1. No category

International Conference on Mechanical And Industrial Engineering (ICMAIE’2015) Feb. 8-9, 2015 Kuala Lumpur (Malaysia)
Optimization of a Two Joint Cross-Ply Laminated
Conical Shells to Minimize the Natural Frequencies
F. Bakhtiari-Nejad, and E. Alavi
Abstract -This study deals with the vibrational behavior of two
joined cross-ply laminated conical shells. Natural frequencies and
mode shapes are investigated. Joined conical shells can be considered
as a general case for joined cylindrical-conical shells, joined cylinderplates or cone-plates, cylindrical and conical shells with stepped
thicknesses and also annular plates. Governing equations are obtained
using thin-walled shallow shell theory of Donnell type and Hamilton’s
principle. The equations are solved assuming trigonometric response
in circumferential directions and series solution in meridional
directions. All combinations of boundary conditions can be assumed
in this method. The effects of semi-vertex angle, meridional length
and shell thickness on the natural frequencies and circumferential
wave number of joined shells are investigated. The finite element
analysis is conducted to predict the natural frequencies of isotropic
and composite specimens.
Index Terms— Optimization; Two joint cross play; conical; shell;
Vibration behavior.
I. INTRODUCTION
The joined shells of revolution have many applications in
various branches of engineering such as mechanical,
aeronautical, marine, civil, and power engineering. The
research on their mechanical behavior such as vibration
characteristics under various external excitations and boundary
restrictions has great importance in engineering practice.
Although the results of many investigations on the
vibration analysis of rotating and non-rotating conical and
cylindrical shells are available [1], a few publications exist on
the vibration analysis of joined conical-cylindrical shells. A
numerical and experimental work was performed by Lashkari
and Weingarten [3]. They employed finite element method to
determine the natural frequencies and mode shapes of joined
conical–cylindrical shells. Irie et al. [4] used the transfer matrix
approach to solve the free vibration of joined isotropic
cylindrical–conical shells. Efraim and Eisenberger [5] applied
a power series solution to calculate the natural frequencies of
segmented axisymmetric shells using the theory of Reissner.
Patel et al. [6] presented results for laminated composite joined
conical–cylindrical shells with first order shear deformation
theory (FSDT) using finite element method (FEM). The free
vibration of joined complete cone-cylinder was also
investigated using FEM by Ozakca and Hinton [7] with a 305
DOF cubic four-nodded C0 Mindlin–Reissner element model.
Firooz Bakhtiari-Nejad, Professor, Amirkabir University of Technology,
Tehran Iran
Ehsan Alavi , MS Student Amirkabir University of Technology, Tehran
Iran, [email protected]
http://dx.doi.org/10.15242/IAE.IAE0215204
32
El Damatty et al. [8] performed experimental and numerical
investigation to assess the behavior of the joined conicalcylindrical shells. Recently, Caresta and Kessissoglou [9]
analyzed the free vibrations of joined truncated conicalcylindrical shells.. Kamat et al. [10] studied the dynamic
instability of a joined conical-cylindrical shell subjected to
periodic in-plane load using C0 two-nodded shear
deformable shell element. Sivadas and Ganesan [11] have
analyzed cylinder-cone, cylinder-plate and stiffened shells
for their free vibration characteristics using a high-order
semi-analytical finite element solution. Lee et al. [12]
studied the free vibration characteristics of the joined
spherical–cylindrical shell with various boundary conditions
using Flügge shell theory and modal testing.
In this study, joint cylindrical shells were investigated.
Joint conical shells can be considered as a general mode for
joint conical-cylindrical shells [13], joint cylindrical-plate
[14],, cylindrical shells with various thicknesses [16], conical
shells with various thicknesses [17] and annular plates [18],
can be considered. In addition, the changes in the levels of
conical and cylindrical shells were considered. Governing
equations are obtained using thin-walled shallow shell theory
of Donnelly type and Hamilton’s principle. The equations
were solved assuming trigonometric response in
circumferential directions and series solution in meridional
directions. All combinations of boundary conditions can be
assumed in this method. The effects of semi-vertex angles,
meridional lengths and shell thicknesses on the natural
frequency and circumferential wave number of joined shells
are investigated. Finite element analysis is conducted to
predict the natural frequencies of isotropic and composite
samples taken and the results are obtained in good agreement
with experimental values.
II. CONSTRUCTIVE EQUATIONS FOR JOINED CONICAL
SHELLS
Consider a set of two joined conical shells with (x,θ,z)
coordinates, as shown in Fig. 1, where x is the coordinate
along the cones’ generators with the origin placed at the
middle of the generators, θ is the circumferential coordinate,
and z is the coordinate normal to the cones’ surfaces. R1, R2
and R3 are the radii of the system of cones at its first, middle
and end, respectively. The angles α1 and α2 are the semivertex angles of cones and L1 and L2 are the cone lengths
along the generators. The thicknesses of cones are h1 and h2.
The displacements are denoted by u, v and w along x, θ and z
directions, respectively.
International Conference on Mechanical And Industrial Engineering (ICMAIE’2015) Feb. 8-9, 2015 Kuala Lumpur (Malaysia)
w
,
x
(6)
vcos  1 w
 

