value effect on the natural 

frequency results. For the normal stresses ( x  and  ) and the shear stresses ( ,x z   and xz ) in the thick FGM circular 

cylindrical shells under temperature difference T   for the 𝑘 th layer are in the following equations [20-21], 

11 12 16

12 22 26

( ) ( )16 26 66
( )

44 45

( ) ( )45 55 ( )

 ,

,

x x x

x x xk k
k

z z

xz xzk kk

Q Q Q T

Q Q Q T

TQ Q Q

Q Q

Q Q

  

  

 

  

  

  

 

 

 
   

 
   

    
    
    
 

       
    
        

 

  

 



 
(3) 

where x  and   are the coefficients of thermal expansion, x  is the coefficient of thermal shear, ijQ  is the stiffness of 

FGM circular cylindrical shells. x ,  and x  are in-plane strains, not negligible z  and xz  are shear strains. 



Advances in Technology Innovation, vol. 5, no. 2, 2020, pp. 84-97 

 

87 

The dynamic equations of motion with TSDT for a thick FGM circular cylindrical shells are given as follows [22], 

2 2 2
0

0 1 1 32 2 2

2 2 2
0

0 1 1 32 2 2

2 2 2

1 2 2 2

2 2 2 2
2

0 1 6 1 32 2 2 2 2

1
+ ( )

1
+ ( )

1 2 1
+ ( )

1
( ) [

xx x x

x

x xx x

N N u w
I J c I

x R xt t t

N N v w
I J c I

x R Rt t t

Q Q P P P
c q

x R R xx R

w w w
I c I c I

t t x R



  

  







 

  



     
  

    

     
  

    

    
   

    

    
   

   

2 2
0 0

42 2

2

1 0 2 1 42

2

1 0 2 1 42

1 1
( ) ( )]

1
+ ( )

1 1
+ ( )

x

xx x

xx

x

u v
J

x R x Rt t

M M w
Q J u K c J

x R xt

M M w
Q J v K c J

x R Rt





 


 

 





 

   
  

    

   
   

  

   
   

  

 (4) 

where 

*

*

*

*

*

*

1

1

2

2

2

2

32

2

, ( , , )

hxx x

h

x x

hxx x

h

x x

hxx x

h

x x

z

x xz

M M c P

Q Q c P x

N

N dz

N

M

M zdz

M

P

P z dz

P

Q

Q

  

 

 

 

 

 

 

 

 

  





























 

  

   
   

   
   
   

   
   

   
   
   

   
   

   
   
   

  
  

  







*

*

*

2

2

1
( )

1

, ( 0,1, 2,..., 6)

h

h

N
k

k i
i

k
k

dz

I z dz i











 




 

(5) 

where 
*

N  is the total number of layers, 
 k

  is the density of k th ply.   i i 1 i 2J I c I , i 1,4   ,
2

2 2 1 4 1 6K I 2c I c I    

There are some assumptions for the terms in the strains, e.g. the higher order terms 2( / )w x  , 2[ / ( )]w R    and 

( / )[ / ( )]w x w R      can’t be neglected. The Von Karman type of strain-displacement relations with 0 0/ /v z v R    , 

0 0/ /u z u R     and / / / 0xw z z z           are used as Eq. (6), 

By substituting equations (3) and (6) into equation (4), the dynamic equilibrium differential equations with TSDT of thick 

FGM circular cylindrical shells in terms of partial derivatives of displacements and shear rotations subjected to the expressions 

terms ( 1 5, ...,f f ) in partial derivatives of thermal loads ( , ,N M P ), mechanical loads ( 1 2, ,p p q ) and inertia terms can be 

derived and expressed in matrix forms. By assuming that mid-plane strain terms   
2

1 / 2 w / x  ,    w / x 1 / R w /      

and     
2

1 / 2 1 / R w /     are in constant values, the 1 5, ...,f f  can be expressed in the derivative terms as Eq. (7)-(8), 



Advances in Technology Innovation, vol. 5, no. 2, 2020, pp. 84-97 88 

2 22
30 x x

x 1 2

2 22
30

1 2 2 2

zz

x

u 1 w u w 1 w
z c z

x 2 x x x x x 2 x

1 v 1 1 w 1 v 1 1 1 w 1 1 w
z c z

R 2 R R R R R 2 R

0

1 u v w 1 w

R x x R

 




 


 


      




 

          
         
          

          
         

          



 



   
    

    





 
2 2

3 30 x x 0
1 1

20
z 1

20
xz x 1 x

1 u w v 1 w w 1 w
z c z z c z

R x x x x R x x R

v 1 w v 1 w 1 w
3c z

z R R R R

u w u w
3c z

z x R x

 

  

   

     

  
  

  



               
               

                 

     
      

    

    
    

  






　　

w

x





 

(6) 

1 1

2 2

2 2 2

3 1 2 2 2

4 1

5 1

1
+

1
+

2 1
( )

