vibration analysis of a cylindrical sandwich panel with … · frostig's second model. in this...

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Abstract This paper deals with free vibration analysis of thick cylindrical composite sandwich panels with simply supported boundary condi-tions based on a new improved higher-order sandwich panel theory. The formulation used the third-order polynomial description for the displacement fields of thick composite face sheets and for the dis-placement fields in the core layer based on the displacement field of Frostig's second model. In this case, the unknowns were coefficients of the polynomials in addition to displacements of the top and bot-tom face sheets. The fully dynamic effects of the core layer and face sheets were also considered in this study. Using Hamilton's principle, the governing equations were derived. Moreover, the effect of some important parameters such as those of thickness ratio of the core to panel, the length to radius ratio of the core and composite lay-up sequences were investigated on free vibration response of the panel. The results were validated by those published in the literature and with the finite element results obtained by ABAQUS. It was shown that thicker panels with thicker cores provided greater resistance to resonant vibrations. Moreover, the effect of increasing face sheets’ thicknesses in general was the significant increase in fundamental natural frequency values. Keywords Cylindrical sandwich panel, Free vibration, Improved higher-order theory, Flexible core, Analytical analysis.

Vibration Analysis of a Cylindrical Sandwich Panel with Flexible Core Using an Improved Higher-Order Theory

1 INTRODUCTION

The use of sandwich structures has increased in recent years in aerospace, naval, civil, transportation, and other industries which require stiff and light-weight structural ingredients. Sandwich structures are constructed of three layers. They are usually composed of two metallic or composite laminated materials: face sheets and a foam core or a low-strength honeycomb core. The materials with a high

A.R. Pourmoayed 1

K. Malekzadeh Fard *

M. Shahravi 2 Department of Structural Analysis and Simulation, Aerospace Research Institute, Malek Ashtar University of Technology, Post Box: 13445-768, Tehran, IRAN. email: 1 pourmoayed @mut.ac.ir * [email protected] 2 [email protected] http://dx.doi.org/10.1590/1679-78253410 Received 02.10.2016 Accepted 21.02.2017 Available online 04.03.2017

A.R. Pourmoayed et al. / Vibration Analysis of a Cylindrical Sandwich Panel with Flexible Core Using an Improved Higher-Order Theory 715

Latin American Journal of Solids and Structures 14 (2017) 714-742

strength are usually used for face sheets, whereas the core layer is made of a low-specific-weight mate-rial which may be much less stiff or strong than the face sheets (Librescu and Hause, 2000). NOMENCLATURE

, ,x zq Longitudinal, circumferential and radial coordinates ( , , )i c t b= Indices for core, outer (top) and inner (bottom) face sheets

, ,i i iu v w Displacements in longitudinal, circumferential and radial directions

0 0 0, ,i i iu v w Mid-plane displacements in longitudinal, circumferential and transverse di-rections

, ,i i ixx zz Strains in face sheets

0 0, ,i i ixx zz Mid-plane strains in face sheets and the core

, ,i i ix xz zq q Shear strains in face sheets and the core

ijQ Reduced stiffnesses referring to the principal material coordinates

ijQ

Transformed reduced stiffnesses

jii Normal and transverse stresses in the face sheets, ( , , ), ( , )i x z j t bq= =

cii Normal and transverse stresses in the core, ( , , )i x zq=

j j jx xz z, , Shear stresses in the face sheets, ( , )j t b=

, ,c c cx xz z Shear stresses in the core

, , , ,...i i i ixx xx xx xxN M P H

Stress resultants in the face sheets, ( , )i t b=

, , , ,...c c c cxx xx xx xxN M P H

Stress resultants in the core

ljkQ The reduced stiffness coefficients of the lth composite layer of each face sheet

K Kinetic energy U Strain energy

il the number of composite layers in each face sheet

jiI Moment of inertia

i Mass density , ,t c bdV dV dV Volume element of the top face sheet, the core and the bottom face sheet,

respectively L Length of cylindrical sandwich panel h Total thickness of cylindrical sandwich panel

th Thickness of the top face sheet

bh Thickness of the bottom face sheet M Mass matrix

716 A.R. Pourmoayed et al. / Vibration Analysis of a Cylindrical Sandwich Panel with Flexible Core Using an Improved Higher-Order Theory


K

Stiffness matrix

Natural frequency

In recent years, to describe the dynamic and static behavior of these structures, various theoret-ical models have been developed. Understanding the free vibration behavior of structures is essential for preventing the occurrence of resonance, and for optimal designing. Moreover it is important to Recovering new types of materials for the face sheets and core in order to reduce failure modes and obtain optimum weight sandwich structural. Different approaches may be used for modeling sand-wich panels. The first approach is equivalent single layer (ESL) models and classical models (see Reddy, 1997; Mindlin, 1951; and Wang, 1996). These theories often obtain inaccurate results when used for the analysis of sandwich panels with flexible cores that are soft in the vertical direction (Frostig and Thomsen, 2008). The second approach is the first-order shear deformation theories (FSDT). When the face sheets in the sandwich panel are thin, the FSDT model obtains good results for the analysis of sandwich panels with flexible cores (see Malekzadeh et al., 2015). Reissner (1985), Noor and Burton (1989), Reddy (1990), and Kant and Swaminathan (2001) have reviewed these developments. There are also other theories studied by some researchers. Biglari and Jafari (2010) studied the free vibration of doubly curved composite sandwich panels with soft cores using the mixed theory. Singh (1999) applied the Rayleigh–Ritz method for free vibration analysis of doubly curved open deep sandwich shells. Meunier and Shenoi (2001), Nayak et al. (2002) and Carrera (2004) used a ‘‘zig-zag’’ displacement pattern for the modeling of layered plates and shells. Garg et al. (2006) investigated the free vibration analysis of simply supported composite and sandwich dou-bly curved shells. Their formulation included Sander's theory based on an equivalent single-layer approach. In order to include the three-dimensional fully dynamic modeling of the flexible thick core of the panels, researchers have usually used a High-order Sandwich Panel Theory (HSAPT). In this context, researchers such as Frostig and Thomsen (2004), Bozhevolnaya and Frostig (2001), and Rabinovitch et al. (2003) have used HSAPT in various structural problems. Malekzadeh et al. (2014) studied the improved high-order bending analysis of doubly curved sandwich panels subject-ed to multiple loading conditions. Rahmani et al. (2010) applied a higher-order sandwich panel the-ory in order to study the free vibration analysis of an open single curved composite sandwich panel with a flexible core.

Unfortunately, the free vibration analysis of cylindrical sandwich panels with thick face sheets and thick flexible cores has not been reported in the literature. In order to include the thickness effects of the top and bottom face sheets of the panel in the current formulation, the third-order polynomial description for the displacement fields was used. Buckling analysis of sandwich panels with a flexible core was investigated by Frostig (1998) using classical plate theory in the face sheets. Damped free vibrations of sandwich plates with a viscoelastic soft flexible core, and low-velocity impact analysis of sandwich panels were investigated by Malekzadeh et al. (2004, 2005) using First Shear Deformation Theory (FSDT) in the face sheets. The free vibration analysis of thick doubly curved sandwich panels with transversely flexible cores were investigated by Malekzadeh et al. (2014) using FSDT in the face sheets. In order to determine the bending response of composite sandwich plates, Kheirikhah et al. (2012) applied the third-order plate theory for the face sheets and quadratic and cubic functions for the transverse and in-plane displacements of the core.



This study investigated the free vibration analysis of doubly curved thick composite sandwich panels using a new improved higher-order sandwich panel theory based on the second computation-al model of Frostig (2004). In the present formulation, the top and bottom face sheets can be thick or thin. The in-plane stresses of the core were also considered. In this study, the analytical solution of the displacement field of the core was presented in terms of polynomials with unknown coeffi-cients according to the second computational model of Frostig (2004). Furthermore, the formulation included accurate stress-resultant equations for composite sandwich structures, in which the term (1 / )c cz R was imported in Eq.3 and exactly integrated. These coefficients could be very important

in the structural analysis of thick cylindrical sandwich panel structures. 2 THEORY AND FORMULATION

2.1 Basic Assumptions

Consider a cylindrical thick composite sandwich panel which is composed of two composite laminat-ed thick face sheets and a flexible core layer. The geometry of the panel and the coordinates are shown in Fig. 1. In this figure, indices t and b refer to the top and bottom face sheets of the sand-wich panels, respectively. Moreover, the thicknesses of the top face sheet, the bottom face sheet, the core layer, the total thickness of sandwich panel, the intermediate radius of the core, the intermedi-ate radius of the top face sheet, the intermediate radius of the bottom face sheet, and the length of the sandwich panel are presented by, th , bh , ch , h, cR , tR , bR and L , respectively. The displacement

fields in face sheets are u , v and w in the directions of x (longitudinal), (circumferential) and z(radial), respectively. They are measured upward from the midplane of the face sheets (Reddy, 2003). Face sheets are thick laminated composite and orthotropic structures. The face sheets and the core are assumed to be perfectly bonded, i.e. there is no relative displacement between the face sheets and the core interfaces.

Figure 1: Geometry and coordinates of the coomposite cylindricl sandwich panel.



