FREE VIBRATION OF ANGLE-PLY CONICAL SHELLS WITH
LINEAR THICKNESS VARIATION
K.K.Viswanathan*, A.K. Nor Hafizah, Saira Javed and Z. A. Aziz
UTM Centre for Industrial and Applied Mathematics (UTM-CIAM), Department of Mathematical
Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor, Malaysia
*Corresponding author e-mail: [email protected]
This paper aims to study the effect of arbitrarily linearly varying thickness in the axial direction and edge conditions on the vibration characteristics of shear deformable conical shells.
The application of spline function technique for predicting the free vibration of composite
laminated conical shells is also investigated. The solutions of displacement functions are assumed in a separable form to obtain a system of coupled differential equations in terms displacement and rotational functions and these displacements and rotational functions are approximated by cubic spline. A generalized eigenvalue problem is obtained and solved numerically for an eigenfrequency parameter and an associated eigenvector of spline coefficients. The effect of cone angle, aspect ratio, thickness variation, number of lay ups, stacking
sequence, materials used on the stability of conical shells are also discussed. New results
presented may be fruitful for the related fields.
1.
Introduction
To build stable and sturdy construction in engineering industry it is necessary to study the vibration analysis of the structures. This means that the structures should be designed by considering the
factors such as frequency parameter, material and its orientation in order to construct highly reinforced structures. Thin shells play important role as structural elements in industries because of
their great range of desirable properties such as high degree of reserved strength and structural integrity. Large-span roofs, water tanks, aircraft and submarine are the examples of shell structures
that can be found in engineering industry. Shells structures made up of composite materials have
been used significantly since they have high specific stiffness, better damping and shock absorbing
characteristics.
Different researchers have conducted research on the vibration of shells. [1] studied the free vibration of angle-ply laminated truncated conical shells using a spline method. A study on free vibration analyses of multiple delaminated angle-ply composite conical shells using finite element approach has been done by [2] using QR iteration algorithm. In order to study the free vibration of
simply supported circular cylindrical shells, a semi-analytical procedure was introduced by [3] for
simply supported boundary conditions in the study. A modified fourier series had been developed
by [4] to analyse the vibration analysis of truncated conical shells with general boundary conditions
and the effects of elastic restraint parameters, semi-vertex angle and the ratio of length to radius
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were considered in solving the problem. [5] performed a study on the dynamic behavior of functionally graded (FG) truncated conical shells subjected to asymmetric internal ring-shaped moving
loads. He used FEM together with Newmark's time integration scheme to discretize the equations of
motion in the spatial and temporal domain, respectively to solve the problem. Recently, [6] used
Fourier–Ritz method to solve the problem for free and forced vibration analysis of coupled conical–
cylindrical shells with arbitrary boundary conditions. The results from their investigation on vibration of the joined conical–cylindrical–conical shell showed the extensive applicability of the method
for more complex shell combinations. Free vibration analysis of fiber reinforced composite (FRC)
conical shells resting on Pasternak-type elastic foundation was investigated by [7] using Galerkin
and Ritz methods.
In this study, spline method is used to solve free vibration of antisymmetric angle-ply conical
shells having linear variation in thickness under shear deformation theory. The governing differential equations are derived and approximated by cubic spline to get a set of field equations. The field
equations along with the equations of boundary conditions reduced into a system of homogenous
simultaneous algebraic equations on the assumed spline coefficients. Two boundary conditions
clamped-clamped and simply supported are considered. The frequencies are obtained using eigensolution techniques since the problem is considered as eigenvalue problem with spline coefficients as
eigenvectors. The thickness variation, cone angle, aspect ratio, circumferential node number and
boundary conditions are taken into consideration in calculating the frequency parameters. The findings are presented and discussed in terms of graphs and tables.
2.
Theoretical formulation
Consider a composite laminated truncated conical shell with an arbitrary number of layers that
supposed to be perfectly bonded together. The coordinate system and geometric parameters of the
shell are demonstrated in [1]. From the figure, ra and rb are the radii of the cone at its small and
large end,  is the semi-vertical angle and  is the length of the cone along its generator. The orthogonal coordinate system ( x,  , z ) is fixed at its reference surface, which is assumed to be at the
middle surface. The radius of the cone at any point along its length is r  x sin  . The radius at the
small end of the cone is ra  a sin  and the other end is rb  b sin  .
The displacement components are considered as
(1) u ( x,  , z, t )  u0 ( x,  , t )  z x ( x,  , t ), v( x, , z , t )  v0 ( x, , t )  z  ( x, , t ), w( x,  , z , t )  w0 ( x, , t )
where u0 , v0 and w0 are the displacements of the shell in the mid-plane, and  x and   are the
shear rotations of any point on the middle surface of the shell. The strain-displacement relations of
conical shell are expressed as
u