1 1
+ ( )

1 1
+ ( )

xx x

x

xx x

xx x xx x

x x

N N
f p

x R

N N
f p

x R

P P P
f q c

R xx R

M M P P
f c

x R x R

M M P P
f c

x R x R



 

 

 

   





 

 

 

 
 

 

 
 

 

  
    

  

   
  

   

   
  

   

 
(7) 

and 

*

*

*

*

*

*

32
11 12 16

2

32
12 22 26

2

32
16 26 66

2

( , , ) ( ) (1, , )

( , , ) ( ) (1, , )

( , , ) ( ) (1, , )

( , , , , , )s s s s s s s s s s s s

h

xx xxxx x xh

h

x xh

h

x xx x xh

i j i j i j i j i j i j

N M P Q Q Q T z z dz

N M P Q Q Q T z z dz

N M P Q Q Q T z z dz

A B D E F H

 

   

   

  

  

  







   

   

   








*

*

*

* ** * * * * * * * * * * * *

2 3 4 62

2

2 3 4 5 * *2

2

(1, , , , , ) , ( , 1, 2, 6)

( , , , , , ) (1, , , , , ) , ( , 4, 5)

s s

h

s s
i jh

h

i ji j i j i j i j i j i j h

Q z z z z z dz i j

A B D E F H k Q z z z z z dz i j







 





 (8) 

where 1p  and 2p  are external in-plane distributed forces in x  and   direction respectively. q  is external pressure load. k  

is the shear correction coefficient. 
*

h  is the total thickness of FGMs circular cylindrical shells. The computed and varied 

values of k  are usually functions of total thickness of circular cylindrical shells, FGM power law index and environment 

temperature [23]. The s s
i j

Q  and * *
i j

Q  for thick FGM circular cylindrical shells with /z R  terms cannot be neglected are used in 

the following simple forms in 2014 by Hong [23], in 2010 by Sepiani et al. [24], 



Advances in Technology Innovation, vol. 5, no. 2, 2020, pp. 84-97 

 

89 

2
11 22

2
12 21

44

55 66

16 26 45

/ (1 )

( ) / [(1 / )(1 )]

/ [2(1 )]

/ [2(1 / )(1 )]

0

fgm fgm

fgm fgm fgm

fgm fgm

fgm fgm

Q Q E

Q Q E z R

Q E

Q Q E z R

Q Q Q



 





  

   

 

   

  

 (9) 

where 1 2=( ) / 2fgm    is the Poisson’s ratios of the FGM circular cylindrical shells, 
* *

2 1 1( )[( ) / ]
nR

fgmE E E z h h E     is the 

Young’s modulus of the FGM circular cylindrical shells, nR  is the power-law exponent parameter, 1E  and 2E  are the Young’s 

modulus, 1 and 2  are the Poisson’s ratios of the FGM constituent material 1 and 2, respectively. The simpler stiffness forms of 

s s
i j

Q  and * *
i j

Q  are used to calculate the stresses, , , , , ,s s s s s s s s s s s s
i j i j i j i j i j i j

A B D E F H  and * * * * * * * * * * * *, , , , ,
i j i j i j i j i j i j

A B D E F H . 

For example, by using change of variable in integration calculation, the 11A , 11E , 11F , 11H  and 44H  of thick FGM circular 

cylindrical shells are given as follows, 

*
1 2

11
21 2

* 4
2 1

11
21 2

* 5
1

11 2 1
21 2

* 7

11 2 1
21 2

( )
1

1 ( )
2

( ) ( ) 1 3 3 1
[ ]

4 2( 3) 4( 2) 8( 1)
1 ( )

2

( ) 1 2 1 1 1
{( )[ ] }

5 4 3 2( 2) 16( 1) 80
1 ( )

2

( ) 1 3
{( )[

7
1 ( )

2

n

n

n n n n

n n n n n

n

R E Eh
A

R

h E E
E

R R R R

h E
F E E

R R R R R

h
H E E

R R

 

 

 

 




 



   

    


      
     



  
 



1

* 6
2 1

44
1 2

13 2 13

6 4( 5) 4 16( 3)

3 1
] }

16( 2) 64( 1) 448

( ) ( ) 1 5 2 1 5 1
[ ]

6 2( 5) 4 3 64( 2) 32( 1)
2(1 )

2

n n n n

n n

n n n n n n

R R R

E

R R

k h E E
H

R R R R R R



 

  
   

  
 


     

      


 (10) 

3. Vibration Frequency 

The thick FGM circular cylindrical shells with layers in the stacking sequence (0 / 0 )
 

 are used to study the free 

vibration frequency results with the effects of environment temperature and varied shear correction coefficient calculations,  

under four sides simply supported boundary condition, no thermal loads ( T 0  ), no in-plane distributed forces ( 1p = 2p =0) 

and no external pressure load ( q =0) . The free vibration frequency mn with mode shape numbers 𝑚 and 𝑛 for four sides 

simply supported boundary condition can be derived by simply assuming that 31 1 0I I J   , 0ij ijB E  , 16 26 0A A  , 

16 26 0D D   and 45 45 45 0A D F    under the following time sinusoidal displacement and shear rotations forms with 

amplitudes mna , mnb , mnc , mnd  and mne . 

