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 cylinder-

plates 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, e.alavi@aut.ac.ir

El Damatty et al. [8] performed experimental and numerical

investigation to assess the behavior of the joined conical-

cylindrical shells. Recently, Caresta and Kessissoglou [9]

analyzed the free vibrations of joined truncated conical-

cylindrical 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 semi-

vertex 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.

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Fig.1 Geometry Of Two Joined Conical Shells

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 𝑁𝐿layers 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:

Fig. 2 Geometry of cross-ply layers

(2) x x x

x x x

e K

e z K

e K

(3)

1

1 1

x

x

u

x

vusin wcos

R x

u vvsin

R x R x x

(4)

1

1 1

x

x

x

x

x

xK

K K sinR x

K

sinR x x R x

whereR(x) is radius of the cone at any point along its length

expressed as

(5) 0R x R xsin

(6)

,  

1 

x

w

x

vcos w

R R

The parameters (휀𝑥, 휀𝜃 , 𝛾𝑥𝜃) are membrane strains, and

(𝐾𝑥 , 𝐾𝜃 , 𝐾𝑥𝜃) are flexural (bending) strains, known as the

curvatures.

A. Constitutive relations

The stress-strain relation for cross-ply laminated conical

shell can be shown as [20]

(7)

11 12 11 12

12 22 12 22

66 66

11 12 11 12

12 22 12 22

66 66

0 0

0 0

0 0 0 0

0 0

0 0

0 0 0 0

xx xx

x x

xx xx

x x

N A A B B

N A A B B

N A B

M B B D D K

M B B D D K

B DM K

in which (𝑁𝑥𝑥 , 𝑁𝜃𝜃 , 𝑁𝑥𝜃) and (𝑀𝑥𝑥 , 𝑀𝜃𝜃 , 𝑀𝑥𝜃) are stress and

moment resultants measured per unit length, respectively and

defined as

(8)

/ 2

/2

x x

hx x

x xh

x x

N

N

Ndz

M z

M z

M z

where(𝜎𝑥 , 𝜎𝜃 , 𝜎𝑥𝜃) are normal and shear stresses and

(𝐴𝑖𝑗 , 𝐷𝑖𝑗 , 𝐵𝑖𝑗) are extensional, bending and bending-

extensional coupling stiffnesses which are defined in terms

of the lamina stiffnesses 𝑄𝑖𝑗as

(9)

1

1

2 2

1

1

3 3

1

1

    ,  1, 2,6

1    ,  1,2,6

2

1    ,  1,2,6

3

L

L

L

N

ijij k kk

k

N

ijij k kk

k

N

ijij k kk

k

A z z i jQ

QB z z i j

D zQ z i j

The subscript k denotes the kth layer of the laminate and

�̅�𝑖𝑗s are transformed stiffnesses expressed for cross-ply

laminates as

(10)

4 4 4 4

11 1211 22 12

4 4

22 16 2611 22

4 4

66 66

,

, 0,     0 

 ,   0    90

Q cos Q sin Q sin cos

Q sin Q cos

Q sin cos

Q Q

Q Q

Q r

Q

o

andφis the angle of fibers in each ply and 𝑄𝑖𝑗are known in

terms of the engineering constants:

(11) 1 12 2 2

11 12 22 66 12

12 21 12 21 12 21

,   ,   ,  1 1 1

E E EQ Q Q Q G

International Conference on Mechanical And Industrial Engineering (ICMAIE’2015) Feb. 8-9, 2015 Kuala Lumpur (Malaysia)

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whereE, G and ν are elastic modulus, shear modulus and

Poisson’s ratios, respectively.

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

(12)

0

T

U V K dt

Where e𝛿𝑈denotes the virtual strain energy, 𝛿𝑉is the virtual

potential energy due to the applied loads, and 𝛿𝐾denotes the

virtual kinetic energy.

( (13)

(14)

/ 2

/2

h

ij ij ij ij

V A h

x x x x

A

x x x x

U dV Rdsd dz

N M K N

M K N M K Rdsd

(15) Γ

x xx x

wV N u T v S w M Rd

x

Where parameters�̂�𝑥 , �̂�𝑥, �̂�𝑥 , �̂�𝑥are stress resultants at

boundaries, ρ is the density of the shell material, and 𝐼𝑖s 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

2

0 2

2

0 2

2

0 2

1 1:

1 2 1:

1 1:

xx

x

x

x

x x

NN uu N N sin I

x R R t

N N vv N sin Q cos I

R x R R t

QQ Q sin ww N cos I

R x R R t

(16)

Where 𝑄𝑥 and 𝑄𝜃denote the shear resultants at x and θ

directions, respectively andare defined as

(17)