1
1 v0
1
1  0 
1
,
 x  0  z x ,    u0 

w z x 
x
x
x
x sin   x tan 
x sin   
x
1 u0 v0 1
 1  x   1 

 v0  z 

  ,
 x 
x sin   x x
x
x 
 x sin  
(2)
 xz   x 
w
and   z     1 w  1 v0 .
x
x sin   x tan 
The resultant stress and moments are given by
(3) ( N x , N , N x , Qx , Q )   ( x ,   , x , xz ,  z )dz and ( M x , M  , M x )   ( x ,   , x ) zdz
z
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Substituting Eq. (1) into Eq. (2) and then using stress – strain relations, the following equations
are obtained as :
 N x   A11 A12 A16 B11 B12 B16  
u0 / x


 



 N   A21 A22 A26 B12 B22 B26  (1/ x)u0  (1/ x sin  )v0 /   (1/ x tan  ) w
 N x   A16 A26 A66 B16 B26 B66   (1/ x sin  )u0 /   v0 / x  (1/ x)v0 

(4) 


 x / x
 M x   B11 B12 B16 D11 D12 D16  

M  B



B
B
D
D
D



(1/
x
)

(1/
x
sin

)

/


22
26
12
22
26
x
    12


  (1/ x sin  ) /    / x  (1/ x) 
M 
B
B
B
D
D
D

 
16
26
66
16
26
66
x


 x 
and
w


 1/ x tan  v0 
 Q 
 A44 A45     1/ x sin 
(5)

 K


 Qx 
 A45 A55  
 x  w / x


The coefficients A16 , A26 , A45 , B11 , B12 , B22 , B66 , D16 and D26 are identically zero for antisymmetric
angle-ply laminates [8].
The thickness of the k -th layer is assumed 0in the 0form hk ( x )  h0 k g ( x ) , where h 0 k is a constant
thickness and g ( x)  1  C x , where C is the thickness coefficients of linear variation.
1
Assume C   1 , where  is the taper ratio hk (0) / hk (1) .

Since the thickness is assumed to be varying along the axial direction, we define the elastic coefficients Aij , Bij and Dij (extensional, bending-extensional coupling and bending stiffnesses) corresponding to layers of uniform thickness with superscript ' c ' as Aij  Aijc g ( x) ,
Bij  Bijc g ( x) ,
Dij  Dijc g ( x) , where
1
1
Qij( k ) ( zk2  zk21 ) and Dijc   Qij( k ) ( zk3  zk31 )

2 k
3 k
k
c
(k )
and Aij  K  Qij ( zk  zk 1 ) for i, j  4, 5 ,
Aijc   Qij( k ) ( zk  zk 1 ) , Bijc 
(6)
for i, j  1, 2, 6 ,
k
In which K is the shear correction factor meanwhile zk-1 and zk are boundaries of k-th layer. The value for the shear correction factor K is chosen from the lamination scheme. The procedure for finding the values of shear correction factors have been explored by [9].
Plugging Eqs. (4) and (5) into the equilibrium equations (2) one can obtain a set second order
partial differential equations in terms of displacement functions u0 , v0 , w and rotational function
 x ,  .
The displacement components u0 , v0 , w and shear rotations  x ,   are written in the form of
u0 ( x, , t )  U ( x)e n eit , v0 ( x, , t )  V ( x)e n eit ,
w ( x,  , t )  W ( x)e n eit ,
 x ( x,  , t )   x ( x)en eit ,   ( x, , t )   ( x)en eit
(7)
where  is the angular frequency of vibration, t is the time, and n is the circumferential node
number. The non-dimensional parameters are written as follows:
I
xa
X
, a  x  b and X  [0,1],    1 , a frequency parameter,
A11