0

0

cos( / )sin( / )sin( )

sin( / ) cos( / )sin( )

sin( / )sin( / )sin( )

cos( / )sin( / )sin( )

sin( / ) cos( / )sin( )

mn mn

mn mn

mn mn

x mn mn

mn mn

u a m x L n R t

v b m x L n R t

w c m x L n R t

d m x L n R t

e m x L n R t

  

  

  

   

   











 
(11) 



Advances in Technology Innovation, vol. 5, no. 2, 2020, pp. 84-97 90 

where m is the number of axial half-waves, n is the number of circumferential waves. By substituting equations (11) into 

dynamic equilibrium differential equations under free vibration  1 2 5... 0f f f    with the assumed reasonable 

simplifications of 13 14 15 23 24 25 0FH FH FH FH FH FH       and 3 61 4 0I IJ J    in the elements of homogeneous 

matrix, thus the simply homogeneous equation can be obtained as follows. 

11 12

12 22

33 34 35

2
34 44 45

0

2
35 45 55

0

0 0 0

00 0 0

00 0

,0
0 0

0

0
0 0

mn

mnmn

mnmn

mn
mn

mn

mn
mn

FH FH

aFH FH

bFH FH FH

cK
FH FH FH

I d

eK
FH FH FH

I











 
 

   
 

   
     
    

    
    
    
       

 
 

 (12) 

where 

2
0

2 2
11 11 66

12 12 66

2 2
22 66 22

2 2 2 4 2 2 2 2
33 55 44 1 11 1 12 1 66

2 4
1 22 1 55 1 55

( / ) ( / )

( )( / )( / )

( / ) ( / )

( / ) ( / ) ( / ) (2 +4 )( / ) ( / )

        ( / ) 3 (2 -3 )(

mn mnI

FH A m L A n R

FH A A m L n R

FH A m L A n R

FH A m L A n R c H m L c H c H m L n R

c H n R c D c F

 

 

 

 

    





 

 

 

   

 
2 2

1 44 1 44/ ) 3 (2 -3 )( / )m L c D c F n R 

 (13) 

and 

2 3 2 2 2
34 55 1 11 1 11 1 66 1 66 1 12 1 12

2
1 55 1 55

2 3 2 2 2
35 44 1 22 1 22 1 66 1 66 1 12 1 12

2
1 44 1 44

( / ) ( - )( / ) (2 -3 + - )( / )( / )

(6 -9 )( / )

( / ) ( - )( / ) (2 -3 + - )( / ) ( / )

(6 -9 )(

FH A m L c F c H m L c F c H c F c H m L n R

c D c F m L

FH A n R c F c H n R c F c H c F c H m L n R

c D c F n

   



   

  



  



2 3 2 2 2 2
44 11 1 11 1 11 66 1 66 1 66 55 1 55 1 55

2 2
45 12 66 1 12 1 12 1 66 1 66

2 2 2 2
55 66 1 66 1 66 22 1 22 1 22 44

/ )

( -2 + )( / ) ( -2 + )( / ) 6 9

( + -2 + -2 + )( / )( / )

( -2 + )( / ) ( -2 + )( / ) 6

R

FH D c F c H m L D c F c H n R A c D c F

FH D D c F c H c F c H m L n R

FH D c F c H m L D c F c H n R A



 

 

 

    



   
2

1 44 1 449c D c F

 (14) 

The determinant of the coefficient matrix in equation (12) vanishes for obtaining non-trivial solution of amplitudes can be 

represented in the simply five degree polynomial equation as follows, thus the mn  can be found. 

5 4 3 2
(1) (2) (3) (4) (5) (6) 0mn mn mn mn mnA A A A A A           (15) 

where 

11 12

11 22 12 12 11 12

11 22 12 12 11 22

11 22 12 12 11 22

11 22 12 12

(1)

(2) ( )

(3) [( ) ( ) ]

(4) ( ) ( )

(5) [( ) ( ) ]

(6) ( )

A sd

A FH FH sd sc

A FH FH FH FH sd FH FH sc sb

A FH FH FH FH sc FH FH sb sa

A FH FH FH FH sb FH FH sa

A FH FH FH FH sa

 

  

     

    

    

 

 (16) 



Advances in Technology Innovation, vol. 5, no. 2, 2020, pp. 84-97 

 

91 

in which 

2
2 0

33 44 2 0

33 55 44 55 33 44 35 35 34 34 2 0 45 45

33 44 55 44 34 35 35 34 45 35 35 44 34 34 55

45 45 33

( / )