1 1

1 2            

xx

x x

x

x

MMQ M M sin

x R R

M MQ M sin

s R R

The boundary conditions are then given by

(18)

Γ

2 0

xx x x

x xx

x

x

x

x x

RN R N u RN M cos

wR T v RM R M

x

RMwRN M sin

x x

M wN RV w d

Thus -neglecting nonlinear terms- the primary or essential

variables (i.e., generalized displacements) are 𝑢0 ،𝑣0 ،𝑤0

𝑤0��و 𝜕𝑥⁄ and secondary variables (i.e., generalized forces)

are 𝑁𝑥 ،𝑇𝑥 ،𝑉𝑥و𝑀𝑥in which

1,x x

x x x x

M MV Q T N cos

R R

(19)

III. SOLUTION PROCEDURE

By the use of trigonometric solution in θ direction

(20)

, , cos  

, ,     

, , cos

i t

i t

i t

u x t u x n e

v x t v x sin n e

w x t w x n e

and series solution in x direction and with the approach

described by Tong [31] we have

0 0 0

, ,m m m

m m m

m m m

u x a x v x b x w x c x

(21)

where 𝑎𝑚 ،𝑏𝑚and 𝑐𝑚coefficients can be determined using

boundary and continuity conditions.

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)

(23)

(24)

0

0 

0

x x

x x x x

v N M w

wv u w

x

T N M V

Simply-supported:

clamped:

free:

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:

(25)

1 1 1 1 2 2 2 2

1 1 1 1 2 2 2 2

1 2

1 2 1 2 1 2

1 2

1 1 1 1 2 2 2 2

1 1 1 1 2 2 2 2

, ,

cos sin cos sin

sin cos sin cos       

,

cos sin cos sin

sin cos sin cos

x x x x

x x x x

x x x x

u w u w

u w u w

w wv v T T M M

x x

N V N V

N V N V

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)

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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.

A. Results of limit elements for isotropic shells

The input data for the simulation are as follows:

1 2 3

1 2 1 2

200  ,   180  ,   0 , υ 0.3 ,

68.97  , 2  , 11.77 90

     

R mm R mm R

E GPa h h mm

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.

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.

1 1 2 2 1200    2  , 180  ,   11.77R mm،hhmmRmm

1 1 2 2 1200    2  , 180  ,   11.77R mm،hhmmRmm

3

3 2 12

12 2 1

100 , 90 , 1600 / , 0.33,

4.47 , 8.8 , 135 , 0.5

R mm Kg m

G GPa E GPa E GPa h mm

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.

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 TABLE I

COMPARISON OF FINITE ELEMENT ANALYSIS AND ANALYTICAL

SOLUTION FOR ISOTROPIC AND COMPOSITE SHELLS

Vibration

mode

Isotropic shell Composite shell

FEM Analytical FEM Analytical

1 0.2433 0.2432 0.2513 0.2519

2 0.3498 0.3505 0.2893 0.2932 3 0.5778 0.5778 0.4205 0.4253

4 0.9279 0.9251 0.6449 0.6482

5 1.3993 1.3924 0.9625 0.9619

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

International Conference on Mechanical And Industrial Engineering (ICMAIE’2015) Feb. 8-9, 2015 Kuala Lumpur (Malaysia)

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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 Cross-

Ply Joint Conical Shells With Free-Clamped Boundary Conditions

[𝛼 = 30°, ℎ 𝑅 = 0.01, 𝑁𝐿 = 4⁄ ]

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 anti-

symmetric 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.

Fig. 7 Variation Of Lowest Dimensionless Frequency With

Change Of Semi-Vertex Angles For Free-Clamped Joint Anti-

Symmetric Conical Shells With

[N = 4 , L R = 1⁄ , h R = 0.01⁄ ]

Fig 8 Variation Of Lowest Dimensionless Frequency Parameter

With Change Of Layer Thiknesses For Free-Clamped Joint Anti-

Symmetric Conical Shells With [𝑵 = 𝟒 , 𝑹 = ⁄ , 𝒉 𝑹 = 𝟎. 𝟎 ⁄ ]

VI. CONCLUSION

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.