2.2 Sandwich Kinematics

The displacements fields in face sheets, , , , , , , , and , , , in the direction of x , and z , respectively are explained as follows:

2 30 1 2 3

2 30 1 2 3

20 1 2

( , , , ) ( , , ) ( , , ) ( , , ) ( , , )

( , , , ) ( , , ) ( , , ) ( , , ) ( , , )

( , , , ) ( , , ) ( , , ) ( , , )

, , t : top f

i i i ii i i i

i i i ii i i i

i i ii i i

u x z t u x t z u x t z u x t z u x t

v x z t v x t z v x t z v x t z v x t

w x z t w x t z w x t z w x t

i t b

q q q q q

q q q q q

q q q q

= + + +

= + + +

= + += ace sheet , : bottom face sheet

( (

) )

2 2t c b c

t b

b

h h h hz z z z

+ += - = +

(1)

where 0iu , 0

iv denote inplane displacement and 0iw out of plane displacement, respectively, in midle

surface. The kinematic equations for the strains in the face sheets are as follows (Bert, 1967, and Soykasap et al., 1996):

i i, , ,

2 ,

2 ,

12

i i ii ixx zz

i i

i i i i

i

i i i ixz xz

i i i i iz z

i i i

x x

u v w

x zv u

xw u

x zw v

z

w

R R

R

v

R R

qq

q q

q q

e e eq

g eq

g e

g eq

¶ ¶ ¶= = + =

¶ ¶ ¶¶ ¶

= = +¶ ¶¶ ¶

= = +¶ ¶

¶ ¶= = - +

¶ ¶

(2)

Based on the second Frostig’s model, the displacement field components of the core layer are derived as:

2 32 3

2 c 30 1 2 3

20 1

0

2

1( , , , ) ( , , ) ( , , ) ( , , ) ( , , )

( , , , ) 1 ( , , ) ( , , ) ( , , ) ( , , )

( , , , ) ( , , ) ( , , ) ( , , )

c c c cc c c c

c c ccc c c c

cc c c

c c c

x z t u x t z x t z x t z x t

v x z t v x t z x t z x t z x t

w

u u u u

zv v v

R

w wx z t w x t z x t z x t

q q q q q

q q q q q

q q q q

= + + +æ ö÷ç ÷= + + + +ç ÷ç ÷çè ø

= + +

(3)

In Eq.3, it is seen that the number of unknown variables is 11. Based on small deformations, the kinematic relations of the core are as follows:

æ ö¶ ¶ ¶÷ç ÷= = + =ç ÷ç ÷æ öç¶ ¶ ¶è ø÷ç ÷+ç ÷ç ÷çè ø

1, ,

1

c c cc c c cxx zz

c c c

c

u v w w

x z R R z

R

qqe e eq

¶ ¶ ¶ ¶= = + = = +

æ ö¶ ¶ ¶ ¶÷ç ÷+ç ÷ç ÷çè ø

12 , 2

1

c c c cc c c cx x xz xz

c c

c

v u w u

x z R x z

R

q qg e g eq

(4)



æ ö¶ ¶÷ç ÷= = - +ç ÷ç ÷æ öç ¶ ¶è ø÷ç ÷+ç ÷ç ÷çè ø

12

1

c c c c cz z

c c c c

c

w v v

z R R z

R

q qg eq

2.3 Compatibility Conditions

Based on the positive upward direction, the compatibility conditions in the interface of the core and the top and bottom face sheets could be written as below:

2 2 2 2

2 2 2 2

2 2 2 2

,

t cb ct c

b c

t cb ct c

b c

t cb ct c

b c

h ht c h hz z b cz z



u u u u

v v v v

w w w w

- -= = = =

- -= = = =

- -= = = =

= =ìïìï ïï ïï ïï ïï ïï ïí íï ïï ïï ïï ïï ïï ïî ïî

= =

= =

(5)

According to Eq.5, the relation between displacement dependent parameters is extracted in the core:

2 2 3 30 0 1 1 2 2 3 3 0

2 2

2 2 3 30 0 1 1 2 2 3 3 1

3 3

2 2 30 0 1 1 2 2 3

2

8( ) 4( ) 2( ) ( ) 16

4

8( ) 4( ) 2( ) ( ) 8

2

8( ) 4( ) 2( ) (

b t b t b t b t cc b t b t b t

ct b t b t b t b c

c t b t b t b c

ct b b t b t b

c b t b t b

u u h u h u h u h u h u h u uu

h

u u h u h u h u h u h u h u h uu

h

v v h v h v h v h v h vv

+ + - + + + - -=

- - + + - - + -=

+ + - + + + -=

33 0

2

2 2 3 30 0 1 1 2 2 3 3 0 1

3 3

2 20 0 1 1 2 2

1

2 20 0 1 1 2 2 0

2

) 16

4

8( ) 4( ) 2( ) ( ) 8 8

2

8( ) 4( ) 2( )

8

8( ) 4( ) 2( ) 16

t ct

c

t b t b t b t b c cct b t b t b c

c c

ct b t b t b

c t b t b

cb t b t b t c

c b t b t

h v v

h

hv v h v h v h v h v h v h v v h v

Rv

h

w w h w h w h w h ww

h

w w h w h w h w h w ww

-

- - + + - - + - -=

- - + + -=

+ + - + + -=

24 ch

ìïïïïïïïïïïïïïïïïïïïïïïïïíïïïïïïïïïïïïïïïïïïïïïïïïî

(6)

According to this equation, the number of unknown variables in the core is decreased from 11 to 5. Therefore, the total number of unknowns in the core and the face sheets is reduced to 27:

0 0 0 0 0 1 1 1 1 1 1 2 2 2t c c c

2 2 2 3 3 3 3 0 1 0 1 0

0, , , , , , , , , , , , , , ,

, , , , , , , , , , ,

b b b t t t b b b t t t b b b

t t t b b t c c

v w u v w u v w u v w u v w

v w u v u v v

u

u u u v w

ìïïïíïïïî (7)

2.4 The Stress-Strain Relations and Stress Resultants

The stress-strain relations for the orthotropic composite face sheets in the global coordinate system are given as follows (Loy and Lam, 1999; Afshin et al., 2010):



11 12 13

12 22 23

13 23 33

44

55

66

0 0 0

0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

i i ixx xx

i i i

i i izz zz

ix x

ixz xz

iz z

kk Q Q Q

Q Q Q

Q Q Q

Q

Q

Q

qq qq

q q

q q

s es es es gs gs g

é ùì ü ìï ï ïê úï ï ïï ï ïê úï ï ïê úï ï ïï ï ïê úï ï ïï ï ïê úï ï ïê ú=í ý íê úï ï ïï ï ïê úï ï ïï ï ïê úï ï ïê úï ï ïï ï ïê úï ïï ï ê úî þ îë û

küïïïïïïïïïýïïïïïïïï ïï ïþ

(8)

where ( , 1, 2, 4)mnQ m n = is the reduced stiffness coefficients and ( , 3,5, 6)mnQ m n = is the trans-

verse shear stiffness coefficients. The values of stiffness coefficients as presented by Malekzadeh et al. (2010) and Garg et al. (2006) are as follows:

4 2 2 411 11 12 44 22

2 2 4 412 11 22 44 12

2 213 13 23

cos 2( 2 )sin cos sin

( 4 )sin cos (cos sin )

cos sin ,

i i i i i

i i i i i

i i i

Q Q Q Q Q

Q Q Q Q Q

Q Q Q

q q q q

q q q q

q q

= + + +

= + - + +

= +

4 2 2 422 11 12 44 22

2 223 13 23

21 12

31 13

32 23

33 332 2 2 2 2

44 11 22 12 442 2

55 55 662

66 55

sin 2( 2 ) sin cos cos

sin cos

( 2 )sin cos (cos sin )

cos sin

sin

i i i i i

i i i

i i

i i

i i

i i

i i i i i

i i i

i i

Q Q Q Q Q

Q Q Q

Q Q

Q Q

Q Q

Q Q

Q Q Q Q Q

Q Q Q

Q Q

q q q q

q q

q q q q

q q

q

= + + +

= +

=

=

=

=

= + - + -

= +

= + 266 cos

,

iQ

i t b

q=

(9)

Stress-strain relations in the core are as follows (Khalili et al., 2012):

11 12 13

12 22 23

13 23 33

44

55

66

0 0 0

0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

c cc c cxx xxc cc c c

c c c c czz zz

cc cx x

cc cxz xz

cz

k k

c cz

Q Q Q

Q Q Q

Q Q Q

Q

Q

Q

qq qq

q q

q q

s e

s e

s e

t g

t g

t g

ì ü ìé ùï ï ïï ï ê úï ï ê úï ïï ï ê úï ïï ï ê úï ï ê úï ïï ï ê ú=í ý íê úï ïï ï ê úï ïï ï ê úï ï ê úï ïï ï ê úï ïï ï ê úï ï ë ûî þ

küïï ïï ïï ïï ïï ïï ïï ïï ïï ïýï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïî þ

(10)

The values of stiffness coefficients ( )Q in Eq. 10 are as follows (George, 1999):



( ) ( ) ( )

( ) ( ) ( )

1 23 32 1 21 31 23 1 31 21 3211 12 13

2 13 31 2 32 12 31 2 12 2122 23 33

44 12 55 13 66 23

1 , ,

1 1, ,

, ,

1

j J J j J J J j J J Jj j j

J J J

j J J j J J J j J Jj j j

J J J

j j j j j j

J

E E EQ Q Q

E E EQ Q Q

Q G Q G Q G

J J J J J J J J

J J J J J J J

J

- + += = =

- + -= = =

= = =

= - 12 21 23 32 31 13 21 32 132

, ,

J J J J J J J J J

j t b c

J J J J J J J J- - -=

(11)