/

( ) /

sd K I

sc FH sd FH K I

sb FH FH FH FH FH FH FH FH FH FH K I FH FH

sa FH FH FH FH FH FH FH FH FH FH FH FH FH FH FH

FH FH FH



 

     

    



 
(17) 

4. Results and Discussion 

Table 1 *f  for SUS304/Si3N4 

*
/L h  nR  

1
c

 
(1/ mm

2
) 

*
f  

Present solution, 
*

1.2h  mm, varied k  

1T K  100T K  300T K  600T K  1000T K  

5 

0.5 
0.925925 

0 

3.178485 

8.567362 

3.434401 

9.391482 

3.873990 

10.984527 

4.111627 

12.617598 

3.662770 

13.538765 

1 
0.925925 

0 

3.318237 

7.958229 

3.570262 

8.647485 

4.005848 

9.985663 

4.253132 

11.39358 

3.862303 

12.597763 

2 
0.925925 

0 

3.477849 

7.132526 

3.723252 

7.690617 

4.151720 

8.763615 

4.409905 

9.831241 

4.098108 

10.888006 

10 
0.925925 

0 

3.756482 

6.452526 

3.984509 

6.885567 

4.394140 

7.695008 

4.670961 

8.305713 

4.532945 

8.563466 

8 

0.5 
0.925925 

0 

2.238668 

6.204801 

2.419486 

6.788758 

2.730914 

7.907682 

2.897884 

9.025515 

2.576800 

9.594029 

1 
0.925925 

0 

2.336488 

5.856612 

2.514541 

6.350593 

2.823131 

7.296222 

2.996852 

8.251528 

2.716518 

8.981841 

2 
0.925925 

0 

2.448283 

5.856612 

2.621661 

6.350593 

2.925246 

6.587167 

3.106589 

7.313977 

2.881696 

7.905719 

10 
0.925925 

0 

2.643672 

5.107512 

2.804862 

5.428395 

3.095301 

6.021389 

3.289678 

6.482052 

3.186499 

6.638852 

10 

0.5 
0.925925 

0 

1.926014 

5.478031 

2.081703 

5.986835 

2.349735 

6.954710 

2.830540 

7.897100 

2.217255 

8.325227 

1 
0.925925 

0 

2.010152 

5.224647 

2.163467 

5.659943 

2.429062 

6.485115 

2.578504 

7.288469 

2.337450 

7.833433 

2 
0.925925 

0 

2.106326 

4.906995 

2.255628 

5.272914 

2.663339 

5.956491 

2.672919 

6.573925 

2.479556 

6.987514 

10 
0.925925 

0 

2.274471 

4.730258 

2.413313 

5.024359 

2.663339 

5.565807 

2.830540 

5.980343 

2.741827 

6.078461 

The composited thick FGM SUS304/Si3N4 material is used to implement the numerical computation of vibration under 

environment temperature T (free stress assumed). The FGM material 1 at inner position of circular cylindrical shells is SUS304 

(stainless steel), the FGM material 2 at outer position of circular cylindrical shells is Si3N4 (silicon nitride) used for the free 

vibration frequency computations with simply homogeneous equation. For the preliminary FGM circular cylindrical shells 

study, it did not considered the effect of nonlinear coefficient term on the calculation of varied shear correction coefficient. The 

varied values of k  are usually functions of
*

h , nR  and T in the thick FGM circular cylindrical shells ( 0ijB  ). For L/R=1, 

1 2h h , 
*

1.2h  mm, calculated values of k  are increasing with nR  (from 0.1 to 10). Thus values of k  are used for 

frequency calculations of the free vibration (no thermal loads under no temperature difference ( T =0) including the effects of 

nonlinear coefficient 
1

c  term. Firstly, for the frequency parameter * 11 2 114 /f R I A  values under the effects of 1c = 

0.925925/mm
2
 and 

1
c = 0/mm2 for */L h = 5, 8 and 10 are shown in Table 1, where 11  is the fundamental first natural 

frequency (m = n = 1). For SUS304/Si3N4 thick circular cylindrical shells under free vibration with
*

1.2h  mm, the *f values 

under 1T K , 100K, 300K, 600K and 1000K with varied k  and 1c  effects are in the values not greater than 13.538765. 



Advances in Technology Innovation, vol. 5, no. 2, 2020, pp. 84-97 92 

The frequency parameter 2 *11 1 1( / ) /L h E   values under the effects of 1c = 0.925925/mm
2
 and 

1
c = 0/mm2 for */L h = 

5, 8 and 10 are shown in Table 2, 1 is the density of FGM material 1, for SUS304/Si3N4 thick circular cylindrical shells under 

free vibration with 
*

1.2h  mm, the Ω values under 1T K , 100K, 300K, 600K and 1000K  with varied k  and 1c   effects are 

in the values not greater than 32.380783. 