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h
h
,  '  , ratios of thickness to radius and to a length,
a
ra
a a length ratio,
h
 ,
 k  k , relative layer thickness of the k-th layer.
b
h

(8)
To obtain the differential equations depending on a single variable x , Eq. (7) is substituted into
the governing equations and then the non-dimensional parameters are applied to the equation resulting a new matrix equation
 L11
 
 L 21
 L31
 
 L 41
 
 L 51
(9)
L12
L13
L14
L22
L23
L24
L32
L42
L33
L43
L34
L44
L52
L53
L54
L15   U  0 
   
L25   V  0 
L35   W   0  ,
   
L45    x  0 
L55     0 
where where L*ij are differential operators in x .
3.
Solution procedure
3.1 Spline collocation method
In this paper, numerical approach which is spline collocation method is used to solve the problem since it has a chain of lower order approximations that has the ability to provide better accuracy
than a global higher order approximations. The study about spline collocation method over a two
point boundary value problem with cubic splines has been successfully carried out by [10].
The displacement and rotational functions U  X  , V  X  , W  X   X  X  ,   X  are approx-
imated by the cubic spline functions as
cubic spline functions in the range of X [0,1] as
U(X ) 
i 0
W (X ) 
i
2
e
i0
(10)
N 1
2
a X
i
i
  bj ( X  X j ) 3 H ( X  X j ) ,
V (X ) 
i0
j 0
Xi 
N 1
f
j0
j
(X  X j )3 H (X  X j ),
 (X ) 
2

i0
li X i 
2

N 1

j0
x (X ) 
ci X i 
2

i 0
N 1

j0
gi X i 
dj (X  X j )3 H (X  X j )
N 1

j0
pj ( X  X j ) 3 H ( X  X j )
qj ( X  X j ) 3 H ( X  X j )
where H  X  X j  is the Heavy side step function and N is the number of sub-intervals within
the range [0,1] of X divided. The collocation points are the knots of the splines at X = Xs = s / N ,
where s = 0, 1, . . . , N. Considering the condition that the differential equations given by Eq. (9) are
satisfied by these splines at the knots, a set 5N + 5 homogenous equations in 5N + 15 unknown
spline coefficients, ai , ci , ei , gi , li , b j , d j , f j , p j , q j  i  0,1, 2; j  0,1, 2,...N  1 , are secured.
3.2 Boundary conditions
Two boundary conditions are used to analyse the problem.
(i) Clamped-Clamped (C-C) (both the ends are clamped)
(ii) Simply-supported (S-S) (both ends are simply supported)
The boundary condition gives 10 equations on spline coefficients. Gathering them with those
obtained earlier, we get 5N + 15 homogenous equations, with the same number unknown. Hence,
the system of equations can be written in the form
(11)
 P  q    Q  q  ,
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here [P] and [Q] are square matrices, [q] is a column matrix, and   1/  2 , where  is the frequency parameter.  or  2 is the eigenparameter and [q] is the eigenvector for this generalized
eigenvalue problem.
4.
Result and discussion
For this work, convergence study for the frequency parameter  has been conducted to choose
the number of subintervals N of the spline function. The program is performed for N (number of
knots) = 2 onwards and finally it is seen that N =18 would be enough to achieve the change in percentage. The materials Graphite Epoxy (AS4/3501-6) (AGE) and Kevler-49 Epoxy (KGE) are used
which are oriented in order of KGE-KGE and AGE-KGE-KGE-AGE for two and four layers of
antisymmetric angle-ply shells having ply-angles as 300/-300, 450/-450, 600/-600 , 300/-300/300/-300,
450/-450/450/-450 and 600/-600/600/-600. The C-C and S-S boundary conditions are considered.
(a)
  1.75
  0.5
 '  0.5
n 1
(b)   0.5   1.75
C-C
KGE/KGE
C-C
 '  0.5 n  1 AGE/KGE/KGE/AGE