Based on the relations of strain potential energy of face –sheets, the stress resultants in the face –sheets are calculated as follows:

2 2

2 2

3 32 2

1 1

,

i i

i i

i ih hxx

i ii ixx i i

xx ii ii ixx h h

i ii ixx

N N

z zM Mdz

z zP P

z zH H

qq

qqqq

qq

qq

s s- -

ì ü ì üì ü ì üï ï ï ïï ï ï ïï ï ï ïï ï ï ïï ï ï ïï ï ï ïï ï ï ïï ï ïï ï ï ïï ï ïï ï ï ïï ï ï= =í ý í ý í ý í ýï ï ï ï ï ï ïï ï ï ï ï ï ïï ï ï ï ï ï ïï ï ï ï ï ï ïï ï ï ï ï ï ïï ï ï ï ï ï ïî þ îî þ î þ

ò ò ,idzïïïïïïïïïþ

2

2

2

2

32

,

1

,

1

i

i

i

i

h

izz i

zz iiizz h

ihx i

xziix i i

x i xziix ih

xziix

Ndz

zM

NQ

zMdz S

zPR

zH

q

qq

q

q

s

s

-

-

ì ü ì üï ï ï ïï ïï ï ï ï=í ý í ýï ï ï ïï ï ï ïî þï ïî þ

ì ü ì üï ï ï ïï ï ï ï ì üï ïï ï ï ï ï ïï ï ï ï ï ïï ï ï ï ï ïï ï ï ï ï ï=í ý í ý í ýï ï ï ï ï ïï ï ï ï ï ïï ï ï ï ï ïï ï ï ï ï ïïî þï ï ï ïï ï ï ïî þî þ

ò

ò2

2

2

2

2

32

1

,

1

, ,

i

i

i

i

h

ixz i i

hi

ihz

iiz i

z iiiz h

iiz

z dz

z

Q

zS i t bdz

zR

zV

q

qq

q

q

s

s

-

-

ì üï ïï ïï ïï ïï ï= í ýï ïï ïï ïï ïï ïï î þ

ì ü ì üï ï ï ïï ï ï ïï ï ï ïï ï ï ïï ï =ï ïï ï ï ï=í ý í ýï ï ï ïï ï ï ïï ï ï ïï ï ï ïï ï ï ïï ï ï ïî þî þ

ò

ò

(12)

And based on the relations of strain potential energy of the core, the stress resultants in the core are calculated as follows:

- -

ì ü ì üì ü ìï ï ï ïï ï ïï ï ï ïï ï ïï ï ï ïï ïï ï ï ïï ïæ öï ï ï ïï ïï ï ï ï÷ï ïç ÷= + =çí ý í ý í ý í÷ç ÷ï ï ï ï ï ïçè øï ï ï ï ï ïï ï ï ï ï ïï ï ï ï ï ïï ï ï ï ï ïï ï ï ï ï ïî þî þ î þ

ò ò2 2

2 2

3 32 2

1 1

1 ,

c c

c c

c ch hxx

c cc cxx c cc

xx cc cc ccxx h h

c cc cxx

N N

z zM Mzdz

z zRP P

z zH H

qq

qqqq

qq

qq

s s

- -

üïïï ïï ïï ïï ïýï ïï ïï ïï ïï ïï ïî þì ü ìï ïï ïï ïï ïì ü ì üï ï æ ö æ öï ïï ïï ï ï ï÷ ÷ï ï ï ïç ç÷ ÷= + = +ç çí ý í ý í ý í÷ ÷ç ç÷ ÷ï ï ï ï ï ïç çè ø è øï ï ï ï ï ïî þï ïî þ ï ïï ïï ïï ïî þ

ò ò

*

*2 2

2*

3*2 2

1

,

1

11,

c c

c c

c

ch hx

c cczz xc cc c

zz c xc cc cc czz xh h

ccx

dz

N

zN Mz zdz

z zR RM P

zH

q

qq

q

q

s s

üï ïï ïï ïï ïï ïï ïýï ïï ïï ïï ïï ïï ïî þ

,dz

(13)



- -

ì ü ì üï ï ï ïï ï ï ï ì ü ì üï ï ï ïï ï ï ï ï ï ï ïï ï ï ï ï ï æ öï ïï ï ï ï ï ï ï ïï ï ÷ï ï ï ï ïç ÷= = +çí ý í ý í ý í ý÷ç ÷ï ï ï ï ï ï ïçè øï ï ï ï ï ï ïï ï ï ï ï ï ïï ï ï ï ï ï ïïï ï îî þï ï ï ïï ï ï ïî þî þ

ò ò

*2 2

*2

* 23

2 2

,

11

1

c c

c c

ch hx c

xzccx c c c c

x c xz xz ccc cx ch h

xz cccx

NQ

zM zdz S z

z RPR z

zH

q

qq

q

q

s s

- -

ïïïïïïþ

ì ü ìï ï ïï ï ïì ü ì üï ï ï ï ï ï ïï ï ï ï ï ï ïï ï æ öï ï ï ï ïï ï ï ï ï ï÷ï ï ï ï ïç ÷= + =çí ý í ý í ý í÷ç ÷ï ï ï ï ï ï ïçè øï ï ï ï ï ï ïï ï ï ï ï ïï ï ï ï ï ïï ïï ï î þî þ ï ïï ï îî þ

ò ò

*

*2 2

2*2

3*2 2

,

,

11

1

c c

c c

c

ch hzc

xz cczc c cc

xz xz c c zccc zc h h

xz c ccz

dz

QQ

zSzS z dz

zR RR z

zH

q

qq

q

q

s s

-

üïïïïïïýïïï ïï ïï ïï ïþì ü ì üï ï ï ïï ï ï ïï ï æ öï ïï ï ï ï÷ï ï ï ïç ÷= +çí ý í ý÷ç ÷ï ï ï ïçè øï ï ï ïï ï ï ïï ï ï ïï ïï ï î þî þ

ò2

2

2

,

1

1

c

c

c

hczc c cz z c c

cc hz c

dz

Qz

S z dzR

R z

q

q q

q

s

3 GOVERNING EQUATIONS

The governing equations of motion for the face sheets and the core are derived using Hamilton's principle of minimization of the Lagrangian L of the deformed system:

0 0

dt dt 0T T

L K Ud d dé ù= - =ë ûò ò (14)

where K is the kinetic energy, U is the strain energy and denotes the variation operator. The variation of kinetic energy is extracted as follows:

2

i

2

, , 2

c

2

( )

( )

,

,

i

i i

c

c c

h

i iA h

hi t b c

c cA h

i i

i i i i i i

c c

i

c c c

i i i i i i

c

c c c c

dz dA

K

dz dA

dA Rdx d i t b

dA R dx d

dV Rdx

u u v v w w

u u v v

d dz dAdz

dV

w w

r

d

r

qq

d d d

d d

q

d

-

=

-

é ùê úê úê úê úê úê ú= - ê úê úê úê

+ +

+ + ú+ê úê úê úë û

= ==

= =

=

ò

å

ò

∬

∬

1 1c cc c c c c

c c

z zR dx d dz dA dz

R Rq

æ ö æ ö÷ ÷ç ç÷ ÷+ = +ç ç÷ ÷ç ç÷ ÷ç çè ø è ø

(15)

where ir and cr are the density of the face sheets and the core, respectively. The variation of

strain energy can be written as below:



( )

( ), ,

,

i

c

i i i i i i i i i i i ixx xx zz zz x x xz xz z z i

V

i t b c c c c c c c c c c c c cxx xx zz zz x x xz xz z z c

V

i i i

c c

dV

U

dV

dA Rdx d i t b

dA R

qq qq q q q q

qq qq q q q q

s de s de s de t dg t dg t dg

d

s de s de s de t g t dg t dg

q

=

é ùê ú+ + + + +ê úê úê ú= ê úê ú+ + + + + +ê úê úê úë û

= ==

òå

ò

,

1

c

i i i

cc c c

c

dx d

dV dAdz

zdV dAdz

R

q=

æ ö÷ç ÷= +ç ÷ç ÷çè ø

(16)

The governing equations for a cylindrical sandwich panel composed of a flexible core and thick

composite face sheets are derived. Hence, after integration by parts and some algebraic manipula-tion, twenty-seven equations of motion are extracted, some of which are as follows: At the top face sheet:

0

, , , , x, x, xz xz2 3 2 3 2 3t c c

t t t t 2 3 4 50 1 2 3 2 3 4 50

c

3 4 52 3

2 3

4

1 0

:

1 2 4 2 4 4 12N N P P H S R

R R h R h h h

2I 4I 8I 16II I I I

h h h h

2I 4I 8I

h h h

t

xt t c c c c c c

c c c c c cc c c c

t t t t

x x x xx x xx x

c

c c cc c c

c c c

h h

u u u u

u

u

Hq q q q q q+ + + + + - -

æ ö÷ç ÷ç= + + + + + - - +÷ç ÷ç ÷è ø

+ - -

6 4 6 4 5 65 3 6 4 5 6

c c 2b 4 b 6 t 4 t 5 t 6 b 43 6 4 5 6c

1 0 0

1 1

16I 4I 16I 4I 16I 16I

h h h h h h

2h I 8h I 2h I 8h I 8h I h I

h h h h h h

c c c c c cc b t

c c c c c cc c c c

b t

c c c c

u u u

u u

æ ö æ ö æ ö÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç+ - + + +÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷è ø è ø è øæ ö æ ö÷ ÷ç ç÷ ÷ç ç+ - + - - - +÷ ÷ç ç÷ ÷ç ç÷ ÷è ø è ø