It is easy to judge the machining abnormality by the external sensor signal in this experiment. At this time, the internal 

condition data could be combined to determine the type of abnormality as chatter. The accuracy of the monitoring is improved. 

There are a variety of visualization methods in this monitoring system, which could help to quickly locate abnormal positions. 

In addition, it helped to grasp the information of CNC machine tool conditions at all times, which reduced the difficulty of 

subsequent fault analysis and diagnosis. 

Table 2   for SUS304/Si3N4 

*
/L h  nR  

1
c  

(1/ mm
2
) 

  

Present solution, 
*

1.2h  mm , varied k  

1T K  100T K  300T K  600T K  1000T K  

5 

0.5 
0.925925 

0 

5.781036 

15.582337 

6.101545 

16.684873 

6.693336 

18.978658 

7.132169 

21.886919 

7.123131 

26.329359 

1 
0.925925 

0 

5.782522 

13.868399 

6.103089 

14.782213 

6.694982 

16.689058 

7.133930 

19.110862 

7.124928 

23.239543 

2 
0.925925 

0 

5.783707 

11.861481 

6.104272 

12.608768 

6.696166 

14.134532 

7.135265 

15.907033 

7.126575 

18.934150 

10 
0.925925 

0 

5.784393 

9.935876 

6.104699 

10.549432 

6.696107 

11.726208 

7.135397 

12.687871 

7.128233 

13.466383 

*
/L h  nR  

1
c  

(1/ mm
2
) 

  

Present solution, 
*

1.2h  mm , varied k  

1T K  100T K  300T K  600T K  1000T K  

8 

0.5 
0.925925 

0 

6.514712 

18.056488 

6.877522 

19.297414 

7.549395 

21.860160 

8.042832 

25.049551 

8.017926 

29.852609 

1 
0.925925 

0 

6.514685 

16.329633 

6.877464 

17.369365 

7.549288 

19.510704 

8.042762 

22.144931 

8.018013 

26.510597 

2 
0.925925 

0 

6.514442 

16.329633 

6.877144 

17.369365 

7.548845 

16.998745 

8.042382 

18.934526 

8.017992 

21.996770 

10 
0.925925 

0 

6.513346 

12.583626 

6.875763 

13.307022 

7.546949 

14.681323 

8.040542 

15.843253 

8.017431 

16.703767 

10 

0.5 
0.925925 

0 

7.006078 

19.926910 

7.396695 

21.272380 

8.119573 

24.032178 

8.647910 

27.397165 

8.623964 

32.380783 

1 
0.925925 

0 

7.005980 

18.209449 

7.396562 

19.350477 

8.119391 

21.677169 

8.650032 

24.450416 

8.623955 

28.901228 

2 
0.925925 

0 

7.005695 

16.320789 

7.396207 

17.289890 

8.117177 

19.214038 

8.649613 

21.273336 

8.623854 

24.302459 

10 
0.925925 

0 

7.004657 

14.567711 

7.394913 

15.395722 

8.117177 

16.963157 

8.647910 

18.271238 

8.623262 

19.117235 

It is interesting to compare the present vibration values of frequency with some authors' work as shown in the Tables 

(3)-(4). The values of *f  vs.
 

*
h for SUS304/Si3N4 under 

*
/L h =10 and 300T K with varied k  and 1c   

effects are shown in 

Table 3. The compared value *f = 8.426538 at 
*

h = 2mm, nR = 0.5 is greater than 
*

f = 8.0 at n= 13 with silicon nitride-nickel 

under classical shell theory (CST), no external pressure ( 0eK  ) by Sepiani et al. in 2010 [24]. The values of     vs. 
*

h for 

SUS304/Si3N4 under 
*

/L h =10 and  T=700K with varied k  and 1c  
effects are shown in Table 4. The compared value  = 

2.459972 at 
1

c = 0.925925/mm2, * 1.2h  mm, nR = 0.5 is greater than  = 1.71137 with the material variation type A, three 

layers thickness ratio 1-8-1, the L directional radius of curvature is ∞, 
*

/L h =10, nR = 0.5 for the FGM sandwich shell 

presented by Chen et al. in 2017 [25]. 