Figure 1. Effect of cone angle on fundamental frequency parameter of two-and four-layered
conical shells under C-C boundary condition.
Figure 1 indicates the effect of cone angle  on the fundamental frequency parameter  of twoand four layered antisymmetric angle-ply shells under C-C boundary condition. The horizontal axis
represents cone angle ranges from 100 <  <800 and the vertical axis represents the fundamental
frequency parameter. The values for length ratio   0.5 , ratio of thickness to a length  '  0.5 ,
circumferential node number n=1 and taper ratio   1.75 are fixed. The two-layered and fourlayered antisymmetric angle-ply shells have material combination of KGE-KGE and AGE-KGEKGE-AGE respectively. It can be seen from the graphs that the value of fundamental frequency
parameter for both graphs decrease gradually as cone angle increases. However, the values fundamental frequency parameter for four-layered shells are higher than for two-layered shells.
In Fig. 2, the variation of fundamental angular frequency values  with respect to length ratio
are presented for two- and four-layered shells under C-C boundary condition. The values for cone
angle   60 , ratio of thickness to radius   0.05 , n=1 and   1.75 are fixed. The angular frequency for four-layered shells is higher than two-layered shells. Nevertheless, as is illustrated by
both graphs for different number of layers of shells, the angular frequency steadily increases from
0.1 to 0.5 and significantly increases from 0.6 to 0.8. The length ratio 0.1 until 0.6 in Fig. 2(a), the
lamination angle for 600/-600 has the highest angular frequency when compared with the other two
lamination angles but changes into the lowest from 0.6 until 0.8. Lamination angle for 600/600/600/-600 in Fig. 2(b) also shows same pattern since it has the highest frequency from point 0.1
until 0.5 but switches into the lowest from point 0.5 until 0.8.
Figure 3 depicts the effect of cone angle on the fundamental frequency parameter of two-layered
shells. Circumferential node number n=1 and n=2 have been chosen to compare the changes in the
value of fundamental frequency parameter under S-S boundary condition and the material, lamination angle,  ,  ' , and  are fixed for both graphs as shown in Fig. 3.
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(a)
  60
  0.05
(b)
C-C
n 1
  1.75
  60
C-C
  0.05
n 1
  1.75
AGE/KGE/KGE/AGE
KGE/KGE


Figure 2. Effect of length ratio on angular frequency of two- and four-layered conical shells under C-C boundary condition.
(a)

  0.5
 '  0.5
n 1
  1.75
(b)   0.5
S-S
 '  0.5
KGE/KGE
n2
  1.75
S-S
KGE/KGE



Figure 3. Effect of cone angle on the fundamental frequency of two-layered conical shells under
S-S boundary condition. (a) n=1 (b) n=2.
In Fig. 3(a), there is a rapid drop in the value of the fundamental frequency parameter from
  10 until   20 . It then remains stable from   20 onwards. Meanwhile for Fig. 3(b), the
0
0
0
difference in the frequencies for lamination angle 300/30 , 450/45 and 600/60 are slight higher
from   10 until   50 as compare to Fig. 3(a).
(a)
  60
  0.05
S-S
KGE/KGE
n 1
  1.75

(b)   60
S-S
KGE/KGE
  0.05
n2
  1.75

Figure 4. Effect of length ratio on the fundamental frequency of two-layered conical shells under
S-S boundary condition. (a) n=1 (b) n=2.
Figure 4 shows the effect of length ratio on the fundamental angular frequency of two-layered
shells under S-S boundary condition for different circumferential node number n=1 and n=2. The
ply angles considered are 300/-300, 450/-450 and 600/-600 and each layer consists of KGE material.
Referring to the figure, it can clearly be seen that the frequency increases with the increase of length
ratio for both graphs but the values of the fundamental frequency parameter of circumferential node
number n=2 are lower when compared to the corresponding values of circumferential node number
n=1.
The manner of variation fundamental frequency with respect to the cone angle of four-layered
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shells under S-S boundary condition for three different lamination angles having material combinations as AGE-KGE-KGE-AGE is demonstrated in Fig. 5. The fixed parameters are n=2,
  0.5 ,  '  0.5 and   1.75 . In general, frequency values for all lamination angles considered
decreases within the range of cone angle, 100    800 . Moreover, from cone angle   10 until   300 , the frequencies for all lamination angles significantly falls but   300 onwards, the
decrease in frequencies is at a much slower pace as compare to the previous one.
  0.5
 '  0.5
S-S
AGE/KGE/KGE/AGE
n2