2b 6

4 6

2 2 2 3 c 3 3 3 3t 4 t 5 t 6 b 4 b 6 t 4 t 5 t 64

2

2 35 6 4 6 34 5 6c

4h I

h

h I 4h I 4h I h I 2h I h I 2h I 2h I

h h h 2h h 2h h h

cb

c cc c c c c c c

t b t

c c c c c c c

u

u u u

æ ö÷ç ÷ç - +÷ç ÷ç ÷è øæ ö æ ö æ ö÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç+ + + - + - - -÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷è ø è ø è ø

(17)

+ - + - - - -

- +- + + = + + +

1

0

t t t t *c *ct t t t, x ,x z z x ,x x ,x , ,2 3 2 3

t t c

*c *c t t1 2 3

t tt t t tz z z z 1 2 3 42 3 2 3

:

h 2h h 2h1 1M M Q S P H P H

R R h h R h R hh 2h 2h 6h

R H S R I I I IR h R h h h

c c

c c c c c

c c t t t t

c c c c c c

t

v v v v

v

qq q q q q q q qq q qq q

q q q q

(18)



æ æ öö- ÷÷ç ç ÷÷ç ç- + + - + - - ÷÷ç ç ÷÷ç ç ÷÷è è øøæ ö æ ö- -÷ ÷ç ç÷ ÷ç ç- + + + +÷ ÷ç ç÷ ÷ç ç÷ ÷è ø è ø

+

t 2 t 3 t 4 t 5 t 4 5 632 3 4 5 2 2 3

cc

t 3 t 4 t 5 t 6 t 4 t 62 3 4 5 4 6c c

0

1 0

h I 2h I 4h I 8h I h 2I 4I 8II

hh h h h R h h h

h I 2h I 4h I 8h I 2h I 8h I

h h h h h h

c c c c c c cc c

cc c c c c c cc c c c c

c

c c c c

v

v væ ö- ÷ç ÷ç - - ÷ç ÷ç ÷è ø

æ ö æ ö æ ö- -÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç+ + + + + + + +÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷è ø è ø è ø-

-

t 4 t 5 t 64 5 6

2 2 2 2 2b t 4 b t 6 t 4 t 5 t 6 t b 4 t b 6

4 6 4 5 6 4 6

3t 4

0

1

c

2

4

1

2h I 8h I 8h I

h h h

h h I 4h h I h I 4h I 4h I h h I 2h h I

h h h h h 2h h

h I 2

2h

c c cb t

c c cc c c c c c c

b t b

c c c c c c cc

v

v v v

æ ö æ ö æ ö-÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç- + + + + +÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷è ø è ø è ø

3 3 3 3 4 4 4t 5 t 6 t b 4 t b 6 t 4 t 5 t 65 6 4 6 4 5 62 3 3

h I 2h I h h I h h I h I h I h I

h h 4h h 4h h h

c c c c c c ct b t

c c c c c c c

v v v

22 2 2 2

, , , , , ,2 2

2 2 2 2

0 1 22 3 42 2

2 2 21 2

2

1 12

4 42 2

4 42

4 2

:

t t t t *c *c *c *ct t t tzz xz x z xz x xz x z z

t t cc c

t t tt

c c c

c c c c t t t

c c cc c

t t t

c

zz zzc

ct

c c

t t

w

h h h hP M R R S R S R

R R h R hh R h

h h h hM P N M I I I

R h hR h h

h I h

w w

I

h

w

I h

h

qq q q q q q q

qq qq

d

- - + + + + + +

- - - - + +

-

+=

+

t

2 2 2 2 2 23 4 2 4 2 3 4

03 4 2 4 2 3 4

2 2 3 3 32 4 2 3

0

42 4 2 3

0

1

2

4 4

8 2 8 2 2

c c c c c cc b t

c c

ct t t t t t

cc

b t b t t t t

c c c cc c c c

b

c c c c

h I h I h I h I h I h I

h h h h h h h

h h I h h I h

w w

I h I h I

h h hw

h

w

h

æ ö æ ö æ ö-÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç- + + + + +÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷è ø è ø è øæ ö- -÷ç ÷ç + + - -÷ç ÷ç

+

÷è ø

2 2 2 2

2 44 2 4

4 4 42 3 42 3

1 2

24

16 4

16 4 4

cb t b t

cc

t t t

ct b

c cc c

t

c c c

h h I h h I

h h

h I h I h

w

I

h

w

hw

h

æ ö æ ö-÷ ÷ç ç÷ ÷ç ç+ +÷ ÷ç ç÷ ÷ç ç÷ ÷è ø è øæ ö÷ç ÷ç + + ÷ç ÷ç ÷è ø

+

(19)

at the bottom face sheet:

1

b b b b b b bxx,x x , xz xx,x xx,x x, x,2 3 2 3

b

b b b bb b b 2 b 3 b 4 b 5xz xz 1 2 3 42 3 2 3 4 5

c0 1 2 3

c c0

b

:

h 2h h 2h1M M Q P H P H

R h h R h R h

2h 6h h I 2h I 4h I 8h IS R I I I I

h h h h h h

h I

b

c c c c

c c c c c cc c c c

c c b b b b c

c c c

u u u u

u

u

q q q q q q

d

+ - + - + -

æ ö÷ç ÷ç- + = + + + + - - + ÷ç ÷ç ÷è ø

+

+

c c c c3 b 4 b 5 b 6 b 4 b 5 b 6

2 3 4 5 4 4 6c cc c 2 2 2 c

b 4 b 6 b 4 b 5 b 6 t b 44

1

6 4 5 6c

0

0 1

2h I 4h I 8h I 2h I 8h I 8h I

h h h h h h h

2h I 8h I h I 4h I 4h I h h I

h h h h h h

c c cc b

c c c c cc c c

t b

c c c c c

u u

u u

æ ö æ ö÷ ÷ç ç÷ ÷ç ç- - + - +÷ ÷ç ç÷ ÷ç ç÷ ÷è ø è øæ ö æ ö÷ ÷ç ç÷ ÷ç ç- + - + -÷ ÷ç ç÷ ÷ç ç÷ ÷è ø è

+ +

+ø

t b 64 6

3

1

2 2

3 3 2 2 4 4 4b 4 b 5 b 6 b t 4 b t 6 b 4 b 5 b 6

4 5 6 4 6 4 5 6c

3 3b t 4 b 6

4

3

t

4h h I

h

h I 2h I 2h I h h I 2h h I h I h I h I

2h h h 2h h 4h h h

h h I h h I

4h h

ct

cc c c c c c c c

b t b

c c c c c c cc c

c

u

u u u

æ ö÷ç ÷ç + +÷ç ÷ç ÷è øæ ö æ ö æ ö÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç+ - + - - +÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷è ø è ø è ø

+

-

+

+

+

36t

c

uæ ö÷ç ÷ç ÷ç ÷ç ÷è ø

(20)



22 2 2 2

b b b b *c *cb b b b, x ,x z z x ,x x ,x , ,2 3 2 3

b b c2 2 2 2

*c *c c b b b bb b b bz z z z 2 3 4 52 3 2 3

c2 2b 2

1

b

0 2 3

32

:

h h h h1 1P P 2S R P H P H

R R 2h h 2R h R h

h h h 3hR H S R I I I I

2R h R h h h

h I h I

2h

c c

c c c c c

c b b b b

c

b

c c c c c

c

v v v v

v

qq q q q q q q qq q qq q

q q q q

+ - + + - + -

- - + = + ++ +

-

+

2 2 2 c cb 4 b 5 b 4 5 6

33 4 5 2 2 3c

2 2 2 2 c 2 2b 3 b 4 b 6 b 4 b

0

5 b 653 3 4 41 5 6

2h I 4h I h 2I 4I 8II

hh h h 2R h h h

h I h I 2h 2I h I 4h I 4h II

h2h h R h h h h

c c c cc c

cc c c c c cc c c c c

c c

cc c c c c c c

v

v

æ æ öö÷÷ç ç ÷÷ç ç- + + - - + ÷÷ç ç ÷÷ç ç ÷÷è è øøæ æ öö æ÷÷ç ç ç÷÷ç ç- - - + - +÷÷ç ç ÷÷ç ç ÷÷è è øø è

+

2 2 cb 4 b 64 6

3 3 3 c 2 2 4 4 4b 4 b 5 b 6 t b 4 t b 6

0 0

1 1b 4 b 5 b 6

4 5 6 4 6 4 5 6 2

h I 4h I

h h

h I 2h I 2h I h h I 2h h I h I h I h I

2h h h 2h h 4h h h

cb t

c cc c c c c c c

b t b

c c c c c c c c

v v

v v v

ö æ ö÷ ÷ç÷ ÷ç ç+ -÷ ÷ç ç÷ ÷ç ç÷ ÷ø è øæ ö æ ö æ ö-÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç- + + + + - +÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷è ø è ø è ø