Advances in Technology Innovation, vol. 5, no. 2, 2020, pp. 84-97 

 

93 

Table 3 Comparison of frequency *f  for SUS304/Si3N4 and silicon nitride-nickel 

1c  (1/mm
2
) *h  (mm) 

*
f  

Present method , 
*

1/ = 0L h , 300T K , 

varied k , for 3 4SUS304 / Si N  

Sepiani et al. 2010, 

for silicon nitride-nickel, 

n 13  

 0.5nR    1nR    2nR   - 

0.925925 1.2 2.349735 2.429062 2.516924 - 

0.333333 2 8.426538 8.711254 9.026930 8.0 

0.000033 200 842669.2 871142.9 902712.1 - 

0.000014 300 253980.0 262560.1 272073.1 - 

0.000003 600 18903.19 19542.01 20250.14 - 

0.000001 900 43930.59 45414.97 47060.53 - 

Table 4 Comparison of frequency   for SUS304/Si3N4 

1c  (1/mm2) 
*

h  (mm) 

  

Present method, * 0/ 1L h  ,  

700T K , varied k  

Chen et al. 2017, type A,  

1-8-1,  0.5nR   

 0.5nR   
 1nR   

 2nR   
- 

0.925925 1.2 2.459972 2.550420 2.651571 1.71137 

0.333333 2 8.821661 9.146189 9.509387 - 

0.000033 200 882174.8 914629.0 950949.6 - 

0.000014 300 267903.0 277757.0 288785.0 - 

0.000003 600 19897.76 20629.82 21448.99 - 

0.000001 900 46307.33 48010.84 49917.17 - 

Table 5 Fundamental natural frequency 11  for 
*

1.2 h mm  

*
/L h  nR  1c  (1/ mm

2
) 

11  

1T K  100T K  300T K  600T K  1000T K  

5 

0.5 
0.925925 

0 

0.001620 

0.004366 

0.001730 

0.004731 

0.001906 

0.005406 

0.001947 

0.005975 

0.001614 

0.005968 

1 
0.925925 

0 

0.001620 

0.003886 

0.001730 

0.004191 

0.001907 

0.004753 

0.001947 

0.005217 

0.001615 

0.005267 

2 
0.925925 

0 

0.001620 

0.003324 

0.001730 

0.003575 

0.001907 

0.004026 

0.001948 

0.004343 

0.001615 

0.004291 

10 
0.925925 

0 

0.001620 

0.002784 

0.001731 

0.002991 

0.001907 

0.003340 

0.001948 

0.003464 

0.001615 

0.003052 

8 

0.5 
0.925925 

0 

0.000713 

0.001976 

0.000761 

0.002137 

0.000840 

0.002432 

0.000857 

0.002671 

0.000709 

0.002643 

1 
0.925925 

0 

0.000713 

0.001787 

0.000761 

0.001924 

0.000840 

0.002170 

0.000857 

0.002361 

0.000709 

0.002347 

2 
0.925925 

0 

0.000713 

0.001787 

0.000761 

0.001924 

0.000839 

0.001891 

0.000857 

0.002019 

0.000709 

0.001947 

10 
0.925925 

0 

0.000712 

0.001377 

0.000761 

0.001474 

0.000839 

0.001633 

0.000857 

0.001689 

0.000709 

0.001479 

10 

0.5 
0.925925 

0 

0.000490 

0.001396 

0.000524 

0.001508 

0.000578 

0.001711 

0.000590 

0.001870 

0.000488 

0.001835 

1 
0.925925 

0 

0.000490 

0.001275 

0.000524 

0.001371 

0.000578 

0.001543 

0.000590 

0.001668 

0.000488 

0.001637 

2 
0.925925 

0 

0.000490 

0.001143 

0.000524 

0.001225 

0.000578 

0.001368 

0.000590 

0.001452 

0.000488 

0.001377 

10 
0.925925 

0 

0.000490 

0.001020 

0.000524 

0.001091 

0.000578 

0.001208 

0.000590 

0.001247 

0.000488 

0.001083 

Secondly, the natural frequency mn  values (unit 1/s) of free vibration ( T 0  ) according to mode shape numbers m and 

n for the SUS304/Si3N4 FGM thick circular cylindrical shells are calculated. For the values of fundamental first (m=n=1) 

natural frequency 11  vs. nR with 
*

1.2h  mm, varied k  and the effects of 1c = 0.925925/mm
2
 and 

1
c = 0/mm2 for */L h = 5, 

8 and 10 are under 1T K , 100K, 300K, 600K and 1000K are shown in Table 5. For the values of natural frequency mn  vs. 

m,n=1,2,…,9 with nR = 0.5, 300T K , 
*

1.2h  mm under varied k  and the effects of 1c = 0.925925/mm
2
 and 

1
c = 0/mm2 for 

*
/L h = 5 and 10 are shown in Table 6. 