R
R 
The parameters (𝜀𝑥 , 𝜀𝜃 , 𝛾𝑥𝜃 ) are membrane strains, and
(𝐾𝑥 , 𝐾𝜃 , 𝐾𝑥𝜃 ) are flexural (bending) strains, known as the
curvatures.
x  
Fig.1 Geometry Of Two Joined Conical Shells
A. Constitutive relations
The stress-strain relation for cross-ply laminated conical
shell can be shown as [20]
The Kirchhoff hypothesis requires the displacement field
(𝒖, 𝒗, 𝒘) to be such that
(1)
where(𝑢0 , 𝑣0 , 𝑤0 ) and (𝛽𝑥 , 𝛽𝜃 ) represent mid-plane
displacements and rotation of tangents along the x and θ,
respectively.
The shells are made of 𝑁𝐿 layers of laminates with the fibers
in 0 or 90 degrees with respect to the x axis and the stacking
sequences is as shown in Fig. 2.
The strains and curvature changes in the middle surface of
each cone can be written by Donnell thin shell theory [19] as:
Kx 


 K 
K 
 x 
(2)
B 22
0
 
A ij   Q ij
k 1
B ij 
(3)
B 11
B 12
0
D11
B 12
B 22
0
D12
D12
D 22
0
0
0    xx 


0    
B 66    x  


0   K xx 
0   K  


D 66  K x  
(7)
k
 z k 1  z k   i , j
  z
1 NL
 Q ij
2 k 1
k
2
k 1
 1, 2, 6 
 z k2   i , j  1, 2, 6 
(9)
 
1 NL
 Q ij k  z k3 1  z k3   i , j  1, 2, 6 
3 k 1
The subscript k denotes the kth layer of the laminate and
̅
𝑄𝑖𝑗 s are transformed stiffnesses expressed for cross-ply
laminates as
D ij 


 x


x


K
 x  


1 


 x sin   
K    K    
(4)


R x  
 
K  

 x  

1  x 
1



 sin 
x R  x 
 R  x  

whereR(x) is radius of the cone at any point along its length
expressed as
R  x   R 0  xsin
(5)
http://dx.doi.org/10.15242/IAE.IAE0215204
0
0
A 66
0
0
B 66
in which (𝑁𝑥𝑥 , 𝑁𝜃𝜃 , 𝑁𝑥𝜃 ) and (𝑀𝑥𝑥 , 𝑀𝜃𝜃 , 𝑀𝑥𝜃 ) are stress and
moment resultants measured per unit length, respectively and
defined as
NL


u



x


 x 
   1  v
 
 usin  wcos   
       


   R  x   
 x  
v 
1 u
1



vsin 
x 
 R  x   R  x 
A12
A 22
0
B 12
Nx 
 x 




 N 
  
 N x   h / 2   x  
(8)

  
 dz
 M x  h / 2  z  x 
 M 
 z  




M x  
z  x  
where(𝜎𝑥 , 𝜎𝜃 , 𝜎𝑥𝜃 ) are normal and shear stresses and
(𝐴𝑖𝑗 , 𝐷𝑖𝑗 , 𝐵𝑖𝑗 ) are extensional, bending and bendingextensional coupling stiffnesses which are defined in terms
of the lamina stiffnesses 𝑄𝑖𝑗 as
Fig. 2 Geometry of cross-ply layers
 ex   x 
   