REFERENCES [1] Ö. Civalek, Vibration analysis of conical panels using the method of

discrete singular convolution, Commune Numerical Methods Engineering 24 (2008) 169-181.

http://dx.doi.org/10.1002/cnm.961

[2] J. Rose, R. Mortimer, A. Blum, Elastic-wave propagation in a joined cylindrical-conical-cylindrical shell, Experimental Mechanics, 13

(1973) 150-156.

http://dx.doi.org/10.1007/BF02322668 [3] M. Lashkari, V. Weingarten, Vibrations of segmented shells,

Experimental Mechanics 13 (1973) 120-125.

http://dx.doi.org/10.1007/BF02323969 [4] T. Irie, G. Yamada, Y. Muramoto, Free vibration of joined conical-

cylindrical shells, Journal of Sound and Vibration 95 (1984) 31-39.

http://dx.doi.org/10.1016/0022-460X(84)90256-6 [5] E. Efraim, M. Eisenberger, Exact vibration frequencies of segmented

axisymmetric shells, Thin Walled Structures 44 (2006) 281-289.

http://dx.doi.org/10.1016/j.tws.2006.03.006 [6] B.P. Patel, M. Ganapathi, S. Kamat, Free vibration characteristics of

laminated composite joined conical-cylindrical shells, Journal of Sound and Vibration 237 (2000) 920-930.

http://dx.doi.org/10.1006/jsvi.2000.3018

International Conference on Mechanical And Industrial Engineering (ICMAIE’2015) Feb. 8-9, 2015 Kuala Lumpur (Malaysia)

http://dx.doi.org/10.15242/IAE.IAE0215204 36

[7] M. Özakça, E. Hinton, Free vibration analysis and optimization of

axisymmetric plates and shells—I. Finite element formulation, Computers and Structures 52 (1994) 1181-1197.

http://dx.doi.org/10.1016/0045-7949(94)90184-8

[8] A.A. El Damatty, M.S. Saafan, A.M.I. Sweedan, Dynamic characteristics of combined conical-cylindrical shells, Thin Walled

Structures 43 (2005) 1380-1397.

http://dx.doi.org/10.1016/j.tws.2005.04.002 [9] M. Caresta, N.J. Kessissoglou, Free vibrational characteristics of

isotropic coupled cylindrical–conical shells, Journal of Sound and

Vibration 329 (2010) 733-751. http://dx.doi.org/10.1016/j.jsv.2009.10.003

[10] S. Kamat, M. Ganapathi, B.P. Patel, Analysis of parametrically excited

laminated composite joined conical–cylindrical shells, Computers Structures 79 (2001) 65-76.

http://dx.doi.org/10.1016/S0045-7949(00)00111-5

[11] K.R. Sivadas, N. Ganesan, Free vibration analysis of combined and stiffened shells, Computers Structures 46 (1993) 537-546.

http://dx.doi.org/10.1016/0045-7949(93)90223-Z

[12] Y.-S. Lee, M.-S. Yang, H.-S. Kim, J.-H. Kim, A study on the free vibration of the joined cylindrical–spherical shell structures, Computers

Structures 80 (2002) 2405-2414.

http://dx.doi.org/10.1016/S0045-7949(02)00243-2 [13] D.T. Huang, Influences of small curvatures on the modal characteristics

of the joined hermetic shell structures, Journal of Sound and Vibration

238 (2000) 85-111. http://dx.doi.org/10.1006/jsvi.2000.3080

[14] A.J. Stanley, N. Ganesan, Frequency response of shell-plate combinations, Computer Structures 59 (1996) 1083-1094.

http://dx.doi.org/10.1016/0045-7949(95)00337-1

[15] S. Liang, H.L. Chen, The natural vibration of a conical shell with an annular end plate, Journal of Sound and Vibration 294 (2006) 927-943.

http://dx.doi.org/10.1016/j.jsv.2005.12.033

[16] L. Zhang, Y. Xiang, Exact solutions for vibration of stepped circular cylindrical shells, Journal of Sound and Vibration 299 (2007) 948-964.

http://dx.doi.org/10.1016/j.jsv.2006.07.033

[17] Y. Qu, Y. Chen, Y. Chen, X. Long, H. Hua, G. Meng, A Domain Decomposition Method for vibration Analysis of Conical Shells With

Uniform and Stepped Thickness, Journal of Sound and Vibration 135

(2013) 011014-011013.

http://dx.doi.org/10.1115/1.4006753

[18] M.S. Qatu, Vibration of laminated shells and plates, Elsevier, 2004.

[19] J.N. Reddy, Mechanics of laminated composite plates and shells: theory and analysis, CRC Press, 2004.

[20] L. Tong, Free vibration of composite laminated conical shells,

International Journal of Mechanics and Sciences 35 (1993) 47-61. http://dx.doi.org/10.1016/0020-7403(93)90064-2

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, e.alavi@aut.ac.ir

International Conference on Mechanical And Industrial Engineering (ICMAIE’2015) Feb. 8-9, 2015 Kuala Lumpur (Malaysia)

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