  1.75

Figure 5. Effect of cone angle on fundamental frequency parameter of four-layered shells under
S-S boundary condition.
Table 1 and 2 represent the influence of taper ratio on the fundamental frequency parameter for
two- and four-layered antisymmetric angle-ply shells under C-C and S-S boundary condition having
range 0.5    1.9 . The parameters such as n=1,   0.5, and   0.05 are fixed. As is shown in
Table 1 and 2, higher lamination angles result in higher frequencies of two- and four-layered shells.
It also shows, the frequencies for two-layered shells are higher than four-layered shells under C-C
and S-S boundary condition. Moreover, comparing Table 1 and 2 it can be seen that the values of
the fundamental frequency parameter is lower for S-S boundary condition as compare to C-C
boundary condition.
Table 1. Effect of taper ratio on fundamental frequency parameter for two- and four-layered antisymmetric angle-ply shells under C-C boundary condition.


(C-C)
0.5
0.7
0.9
1.1
1.3
1.5
1.7
1.9
300/-300
1.481631
1.485826
1.487381
1.488405
1.488968
1.489232
1.485978
1.489506
450/-450
1.754512
1.761806
1.764158
1.765919
1.766084
1.765937
1.765549
1.764929
600/-600
2.367439
2.377206
2.378084
2.381379
2.382309
2.382683
2.382774
2.381051
300/-300/300/-300
0.659238
0.661136
0.66158
0.661353
0.660755
0.660148
0.659043
0.658064
450/-450/450/-450
1.370617
1.372585
1.373581
1.374065
1.373026
1.373158
1.37097
1.370527
600/-600/600/-600
1.731168
1.730132
1.734706
1.733884
1.733466
1.732866
1.733106
1.731116
Table 2. Effect of taper ratio on fundamental frequency parameter for two- and four-layered antisymmetric angle-ply shells under S-S boundary condition.


(S-S)
0.5
0.7
0.9
1.1
1.3
1.5
1.7
1.9
300/-300
1.300378
1.275566
1.204745
1.146880
1.102348
1.056352
1.020276
0.988152
450/-450
1.372134
1.401198
1.300939
1.211080
1.193662
1.049757
1.049568
1.012192
600/-600
1.633520
1.500187
1.486243
1.458394
1.486897
1.481769
1.468535
1.449139
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300/-300/300/-300
0.436090
0.356914
0.294749
0.239799
0.186436
0.128028
0.033451
0.108888
450/-450/450/-450
0.814206
0.611850
0.458083
0.313514
0.125500
0.220602
0.317456
0.379552
600/-600/600/-600
0.946008
1.072824
0.706600
0.667698
0.473133
0.473133
0.194662
0.194663
7
The 22nd International Congress on Sound and Vibration
5.
Conclusion
This paper aimed to analyse the effect of linear thickness in variation, cone angle, length ratio,
circumferential node number, different lamination materials, ply-angle and two different boundary
condition on the values of the frequency parameter. The values of natural frequencies of the layered
conical shells with different material properties are being different with those of homogeneous
shells of any one of the layered materials. It is concluded from the results that the value of the frequency parameter strictly decreases for certain value of cone angle and become steady afterwards
with the increase of cone angle. Moreover, the frequency parameter value remain steady for certain
value of length ratio and strictly increases afterwards as the length ratio increases. Further, S-S
boundary condition results in lower value of frequency parameter as compare to C-C boundary condition.The results presented in this paper may be fruitful for designers of the relate fields for designing the conical shell structure according to their needs.
Acknowledgment
This work was supported by MOHE, UTM Flagship Project Vote No. 01G40 and FRGS Project
Vote No. R.J130000.7809.4F249, under Research Management Centre (RMC), Universiti Teknologi Malaysia, Johor Bahru, Malaysia.
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ICSV22, Florence, Italy, 12-16 July 2015
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