+

+

2 2 2 2 5 5 c 5 2 3 c 2 3 cb t 4 b t 6 b 4 b 5 b 6 b t 4 b t 6

4 6 4 5 62 43 36

h h I h h I h I h I h I h h I h h I

4h h 8h 2h 2h 8h 2h

c c c ct b t

c c c c c c c

v v væ ö æ ö æ ö-÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç- + - + + +÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷è ø è ø è ø

(21)

22 2 2 2

b b b b *c *c *c *cb b b bzz xz,x z, xz,x xz,x z, z,2 2

b b c c cc c c2 2 2 2 2 c 2 c

c c c c b b bb b b b b 2 b 4zz zz 2 3 42 2 2 4

c c cc c c c c0 1 2

:

h h h h1 1P 2M R R S R S R

R R 4h 4R h2h 2R h

h h h h h I h IM P N M I I I

4R h 4h2R h h 4h hb b bw

w

w w

qq q q q q q q

qq qq

d

- - + + - + - +

æ-ç+ - + - = + + +è

+

+

b

2 c 2 c 2 c 2 c 2 c 2 c 2 c 3 c 3 c 3 cb 1 b 2 b 3 b 4 b 2 b 3 b 4 b 2 b 3 b 4

2 3 4 2 3 4 2 3 4c c c c c c c c c c

0

0 0 1

h I h I h I 2h I h I h I h I h I h I h I

4h 2h h h 4h h h 8h 2h 2h

t

c b b

w

w w w

ö÷÷ç ÷ç ÷ç ÷øæ ö æ ö æ ö- ÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç+ + - + - + + - +÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷è ø

+ +ø ø

+

è è

2 c 2 c 4 c 4 c 4 c 2 2 c 2 2 ct b 2 t b 4 b 2 b 3 b 4 b t 2 b t 4

2 4 2 3 4 2 4c c c c c c c

1 2 2

h h I h h I h I h I h I h h I h h I

8h 2h 16h 4h 4h 16h 4ht b tw w w

æ ö æ ö æ ö-÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç+ - - + + +÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷è ø è ø+

è ø

(22)

And at the core:

0c c

c c c c c c 2 4xx,x xx,x x, x, xz 02 2 2 2 4

cc c c c c cc c c c c c c c

c 3 5 2 3 4 5 2 31 2 4 2 3 4 5 2 3

c c c c c c c

0

1 0c

8I 16I4 1 4 8N P N P S I

Rh R h h h h

12I 32I 2I 4I 8I 16I 2I 4II

h h h h h h h h

:

c

c b

u

u

u

u

q q q q

dæ ö÷ç ÷ç- + - + = - + ÷ç ÷ç ÷è ø

æ ö æ ö÷ ÷ç ç÷ ÷ç ç- + + - - + +

+

+÷ ÷ç ç÷ ÷ç ç÷ ÷è ø è ø

c

c c4 54 5c c

c c c c c c c cb 2 b 3 b 4 b 5 t 2 t 3 t 4 t 52 3 4 5 2 3 4 5c c c c c c c c

2 c 2 c 2 c 2 cb 2 b 3 b 4 b

0

1 1

52 3 4 5c c c c

8I 16I

h h

h I 2h I 4h I 8h I h I 2h I 4h I 8h I

h h h h h h h h

h I h I 2h I 4h I

2h h h h

t

b t

u

u u

æ ö÷ç ÷ç - - ÷ç ÷ç ÷è øæ ö æ ö-÷ ÷ç ç÷ ÷ç ç- + + + - + +÷ ÷ç ç÷ ÷ç ç÷ ÷è ø è ø

+

+

æ öçç - - +ççè

2 c 2 c 2 c 2 ct 2 t 3 t 4 t 5

2 3 4 5c c c c

3 c 3 c 3 c 3 c 3 c 3 c 3 c 3 cb 2 b 3 b 4 b 5 t 2 t 3 t 4 t 5

2 3 4 5 2 3 4 5c c c c c

2

c c c

2

3 3

h I h I 2h I 4h I

2h h h h

h I h I h I 2h I h I h I h I 2h I

4h 2h h h 4h 2h h h

b t

b t

u u

u u

æ ö÷ ÷ç÷ ÷ç+ + - -÷ ÷ç÷ ÷ç÷ ÷ø è øæ ö æ ö-÷ ÷ç ç÷ ÷ç ç- - + + - + +÷ ÷ç ç÷ ÷ç ç÷

+

÷è ø è ø

(23)



1

*c *c c c *c *c c cx ,x x ,x , , z z z z2 2 2 2

c cc c c c c cc c c c c c c

c c C2 3 5 5 6 4 61 4 22 04 2 2 2 2 4

c c c c c c c c c

4 1 4 1 4 12M H M H S H Q R

R Rh R h R h h

I 8I 16I 2I 2I 8I 16I8I I I

R h h R h h h h h

:

cv

v

q q qq q qq q q q q q

d

- + - + - - + =

é æ öù æ ö÷ ÷ç çê ú÷ ÷ç ç+ - + - - - - +÷ ÷ç çê ú÷ ÷ç ç÷ ÷è ø è øê ú+

ë û

c

c c c c c c c c3 4 5 6 3 4 5 62 3 4 5 2 3 4 5c c c c c c c c

c c c c c c c cb 3 b 4 b 5 b 6 t 3 t 4 t 5 t 62 3 4 5 2 3 4 5c c c c c c c c

1

0 0

1

2I 4I 8I 16I 2I 4I 8I 16I

h h h h h h h h

h I 2h I 4h I 8h I h I 2h I 4h I 8h I

h h h h h h h h

c

b t

b

v

v v

v

+

é ù é ùê ú ê ú- - + + + - -ê ú ê úê ú ê úë û ë ûæ ö æ-÷ç ç÷ç ç- - + + - + +÷ç ÷ç ÷

+

è ø è

2 c 2 c 2 c 2 c 2 c 2 c 2 c 2 cb 3 b 4 b 5 b 6 t 3 t 4 t 5 t 6

2 3 4 5 2 3 4 5c c c c c c c c

3 c 3 c 3 c 3 cb 3 b 4 b 5 b 6

2 3 4 5c c c c

1

2 2

3

h I h I 2h I 4h I h I h I 2h I 4h I

2h h h h 2h h h h

h I h I h I 2h I h

4h 2h h h

t

b t

b

v

v v

v

ö÷÷÷ç ÷ç ÷øæ ö æ ö÷ ÷ç ç÷ ÷ç ç- + + + + - -÷ ÷ç ç÷ ÷ç ç÷ ÷è ø è øæ ö -÷ç ÷ç - - + +÷ç ÷ç ÷ø

+

+

è

3 c 3 c 3 c 3 ct 3 t 4 t 5 t 62 3 4 5c c c c

3

I h I h I 2h I

4h 2h h htv

æ ö÷ç ÷ç - + + ÷ç ÷ç ÷è ø

(24)

0

*c *c *c *c c c cxz,x xz,x z, z, zz2 2 2 2

c cc c c c c cc c c c c c c c c c

c 2 4 1 2 3 4 1 2 3 40 2 4 2 3 4 2 3 4

c cc c c c c c0

c0

c

:

4 1 4 1 4 8Q R Q R N P M

R Rh R h R h h

8I 16I I 2I 4I 8I I 2I 4I 8II

h hh h h h h h h hc

c

bw w

w

q q q q qq qq- + - - + + =

æ ö æ ö æ-÷ ÷ç ç ç÷ ÷ç ç ç- + + + - + + - -÷ ÷ç ç ç÷ ÷ç ç÷ ÷è ø è ø è+

0tw

ö÷÷÷÷ç ÷+

ø

c c c c c c c cb 1 b 2 b 3 b 4 t 1 t 2 t 3 t 4

2 3 4 2 3 4c cc c c c c c

2 c 2 c 2 c 2 c 2 c 2 c 2b 1 b 2 b 3 b 4 t 1 t 2 t

2 3 4 2c cc c c c

1 1

2

h I h I 2h I 4h I h I h I 2h I 4h I

2h 2hh h h h h h

h I h I h I 2h I h I h I h I

4h 4h2h h h 2h

b t

b

w w

w

æ ö æ ö- -÷ ÷ç ç÷ ÷ç ç+ + - + - + +÷ ÷ç ç÷ ÷ç ç÷ ÷è ø è øæ ö- ÷ç ÷ç + + - + + -÷ç ÷ç ÷è ø

+

c 2 c3 t 4

3 4c

2c

2h I

h htw

æ ö÷ç ÷ç - ÷ç ÷ç ÷è ø

(25)

where the inertias are given by

( )m n ci i c c c

ci

zI z I z

R

i t b m n

2 2

2 2

, 1

, 0,1, ,6 , 0,1, ,6

r r- -

æ ö÷ç ÷= = +ç ÷ç ÷çè ø

= = ¼ = ¼

ò ò

i i

i i

h h

i cm n

h h

dz dz

,

(26)

The stress resultants of the face sheets in Equations (17)-(26) can be related to the total strains by the following equations. For each face sheet (i = t, b):



0 1 2 3,11 ,11 ,11 ,11 0,1 2 3 4,11 ,11 ,11 ,11 1,2 3 4 5,11 ,11 ,11 ,11 2,3 4 5 6,11 ,11 ,11 ,11 3,

i ixx i i i i xi ixx i i i i x

iii i i i xxx

iii i i i xxx

N K K K K u

M K K K K u

K K K K uP

K K K K uH

ì ü é ù ì üï ï ïï ï ïê úï ï ïê úï ï ïï ï ïê úï ï ï= ê úí ý í ýï ï ïê úï ï ïï ï ïê úï ï ïê úï ï ïï ï ïê úë û îî þ