Advances in Technology Innovation, vol. 5, no. 2, 2020, pp. 84-97 94 

Table 6 Natural frequency mn  vs. m  and n  under varied k , 1c ,  0.5nR   and 300T K  

1c  

(1/ mm
2
) 

*
/L h  

1n  

1n  2n  3n  4n  5n  6n  7n  8n  9n  

0.925925 
5 

10 

0.001906 

0.000578 

0.001347 

0.000409 

0.001098 

0.000334 

0.000949 

0.000289 

0.000847 

0.000258 

0.000772 

0.000236 

0.000713 

0.000218 

0.000666 

0.000204 

0.000627 

0.000193 

0 
5 

10 

0.005406 

0.001711 

0.005285 

0.001621 

0.005244 

0.001602 

0.005209 

0.001595 

0.005170 

0.001592 

0.005126 

0.001590 

0.005077 

0.001588 

0.005022 

0.001587 

0.004963 

0.001586 

1c  

(1/ mm
2
) 

*
/L h  

2n  

1n  2n  3n  4n  5n  6n  7n  8n  9n  

0.925925 
5 

10 

0.001347 

0.000409 

0.000953 

0.000289 

0.000778 

0.000236 

0.000673 

0.000204 

0.000602 

0.000183 

0.000549 

0.000167 

0.000508 

0.000154 

0.000474 

0.000144 

0.000447 

0.000136 

0 
5 

10 

0.002712 

0.000861 

0.002647 

0.000811 

0.002633 

0.000801 

0.002626 

0.000797 

0.002620 

0.000796 

0.002613 

0.000795 

0.002606 

0.000794 

0.002606 

0.000794 

0.002590 

0.000794 

1c  

(1/ mm
2
) 

*
/L h  

3n  

1n  2n  3n  4n  5n  6n  7n  8n  9n  

0.925925 
5 

10 

0.001098 

0.000334 

0.000778 

0.000236 

0.000635 

0.000192 

0.000550 

0.000167 

0.000492 

0.000149 

0.000449 

0.000136 

0.000415 

0.000126 

0.000388 

0.000118 

0.000366 

0.000111 

0 
5 

10 

0.001817 

0.000578 

0.001766 

0.000541 

0.001757 

0.000534 

0.001753 

0.000532 

0.001751 

0.000531 

0.001748 

0.000530 

0.001746 

0.000530 

0.001744 

0.000529 

0.001741 

0.000529 

1c  

(1/ mm
2
) 

*
/L h  

4n  

1n  2n  3n  4n  5n  6n  7n  8n  9n  

0.925925 
5 

10 

0.000949 

0.000289 

0.000673 

0.000204 

0.000550 

0.000167 

0.000476 

0.000144 

0.000426 

0.000129 

0.000389 

0.000118 

0.000360 

0.000109 

0.000337 

0.000102 

0.000317 

0.000096 

0 
5 

10 

0.001373 

0.000438 

0.001326 

0.000407 

0.001318 

0.000401 

0.001316 

0.000399 

0.001314 

0.000398 

0.001313 

0.000397 

0.001312 

0.000397 

0.001311 

0.000397 

0.001309 

0.000397 

1c  

(1/ mm
2
) 

*
/L h  

5n  

1n  2n  3n  4n  5n  6n  7n  8n  9n  

0.925925 
5 

10 

0.000847 

0.000258 

0.000602 

0.000183 

0.000492 

0.000149 

0.000426 

0.000129 

0.000381 

0.000115 

0.000348 

0.000105 

0.000322 

0.000098 

0.000301 

0.000091 

0.000284 

0.000087 

0 
5 

10 

0.001111 

0.000355 

0.001062 

0.000326 

0.001055 

0.000321 

0.001053 

0.000319 

0.001052 

0.000318 

0.001051 

0.000318 

0.001050 

0.000318 

0.001050 

0.000317 

0.001049 

0.000317 

1c  

(1/ mm
2
) 

*
/L h  

6n


 

1n  2n  3n  4n  5n  6n  7n  8n  9n  

0.925925 
5 

10 

0.000772 

0.000215 

0.000549 

0.000167 

0.000449 

0.000136 

0.000389 

0.000118 

0.000348 

0.000105 

0.000317 

0.000096 

0.000294 

0.000089 

0.000275 

0.000083 

0.000259 

0.000079 

0 
5 

10 

0.000941 

0.000300 

0.000886 

0.000272 

0.000880 

0.000267 

0.000878 

0.000266 

0.000877 

0.000265 

0.000876 

0.000265 

0.000876 

0.000265 

0.000875 

0.000265 

0.000875 

0.000265 

1c  

(1/ mm
2
) 

*
/L h  

7 n


 

1n  2n  3n  4n  5n  6n  7n  8n  9n  

0.925925 
5 

10 

0.000713 

0.000218 

0.000508 

0.000154 

0.000415 

0.000126 

0.000360 

0.000109 

0.000322 

0.000098 

0.000294 

0.000089 

0.000272 

0.000082 

0.000254 

0.000846 

0.000240 

0.000073 

0 
5 

10 

0.000826 

0.000263 

0.000761 

0.000234 

0.000754 

0.000229 

0.000752 

0.000228 

0.000751 

0.000227 

0.000751 

0.000227 

0.000751 

0.000227 

0.000750 

0.000227 

0.000750 

0.000227 

1c  

(1/ mm
2
) 

*
/L h  

8n


 