 e        z
e   
 x   x 
 N xx   A11

 
 N    A12
 N x    0


 M xx   B 11
 M    B 12

 
M x    0
Q 11  Q11cos 4  Q 22 sin 4 , Q 12  Q12  sin 4  cos 4 
Q 22  Q11sin 4  Q 22cos 4 , Q 16  0, Q 26  0
(10)
Q 66  Q 66  sin 4  cos 4  ,   0or 90
andφis the angle of fibers in each ply and 𝑄𝑖𝑗 are known in
terms of the engineering constants:
E1
 E
E2
Q11 
, Q12  12 2 , Q 22 
, Q 66  G12
(11)
1  1221
1  1221
1  1221
33
International Conference on Mechanical And Industrial Engineering (ICMAIE’2015) Feb. 8-9, 2015 Kuala Lumpur (Malaysia)
whereE, G and ν are elastic modulus, shear modulus and
Poisson’s ratios, respectively.

  RN
Γ
B. Governing equations
The strain-displacement relations (2)-(4) can be used to
drive the governing equations of conical shells. Using the
dynamic version of virtual work (Hamilton’s principle) we
have
T
 U  V
  K dt
(12)
0
Where e𝛿𝑈denotes the virtual strain energy, 𝛿𝑉is the virtual
potential energy due to the applied loads, and 𝛿𝐾 denotes the
virtual kinetic energy.
x

 R N x   u   RN x   M x  cos 



  w 
R T x  v   RM x  R M x   



  x 
(18)

w   RM x 
  RN x

 M  sin 
x
x

M x 
w

2
 N x
 RV x  w d   0



Thus -neglecting nonlinear terms- the primary or essential
variables (i.e., generalized displacements) are 𝑢0 ،𝑣0 ،𝑤0
‫𝑤𝜕و‬0 ⁄𝜕𝑥 and secondary variables (i.e., generalized forces)
are 𝑁𝑥 ،𝑇𝑥 ،𝑉𝑥 ‫ 𝑥𝑀و‬in which
M
1 M x 
(19)
V x  Qx 
, T x  N x   x  cos 
R 
R
III. SOLUTION PROCEDURE
( (13)
By the use of trigonometric solution in θ direction
u  x , , t   u  x  cos n e i t
v  x , , t   v  x  sin n e i t
(20)
i t
w  x , , t   w  x  cos n e
and series solution in x direction and with the approach
described by Tong [31] we have
U   ij  ij dV  
V
 N
x
h /2

ij
 ij Rdsd  dz 
A h / 2
A
M   K   N x   x   M x   K x   Rdsd 
 w  
V   N x  u T x v  S x w  M x  
 Rd 
 x  
Γ
(15)
̂𝑥 are stress resultants at
̂𝑥 , 𝑇̂𝑥 , 𝑆̂𝑥 , 𝑀
Where parameters𝑁
boundaries, ρ is the density of the shell material, and 𝐼𝑖 s are the
mass.By substitution of Eqs.(2)-(4) into Eqs.(7)-(8), neglecting
𝐼2 due to thin shell assumptions and then imposing all into
Eqs.(13)-(15), we have
N x 1
1 N x 
 2u
  N x  N   sin 
 I0 2
R
R 
x
t
1 N  N x  2
1
 2v
v :

 N x  sin  Q cos   I 0 2 (16)
R 
R
R
x
t
2
Q sin 1 Q
Q
1
w
w :  N  cos   x  x

 I0 2
R
R
R 
x
t
Where 𝑄𝑥 and 𝑄𝜃 denote the shear resultants at x and θ
directions, respectively andare defined as
M x 1
1 M x 
Qx 
  M x  M   sin 
R
R 
x
(17)
M x  1 M  2
Q 

 M x  sin
R 
R
s
The boundary conditions are then given by
u :
http://dx.doi.org/10.15242/IAE.IAE0215204


m 0
m 0
m 0
(21)
where 𝑎𝑚 ،𝑏𝑚 and 𝑐𝑚 coefficients can be determined using
boundary and continuity conditions.
(14)
 x  M x  K x  N   

u  x   am x m , v  x   bm x m , w  x   c m x m
34
A. Boundary and continuity conditions
All types of boundary conditions can be used at both
ends of the joined cones. The simply-supported (shear
diaphragm), clamped and free boundary conditions are
described as Shear-diaphragm (SD):
(22)
v  N x  M x w  0
Simply-supported:
w
(23)
v u 
w  0
clamped:
x
free:
T x  N x  M x V x  0
(24)
All of the above mentioned boundary conditions are
applicable in the current solution method. The continuity
conditions at the conical shell joint can be obtained from
boundary conditions in Eq. (18) as:
u1 cos 1 w 1 sin 1  u 2 cos  2 w 2 sin  2
u1 sin 1  w 1 cos 1  u 2 sin  2  w 2 cos  2
v1 v 2,
w 1 w 2