0, 0

0 1 2 3,12 ,12 ,12 ,12 1, 11 2 3 4,12 ,12 ,12 ,122 3 4 5,12 ,12 ,12 ,12 2, 23 4 5 6,12 ,12 ,12 ,12

3,

i i

i ii i

i i i i

i i i i i ii i

i i i i

i i i i i ii

i

v w

R RK K K K v wK K K K R R

K K K K v w

K K K K R R

v

R

q

q

q

q

ìïïï +ïïïïé ùï ïï ïê úï ï +ê úï ïï ïê úï ï+ ê ú íï ïê úï ïï ïê ú +ï ïê úï ïï ïê úþ ë û ïï

î0 1,13 ,13 11 2,13 ,13 22 3,13 ,133 4,13 ,13

0 0

0 0 2,

0 0 0

00 0

iii i i

iii i

ii i

ii i

NK K Kw

K K Mw

K K P

K K H

qq

qq

qq

qq

üïïïïïïïïïïïïï +ýïïïïïïïïï ïï ïï ïï ïï ïþì üé ù ì ü ï ïï ï ï ïê ú ï ï ï ïï ïê ú ï ïï ï ï ïï ïê ú ï ïï ï =ê ú í ý í ýï ï ï ïê ú ï ï ï ïï ï ï ïê ú ï ï ï ïê ú ï ï ï ïï ï ï ïê ú î þë û î þ

0 1 2 3,12 ,12 ,12 ,11 0,1 2 3 4,12 ,12 ,12 ,12 1,2 3 4 5,12 ,12 ,12 ,12 2,3 4 5 6,12 ,12 ,12 ,12 3,

0 1 2 3,22 ,22 ,22 ,221,22 ,22

ii i i x

ii i i i x

ii i i i x

ii i i i x

i i i i

i i

K K K u

K K K K u

K K K K u

K K K K u

K K K K

K K

é ù ì üï ïï ïê ú ï ïê ú ï ïï ïê ú ï ï +ê ú í ýï ïê ú ï ïï ïê ú ï ïê ú ï ïï ïê úë û î þ0, 0

0 1,23 ,231, 1

2 3 4,22 ,22

2 3 4 5,22 ,22 ,22 ,22 2, 23 4 5 6,22 ,22 ,12 ,22

3,

0 0

i i

i ii i

i i

i i i ii i

i i i i

i i i i i ii

i

v w

R RK Kv w

K K R R

K K K K v w

K K K K R R

v

R

q

q

q

q

ì üï ïï ïï ï+ï ïï ïï ïï ïé ù ï ïï ïê ú ï ï+ê ú ï ïï ïê ú ï ï +ê ú í ýï ïê ú ï ïï ïê ú +ï ïê ú ï ïï ïê úë û ï ïï ïï ïï ïï ïï ïï ïî þ

11 2,23 ,23 22 3,23 ,233 4,23 ,23

0 0 2,

0 0 0

00 0

i

ii i

i i

i i

w

K K w

K K

K K

é ù ì üï ïê ú ï ïï ïê ú ï ïï ïê ú ï ïê ú í ýï ïê ú ï ïï ïê ú ï ïê ú ï ïï ïê ú î þë û

ì üï ïï ïï ïï ïì ü é ù é ùï ï ï ïï ï ï ïê ú ê úï ï = +í ý í ýê ú ê úï ï ï ïê ú ê úï ï ï ïï ïî þ ë û ë ûï ïï ïï ïï ïî þ

i

ix

i izz i i i i x i i i ii izz i i i i x i i i i

ix

v

Ru

N K K K K u K K K K

M K K K K u K K K K

u

0,

0,0 1 2 3 0 1 2 3,13 ,13 ,13 ,13 1, ,23 ,23 ,23 ,231 2 3 4 1 2 3 4,13 ,13 ,13 ,13 2, ,23 ,23 ,23 ,23

3,

qì üï ïï ïï ï+ï ïï ïï ïï ïï ïï ïï ï+ï ïï ïï ï +í ýï ïï ïï ï+ï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïî þ

ìì ü ïï ï ïï ï ïï ï ïï ïé ù ïï ïê ú ï ïí ý íê ú ï ïê ú ï ïë û ï ïï ïï ïï ïî þ

i

i ii i

i ii i

i ii

i

iixii

i i xi

i i xix

w

R

v w

R R

v w

R R

v

R

Nw

K K Mw

K K P

H

0

1, 1

2, 2

3,

10 1,33 ,33 21 2,33 ,33

0 0 2,

0 0 0

0

q

q

q

q

q

q

q

ìï+

ü é ùïï ê úï +ê úïï ê úï ï = ê úý íï ï ê úï ïï ï ê ú +ï ï ê úï ïï ï ê úî þ ë û

+

iix

ii

i i i i ix

i i i i ii

i i i i ix

i i i i ii

ix

i

uv

RK K K K u

vK K K K R

K K K K uv

K K K K R

uv

R

0,0,

0 1 2 3,44 ,44 ,44 ,44 1,1 2 3 4 1,,44 ,44 ,44 ,442 3 4 5,,44 ,44 ,44 ,44 2,

2,3 4 5 6,,44 ,,44 ,,44 ,,44

3,3,

q

q

q

q

üïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïýï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïî þ

,

(27)



ìì ü ïï ï+ ïï ïì ü é ùï ï ïï ïï ï ïï ïê úï ï ïï ï+ê úï ï ïï ïï ï =í ý í ý íê úï ï ï ï ï+ê úï ï ï ï ïï ï ï ï ïê úï ï ï ï ïï ïî þ ë û ï ïï ïî þ î

ii izxi

xz i i i ii izxi

xz i i i ii izxi

xz i i i iz

Qw uQ K K K

Sw uS K K K

Rw uR K K K

V

0, 10 1 2,55 ,55 ,55

1, 21 2 3,55 ,55 ,55

2, 32 3 4,55 ,55 ,55

02

0 ,3

00

q

q

q

q

üïïïïïï =ýïïïïï ïï ïþìïïï - +ïï

é ùê ú

+ -ê úê úê ú íê úê ú + -ê úê úë û

-

i ii

i ii ii i i i i

i i i i i ii i

i i i i i

i i i i i ii

i

w vv

R RK K K K w v

vK K K K R RK K K K w v

vK K K K R R

v

R

0, 01

0 1 2 3,66 ,66 ,66 ,66 1, 11 2 3 4 2,66 ,66 ,66 ,662 3 4 5,66 ,66 ,66 ,66 2, 23 4 5 6 3,66 ,66 ,66 ,66

3

2

3

q

q

q

üïïïïïï ïï ïï ïï ïï ïï ïï ïï ïýï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïî þ

The coefficients ,ni jkK can be defined as:

1

,1

lii

li

zln l ni jk jk i i

l z

K Q z dz-=

= òå (28)

and for the core:

é ùì üï ï ê úï ïï ï êï ïï ï êï ï ê=í ý êï ïï ï êï ïï ï êï ïï ï êî þ ë û

c xx c xx c xx c xx c xxcxx

c xx c xx c xx c xx c xxcxxc c xx c xx c xx c xx c xxxxc c xx c xx c xx c xx c xxxx

e e e e eN

M e e e e e

P e e e e e

H e e e e e

( ) ( ) ( ) ( ) ( )0 1 2 3 4( ) ( ) ( ) ( ) ( )

1 2 3 4 5( ) ( ) ( ) ( ) ( )

2 3 4 5 6( ) ( ) ( ) ( ) ( )

3 4 5 6 7

ì üï ïï ïï ïï ïï ïï ïï ï+ï ïï ïï ïï ïï ïú ï ïï ïú +ï ïú í ýú ï ïï ïú ï ïï ïú ï ï+ú ï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïî þ

ì üï ïï ïï ïï ïï ïï ï =í ýï ïï ïï ïï ïï ïï ïî þ

cx

cx c

xccx c

xccx c

xccx

c

c c c cc

c

c

c

u

uu

R

uu

R

uu

R

u

R

e e e eN

M

P

H

0,

0,1,

1,2,

2,3,

3,

( ) ( ) ( )) (0 1 2 3

,

qq qq qq qqqq

qq

qq

qq

ì üï ïï ïï ï+ï ïï ïï ïïé ù ïê ú ïï + +ê ú ïïê ú ïïê ú í ýê ú ïïê ú ï +ïê ú ïê ú ïë û ïïïïïïïïî

c c

c cc c c

c c c cc cc

c c c c c c

c c c cc cc

c

v w

R R

v v we e e e R RR

ve e e e w

R Re e e ev

R

0, 0

)

0, 1, 1( ) ( ) ( ) ( )) 2

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

2,2 3 4 5 2( ) ( ) ( ) ( )

3 4 5 6

3,

q

q qqq qq qq qq

qq qq qq qqq

qq qq qq qq

q

ïïïïï é ùì ü ì üï ï ï ïï ï ï ï ïï ê úï ï ï ï ï=í ý í ýê úï ï ï ï ïê úï ï ï ï ïï ï ï ïî þ î þë ûïïïïïïïïïïïïþ

c zz c zzc czzc c zz c zz czz

N e e w

M we e

( ) ( )0 1 1( ) ( )