1n  2n  3n  4n  5n  6n  7n  8n  9n  

0.925925 
5 

10 

0.000666 

0.000204 

0.000474 

0.000144 

0.000388 

0.000118 

0.000337 

0.000102 

0.000301 

0.000091 

0.000275 

0.000083 

0.000254 

0.000078 

0.000238 

0.000072 

0.000224 

0.000069 

0 
5 

10 

0.000744 

0.000236 

0.000668 

0.000205 

0.000661 

0.000201 

0.000658 

0.000199 

0.000658 

0.000199 

0.000657 

0.000199 

0.000657 

0.000198 

0.000657 

0.000198 

0.000656 

0.000198 

1c  

(1/ mm
2
) 

*
/L h  

9n


 

1n  2n  3n  4n  5n  6n  7n  8n  9n  

0.925925 
5 

10 

0.000627 

0.000193 

0.000447 

0.000136 

0.000366 

0.000111 

0.000317 

0.000096 

0.000284 

0.000087 

0.000259 

0.000079 

0.000240 

0.000073 

0.000224 

0.000069 

0.000212 

0.000064 

0 
5 

10 

0.000684 

0.000215 

0.000596 

0.000183 

0.000588 

0.000179 

0.000586 

0.000177 

0.000585 

0.000177 

0.000584 

0.000176 

0.000584 

0.000176 

0.000584 

0.000176 

0.000584 

0.000176 



Advances in Technology Innovation, vol. 5, no. 2, 2020, pp. 84-97 

 

95 

  

Fig. 2 1n  vs. nR  for 
*
=/ 5L h  Fig. 3 1n  vs. nR  for 

*
1/ = 0L h  

  

Fig. 4 1n  vs. T for 
*
=/ 5L h  Fig. 5 1n  vs. nR  for 

*
1/ = 0L h  

Finally, the natural frequency mn  values (unit 1/s) vs. nR and T of free vibration ( T =0) according to mode shape 

numbers m=1 and n (from 1 to 9) for the SUS304/Si3N4 FGM thick circular cylindrical shells are calculated. Figs. (2)-(3) show 

the values of 1n  vs. nR in FGM circular cylindrical shells for thick 
*

/L h  = 5, 10 respectively, with the effects of varied k    

and
1

c  = 0.925925/mm2
 
under 300T K . Generally the values of  1n are decreasing with values of n (from 1 to 9) for 

*
/L h = 

5, nR = 0.5, 1 and 10. The greatest value of 11 = 0.00191 (unit 1/s) is found for 
*

/L h = 5. The values of  1n  are also 

decreasing with values of n (from 1 to 9) for 
*

/L h = 10, nR = 0.5, 1 and 10. Figs. 4-5 show the values of 1n  vs. T in FGM 

circular cylindrical shells for thick 
*

/L h =5, 10 respectively, under the effects of varied k , 1c = 0.925925/mm
2
 and nR = 0.5. 

Generally the values of  1n are decreasing with values of n (from 1 to 9) for 
*

/L h = 5, 300T K , 600K and 1000K, the values 

of 1n are almost in the same for 300T K  and 600K, but in greater values than that in the  1000T K . The greatest value of 

11 = 0.00191 (unit 1/s) is found for 
*

/L h = 5, 600T K . The values of 1n can stand for higher temperature  1000T K at 

*
/L h = 5. The values of  1n are decreasing with values of n (from 1 to 9) for 

*
/L h = 10, 300T K , 600K and 1000K , the 

values of 1n  are almost in the same for 300T K and 600K, but in greater values than that in the 1000T K . The greatest 

value of 11 = 0.00059 (unit 1/s) is found for
*

/L h  = 10, 600T K . The values of 1n  can stand for higher temperature 

1000T K at 
*

/L h = 10. The values of 1n  at 
*

/L h = 5 are also found in the greater values than that at 
*

/L h = 10. 

5. Conclusions 

The values of natural frequency and frequency parameters are calculated and obtained by using the simply homogeneous 

equation with the polynomial equation in fifth-order of 
mn
  in the free vibration of thick FGM circular cylindrical shells. The 



Advances in Technology Innovation, vol. 5, no. 2, 2020, pp. 84-97 96 

three items of value effects are considered in nonlinear coefficient term
1

c , shear correction coefficient and environment 

temperature. Some of the important results are found as follows. (a) Data investigated in the three kinds of frequency 

parameters under free vibration with and without the effects of 
1

c . (b) Generally the values of 1n  are decreasing with values 

of n (from 1 to 9) for
*

/L h =5 and 10, nR = 0.5, 1 and 10. (c) The values of 1n  can stand for higher environment temperature 

1000T K at
*

/L h = 10. (d) The values of 1n   vs. environment temperature T at 
*

/L h =5 are found in the greater values than 

that at 
*

/L h = 10. 

Conflicts of Interest 

The authors declare no conflict of interest. 

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