,Tx 1 Tx 2 , M x 1  M x 2
x 1
x 2
(25)
N x 1 cos 1 V x 1 sin 1  N x 2 cos  2 V x 2 sin  2
N x 1 sin 1 V x 1 cos 1  N x 2 sin  2 V x 2 cos  2
These continuity conditions are extracted thoroughly
from Hamilton’s principle with no restriction, and guarantee
the continuity of the displacements (translations and
rotations) and load and moment transfer between the cones.
International Conference on Mechanical And Industrial Engineering (ICMAIE’2015) Feb. 8-9, 2015 Kuala Lumpur (Malaysia)
This means that no flexural motion is allowed at the joint
section.
IV. SIMULATION PROCESSES OF THE VIBRATION OF JOINT
CONICAL SHELLS, GEOMETRICAL CHARACTERISTICS OF THE
MODEL
In the simulation procedure, following assumptions have
been applied:
 material properties were considered in two separate
ways, as isotropic and composite orthotropic
 Linear perturbation analysis for several modes which
will be explained later has been done for a number of
given natural frequencies.
Fig .4 Iilustration Of The Isotropic Specimen With Mesh In Abacus
V. COMPARISON OF ANALYTICAL AND FINITE ELEMENT
RESULTS
In order to verify the accuracy of the analytical model,
results obtained by the solution of Equation 26 in the
previous section were compared with the results of finite
element analysis in Table 1
A. Results of limit elements for isotropic shells
The input data for the simulation are as follows:
R1  200 mm , R 2  180 mm , R 3  0 , υ  0.3 ,
TABLE I
COMPARISON OF FINITE ELEMENT ANALYSIS AND ANALYTICAL
SOLUTION FOR ISOTROPIC AND COMPOSITE SHELLS
Composite shell
Isotropic shell
Vibration
mode
Analytical
FEM
Analytical
FEM
0.2519
0.2513
0.2432
0.2433
1
0.2932
0.2893
0.3505
0.3498
2
0.4253
0.4205
0.5778
0.5778
3
0.6482
0.6449
0.9251
0.9279
4
0.9619
0.9625
1.3924
1.3993
5
E  68.97GPa, h1  h2  2 mm , 1  11.77 2  90
Consequently, the mesh generator model in Abacus is as
follows (Figure 3). As seen the model seems a frustum attached
to a circular plate which is rounded at its end. Free vibrations
are also studied, so the sample isn't influenced by external
loads. In addition, the sample at the lower edge of the cone is
tangled and bound in all directions, and the other edge is free.
In the initial conditions, all measurement parameters have a
value of zero.
Results Table 1 shows good agreement between the
results of the analytical model and the finite element.
A. Effect of geometric parameters and materials
The impacts of different parameters on the natural
frequencies of a few specimens have been examined. Figure
5 shows the effect of the length of cone shell (L/R1) on the
lowest dimensionless vibrational parameters (Ω1) and
peripheral wave number (n) of the corresponding jointcone
shells. The first half apex angle of the cone is 30 °, h / R1 =
0.01and NL = 4 . Sharp points on the graph indicate the
points where the peripheral wave number has changed. As
can be seen , the increase in shell length decreases the
natural frequencies Also, this result can be obtained that the
first ( lowest ) natural frequency of the joint shells rise
when α2and α1 get closer together, meaning that higher
amounts of energy are required tor stimulating joint shells
when α2 and α1 are clos to each other. When there is a huge
difference between α1 and α2 , the first natural frequency of
the mode numbers occur in the lower atmosphere (less
energy is needed to excite the structure) and the lowest
natural frequency of the circuit cone-plate occurs (ie, the half
apex angle of one of the cones is 90 °).
The effect of number of layers (NL) on the structural
frequencies (Ω1) of joint asymmetric cone shells has been
shown in Figure 6. The half–apex angle of the first cone is
30 °, h/R1 = 0.01 and L/R1 = 1. It is observed that increasing
the number of layers of constant thickness leads in
increasing the natural frequency of the shell. Increasing the
number of layers in a cross-ply asymmetric laminate results