21 2

, ,2

(29)



éì üï ï êï ïï ï êï ïï ï êï ï ê=í ý êï ïï ï êï ïï ï êï ïï ïî þ ë

c x c x c x c x c xcx

c x c x c x c x c xcxc c x c x c x c x c xxc c x c x c x c x c xx

H H H H HN

M H H H H H

P H H H H H

H H H H H H

( ) ( ) ( ) ( ) ( )*0 1 2 3 4( ) ( ) ( ) ( ) ( )*

1 2 3 4 5* ( ) ( ) ( ) ( ) ( )

2 3 4 5 6* ( ) ( ) ( ) ( ) ( )

3 4 5 6 7

q q q q qq

q q q q qq

q q q q qq

q q q q qq

ì üï ïï ïï ïï ïï ïï ïï ï+ï ïï ïù ï ïú ï ïï ïú ï ïï ïú + +ï ïú +í ýú ï ïï ïú ï ïï ïú ï ï+ê ú ï ïû ï ïï ïï ïï ïï ïï ïï ïï ïï ïî þ

cx

cx c

xc

c cx x c

xcc

cx c

xccx

c

c x c x c x c x

c x c x c x c x

c x

v

vv

R

v vv

RR

vv

R

v

R

g g g g

g g g g

g

0,

0,1,

0, 1,2,2

2,3,

3,

( ) ( ) ( ) ( )0 1 2 3( ) ( ) ( ) ( )

1 2 3 4(

2

2

q q q q

q q q q

ì üï ïï ïï ïï ïï ïï ïï ïé ù ì üï ïï ïê ú ï ïï ï ï ïï ïê ú ï ïï ï ï ïï ïê ú ï ïï ïê ú í ý í ýê ú ï ï ï ïï ï ï ïê ú ï ï ïï ï ïê ú ï ï ïê ú ï ï ïî þë û ï ïï ïï ïï ïï ïï ïï ïî þ

c

cccxcxc

c x c x c x c cxcc x c x c x c xxc

c

c

u

RNu

MR

u Pg g g

HRg g g gu

R

0,

1,

) ( ) ( ) ( )2,3 4 5

( ) ( ) ( ) ( )3 4 5 6

3,

,

q

qq

qq q q q

q qq q q q

q

q

ì üï ïï ïï ïï ïïïïé ù ïê ú ïïê ú ïïê ú ïê ú= í ýê ú ïïê úï ïï ïê úï ïï ê ú ïë û ïïïïïïïî

c

cc x c x c x c x c

c x c x c x c x

cc x c x c x c x c

c x c x c x c xcc

c

u

RH H H H u

H H H H R

uH H H H

RH H H Hu

R

0,

( ) ( ) ( ) ( )0 1 2 3 1,( ) ( ) ( ) ( )

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

2,2 3 4 5( ) ( ) ( ) ( )

3 4 5 6

3,

q

q q q qq

q q q q

q q q qq

q q q q

q

ïïïïïïïïï +ïïïïïïïïïïïïïþ

ì üï ïé ù ï ïï ïê ú ï ïï ïê ú ï ïê ú ï ï+ï ïê ú í ýê ú ï ïï ïê ú ï ïï ïê ú ï ïê ú ï ïë û ï ïï ïî þ

cc x c x c x c x x

cc x c x c x c x x c

xc x c x c x c x c

cxc x c x c x c xcx

vg g g g

vg g g g v

Rg g g g

vg g g g

v

Q( ) ( ) ( ) ( ) 0,

0 1 2 3( ) ( ) ( ) ( ) 0,

1 2 3 4 1,( ) ( ) ( ) ( )

2 3 4 52,( ) ( ) ( ) ( )

3 4 5 63,

,

q q q q

q q q q

q q q q

q q q q

ìïïïïïïï +é ùì ü ïï ï ïê úï ï ïï ï ê ú ïï ïï ï ï= ê úí ý íï ï ïê ú +ï ï ïê úï ï ïï ï ïê úï ïî þ ë û ïï

î

cxcx cc xz c xz c xz c xzc x

xz cc xz c xz c xz c xzc c

xz x cc c xz c xz c xz c xz xxz c

cx

c

wH H H H RH H H H

w

w

Sw

H H H

w

H

R

R

w

R

0,

0,( ) ( ) ( ) ( )* 1,

0 1( ) ( ) ( ) ( )*

2 3 4 1,* ( ) ( ) ( ) ( ) 2,

2 3 4

,

5

2

2 3

1

üïïïïïïïïïïïï +ýïïïïïïï ïï ïï ïï ïï ïï ïþ

ì üï ïï ïï ïï ïï ïï ï+é ù ï ïï ïê ú ïïê ú ï+ ê ú í ýïê ú +ïê ú ïïê úë û ïïïïïïïî

c

ccc xz c xz c xz c xz

cc xz c xz c xz c xz c

cc xz c xz c xz c xz

cc

c

u

uug g g g

g g g g uu

g g g

R

R

g R

u

1

1( ) ( ) ( ) ( )0 1( ) ( ) ( ) ( )

2 3 43( ) ( ) ( ) ( )

2 3 4

3

22 3

1 2

5

32

3

2

ìïïïïï+é ùì üï ï ê úï ïï ï ïï ê úï ïï ï ï = ê úí ý íï ï ï ê ú +ï ï ï ê úï ï ïï ï ï ê úï ïî þ ë ûïïïïïïïþ

cxcx cc xz c xz c xz c xzc x

xz cc xz c xz c xz c xzc c

xz x cc c xz c xz c xz c xz xxz c

cx

c

w

wg g g gQ

S g g g g w

R g g g g

w

wR

wR

R

0,

0,( ) ( ) ( ) ( ) 1,

0 1( ) ( ) ( ) ( )

2 3 4 1,( ) ( ) ( ) ( ) 2,

2 3 4

,

5

2

2 3

1,

üïïïïïï ïï ïï ïï ïï ïï ïï ï +ýï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïî þ



ì üï ïï ïé ù ì üï ï ï ïê ú ï ï ï ïï ï ï ïê ú ï ï ï ïï ï =ê ú í ý í ýï ï ï ïê ú ï ï ï ïê ú ï ï ï ïï ï ï ïï ïê ú î þë û ï ïï ïî þ

c z ccc xz c xz c xz c xz zc

czc xz c xz c xz c xz cczc xz c xz c xz c xz ccz

uH HQ

F F F FS

F F F FR

F F F F

u

uH

( ) (*0 1( ) ( ) ( ) ( )

0 1 2 3 1 *( ) ( ) ( ) ( )

1 2 3 4 *( ) ( ) ( ) ( )

2 3 4 5 *

2

3

2 ,

3

qq

q

q

q

( )

ìïïï -ïïïé ù ïïê ú ïïê ú - -ïïê ú ïê ú íê ú ïïê ú ï -ïê ú ïê ú ïë û ïïïï -ïïïî

c c

c cz c z c zc c c

c z c z c z c z

c ccc xc z c z c z c c

c z c z c z c zc c

c

c

w v

R RH H w v v

H H H H R RR

wH H H H v

R RH H H Hv

R

0, 0

) ( ) ( )2 3 1, 0 1

( ) ( ) ( ) ( )21 2 3 4

( ) ( ) ( )2,2 3 4 5 2

( ) ( ) ( ) ( )3 4 5 6

3

q

q q q

qq q q q

qq q qq

q q q q

üïïïïïïïïïïïïï +ýïïïïïïïïïïïïïþ

ì üïé ù ï +ïê ú ïê ú ïïê ú ïïê ú íê ú ïïê ú ïïê ú ïê ú ïë û ïïî

cc z c z c z c z c

c z c z c z c z cc

c z c z c z c zc

c z c z c z c z

vg g g g vR

g g g gv

g g g gv

g g g g

( ) ( ) ( ) ( ) 00 1 2 3 1( ) ( ) ( ) ( )

1 2 3 42( ) ( ) ( ) ( )

2 3 4 53( ) ( ) ( ) ( )

3 4 5 6

2

3

0

q q q q

q q q q

q q q q

q q q q

-ïïï é ùì üï ï ïï - -ê úï ïï ï ï ê úï ï ïï ï ï = ê úý í ýï ï ï ê úï ï ï ê úï ï ï -ï ï ïï ï ê úî þ ë ûïïïïþ

c c

c cc c cc z c z c z c zc

zc z c z c z c zc c cc

z cc c z c z c z c zz

c

w v

R R

w v vg g g gQR RRS g g g gw vR g g g gR

0, 0

( ) ( ) ( ) ( ) 1, 0 10 1 2 3

2( ) ( ) ( ) ( )

1 2 3 4( ) ( ) ( ) ( ) 2,

2 3 4 5

,

q

q q q q qq

q q q qq

q q q q qq

ì üï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ï +í ýï ïï ïï ïï ïï ïï ïï ïï ïï ïï ï-ï ïï ïï ïî þ

c

cc

c

R

v

R

2

3

0( z) ( z) ( z) ( z) 1

0 1 2 3( z) ( z) ( z) ( z)

21 2 3 4( z) ( z) ( z) ( z)