in convergence of the results in the isotropic case and less
Fig. 3 Illustration Of Sample Mesh Isotropic In Abacus
B. Results of limit elements for composite shells
Lamination of the composite material is [0/90]s and the
input data for the simulation is as follows.
R1  200 mm،hhmmRmm
, 2  180
, 1  11.77
1  2 2
R1  200 mm،hhmmRmm
1  2 2
,
2
, 1  11.77
 180
R 3  100mm ,  2  90,   1600Kg / m , 12  0.33,
3
G12  4.47GPa, E 2  8.8GPa, E 1  135GPa, h  0.5mm
Consequently, a model of the mesh generator will be illustrated
in Figure 4 which is seen as a model of a frustum attached to a
hollow circular plate at the end.
http://dx.doi.org/10.15242/IAE.IAE0215204
35
International Conference on Mechanical And Industrial Engineering (ICMAIE’2015) Feb. 8-9, 2015 Kuala Lumpur (Malaysia)
changes in the natural frequency of more than 4 layers can be
seen.
Fig. 5 Effect Of The L/R1on Lowest Frequency Parameters And
Corresponding Environmental Mode Numbers In Asymmetric CrossPly Joint Conical Shells With Free-Clamped Boundary Conditions
[𝛼 = 30°, ℎ⁄𝑅 = 0.01, 𝑁𝐿 = 4]
Fig 8 Variation Of Lowest Dimensionless Frequency Parameter
With Change Of Layer Thiknesses For Free-Clamped Joint AntiSymmetric Conical Shells With
[𝑵 = 𝟒 , ⁄𝑹 = , 𝒉⁄𝑹 = 𝟎. 𝟎 ]
VI. CONCLUSION
Fig. 6 Effect Of Number Of Layers On The Lowest Frequency
Parameter Of Asymmetric Conical Shells With Free-Clamped
Boundary Conditions
[𝛼 = 30° , 𝐿⁄𝑅 = 1 , ℎ⁄𝑅 = 0.01]
Variation of lowest dimensionless frequency parameter
with change of semi-vertexangles for free-clamped joined antisymmetric cross-ply conical shells with L/R1=1,h/R1=0.01 and
NL=4 is presented in Fig. 7. It can be seen that maximum
values of first natural frequency occurs when α2 is slightly
more than α1. In addition, the cone-platecombination (i.e. α2 or
α1=±90°) has the lowest values of fundamentalfrequencies and
the semi-vertex angle of the cone has no significant effect on
the firstnatural frequenciesin cone-plate combinations. Figure 8
presents the effects of shell thicknesses on natural frequency
parameter ofthe joined shells. The semi-vertex of the first cone
is set to 30o, h/R1=0.01, NL=4.
Joined conical shells can be used to study several types
of problems by changing the semi-vertex angles or thickness
of the cones. Among these types of problem, we can mention
joined cylindrical-conical shells, joined cylinder plates or
cone-plates, conical and cylindrical shells with stepped
thickness or change in the lamination sequence, and also
annular plates.
• The first natural frequency of joined shells and
corresponding circumferential mode number increases
when two semi-vertex angles get close to each other (i.e.
higher values of energy are needed to excite the joined
shells).
• The fundamental frequency of joined shells increases with
increase in the thickness of shells. Thinner shells show
more mode changes (more cusps on the α2-Ω1 graphs)
as we change the vertex angle of the cone.
• Increasing the number of layers in constant thickness has
a small effect on natural frequencies of joined shells
when more than 4 layers are used in cross-ply
lamination.
• Maximum values of the first natural frequency occur
when the first semi vertex angle is slightly greater than
the second one.
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Fig. 7 Variation Of Lowest Dimensionless Frequency With
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Firooz Bakhtiari-Nejad,
Iran, 1951
Education:
PhD, Engineering; Kansas State University, USA ,
Aug. 1983
MS, ME; Kansas State University, USA May 1978
BS, ME; Kansas State University, USA December
1975
BS, EE; Kansas State University, USA May 1975
Research Interests: Automatic Controls, Mechanical Vibrations,
Engineering Reliability, Health Monitoring, Professor,
Amirkabir
University of Technology, Tehran Iran
Ehsan Alavi ,
MS Student Amirkabir University of Technology,
Tehran Iran, [email protected]
http://dx.doi.org/10.15242/IAE.IAE0215204
37