32 3 4 5

2

3

0

cc

c c c c

cc c c c c

cc c c c

vv

F F F F R

vF F F F

vF F F F

q q q q

q q q q

q q q q

ì üï ïï ï+ï ïé ù ï ïï ïê ú ï ïê ú ï ïï ïê ú í ýï ïê ú ï ïê ú ï ïï ïê úë û ï ïï ïï ïï ïî þ

The coefficients in Equation (29) can be defined as:

( )

- - -

- -

æ ö÷ç ÷= = = +ç ÷ç ÷æ ö çè ø÷ç ÷+ç ÷ç ÷çè ø

= =

ò ò ò

ò ò

2 2 2( ) ( ) ( ) c

c c

2 2 2c

2 2( ) (

2 2

( )( ) ,

z. , ( ) 1

z R1

R

( ) ,

,

,( )

c c c

c c c

c c

c c

h h h

cc xx c n c n c zz c nn xx c c n c c n zz c c

h h h

h h

c x c n c x n c cn x c c n c x c n

h h

Ee E z dz e z dz e E z z dz

H G z z dz g z G z d H

z

z

z qq qq

q q qq q

-

- - -

=æ ö÷ç ÷+ç ÷ç ÷çè ø

= = =

ò

ò ò ò

2)

c

2 c

2 2 2( ) ( ) ( )

2 2 2

( )

z1

R

( ) ( ) , ( ),

c

c

c c c

c c c

h

cx nx

c c

h

h h h

c x c n c xz c n c xz c nn x c c n xz c c n xz c c

h h h

G zz dz

g G z z dz H G z z dz g G z z dz

q

qq

(30)



- -

-

æ ö÷ç ÷= + =ç ÷ç ÷çè ø

=æ ö÷ç ÷+ç ÷ç ÷çè ø

ò ò

ò

2 2( ) ( )

2 2

2( )

c

2 c

( ) 1 , ( )

( )

, z1

R

c c

c c

c

c

h h

c xz c n c z n ccn xz c c n c z c

ch h

h

cc z nzn c c c

nh

zF G z z dz g z G z dz

R

G zH z dz

F

qq

q q

-

æ ö÷ç ÷= +ç ÷ç ÷çè øò2

( )

2

( ) 1

c

c

h

z c n cz c c

ch

zG z z dz

Rq

q

Based on simply supported boundary conditions, the essential and natural boundary conditions for the cylindrical sandwich panel obtained from Hamilton's principle are as follows:

0 1 2 3

0 1 2

0 1 2 3 0 1 2

0

0 , 0 , 0 , 0

0 , 0 , 0

0 , 0 , 0 , 0 , ,

0 , 0 , 0 , 0 , 0 , 0 , 0

i i i ixx xx x

i i i i

i i i

c c c c cxx

c cx

at x or L

v v v v

w w w

N M P H i t b

v v v v w w w

=

= = = =

= = =

= = = = =

= = = = = = =

(31)

0 1 2 3

0 1 2 3

0 2

0 , 0 , 0 , 0 , ,

0 , 0 , 0 , 0

i i i i

c c c c

at y or

v v v v i t b

v v v v

=

= = = = =

= = = =

(32)

4 SOLUTION

In order to solve the free vibration problem of simply supported cylindrical sandwich panel, the Navier method is applied (Qatu, 2004). For satisfing the boundary conditions, the displacement fields based on double Fourier series are assumed to be in the following form:

( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )

j j mni ic c mn

c c mn

j j mni ic c mn

c c mn

j j mnl lc c mn

u u x n

u u x n

u u x n

v v x n

v v x n

v v x n

w w x n

w w x n

,

,0 0

,1 1

,

,0 0

,1 1

,

,0 0

cos cos

cos cos

cos cos

sin sin

sin sin

sin sin

sin cos

sin cos

q

q

q

q

q

q

q

q

é ù éê ú êê ú êê ú êê ú êê ú êê ú êê ú êê ú =ê úê úê úê úê úê úê úê úê úë û ë

n m

mi l j t b

l

0 1

0,1,2,3; 0,1,2; , ;

¥ ¥

= =

ùúúúúúúú

ê úê úê úê úê úê úê úê úê úê úû

æ ö÷ç ÷= = = = ç ÷ç ÷çè ø

åå

(33)



Wherem is the axial half-wave number andn is the circumferential wave number. Also, j mniu, ,

c mnu ,0 , c mnu ,

1 , j mniv, , c mnv ,

0 , c mnv ,1 , j mn

lw, , c mnw ,

0 are the constant amplitudes of vibrations asso-

ciated with the natural mode shapes and should be determined. In order to analyze the free vibra-tion of the sandwich panel, the time coefficients in Equation (33) were considered as follows:

( ) ( ) ( ){ } { }( ) ( ) ( ){ } { }( ) ( ){ } { }

, , , , , ,0 1 0 1

, , , , , ,0 1 0 1

, , , ,0 0

, , , ,

, , , ,

, ,

0,1,2,3; 0,1,2; ,

j mn c mn c mn j mn c mn c mn i ti i

j mn c mn c mn j mn c mn c mn i ti i

j mn c mn j mn c mn i tl lw w

u t u t u t u u u e

v t v t v t v v v e

w t t w e

i l j t b

=

=

= =

=

=

(34)

where ω is the natural angular frequencies (rad/s) related to mode number m n( , ) . Substituting

Equations (33) and (34) into the equations of motion, applying the Navier method and collecting the coefficients, the eigen value-eigen vector equation is obtained as follows:

{ }

2

0 1 2 3 0 1 2 3 0 1 2

0 1 2 3 0 1 2 3 0 1 2

0 1 0 1 0

([ ] [ ]){ } {0}

, ,

,

, , , , , , , , ,

, , , , , , , , , , ,

, , ,

t t t t t t t t t t t

b b b b b b b b b b b

c c c c c

T

K M d

u u u u v v v v w w w

u u u u v v v v w w wd

u u v v w

w

ì üï ïï ïï ïï ïï ïï ïï ïí ýï ïï ïï ïï ïï ïï ïï þ

-

î

=

ï

=

(35)

where d , K

and M are the vector of mode shape coefficients, stiffness matrix and the mass matrix of composite panel, respectively. The minimum eigen value of Equation (35) is the fundamental natural frequency of sandwich cylindricl panel which is determined using MATLAB code. 5 RESULTS AND DISCUSSION

In this section, based on the present analytical method, some examples are considered and the re-sults are validated and discussed. The results of the present analysis are validated by FSDT and ABAQUS commercial FE software. In all examples, a cylindrical sandwich panel is simply support-ed. Example 1 The geometric specifications and mechanical properties of a cylindrical sandwich panel composed of fiber glass reinforced polyester matrix face sheets and a PVC foam (named HEREX C70.130) core are given in Table 1. The sandwich panel was made by (0/90/0/core/0/90/0) lay-ups. In this table, the thickness of each top and bottom face sheets is determined to be 3 mm, the thickness of the core to total thicknes 0.8, the radius of the core intermidate to total thickness 10, and the length of the cylinder to the radius of the core intermidate 2. The dimensionless angular



velocity is 2 1/22( / ) / tL E h (Rahmani et al., 2010). In the FE modeling procedure via

ABAQUS, an eight-node C3D8R was selected as the element type for the core layer and Lanczos solver was chosen to carry out the vibration analysis.

E 24.51GPa , E E 7.77 GPa ,G G 3.34 GPa , G 1.34GPa 1 2 3 12 13 2330.078 , 0.49 , 1800 kg/m12 13 23n n n r

= = = = = =

= = = = Face sheets

E E E 0.10363 GPa , G G G 0.05 GPa1 2 3 12 13 2330.32 , 130 Kg/mn r

= = = = = =

= = Core

,/ 0 . 8 , 0 . 3 , 3

0 . 0 0 3 , [ 0 9 0 0 / / 0 9 0 0 ]

c

t b

m

m

h h R L Rc c

h h c o r e

= = =

= = Geometry

Table 1: Geometrical and mechanical properties used for the analysis (Meunier et al., 1999).

2 1/2

2( / ) /tL E hw w r=

Mode (m,n) Present model FSDT model Error (%) ABAQUS Error (%) (1,2) 16.39 16.66 1.62 17.13 4. 32 (1,3) 16.44 16.75 1.85 17.36 5.29 (1,4) 21.00 21.52 2.42 22.03 4.67 (1,5) 27.11 27.79 2.45 28.39 4.51

Table 2: Comparison of the dimensionless natural frequencies for the cylindrical sandwich panel.

In Table 2, the first four dimensionless natural frequencies obtained from the present model,

FSDT model and ABAQUS modeling for the sandwich panel are presented and compared. It is shown that the results obtained from the present method, FSDT, and FE analysis for different mode shapes are very close and in good agreement when face sheets are thin. Moreover, the smallest dimensionless natural frequency of sandwich panel occurs at the mode shape of (m,n)= (1,2). Example 2

In this example, mechanical properties of face sheets and the core for the cylindrical sandwich panel are the same as those in Table 1. To show the accuracy of the present analysis, especially when the face sheets’ thickness is increased, two geometrical models are considered. The first geometrical model is shown in Table 3. It can be observed that the face sheetes are thin in comparison with Table 5.

Geometry

,/ 0.8 , 0.3 , 3

3 , [0 90 0 / / 0 90 0]

mc

t b

h h R L Rc ch h mm core

= = =

= =

Table 3: Geometry of the cylindrical sandwich panel for the first model (thin).



Four natural frequencies for a cylindrical sandwich panel based on Table 3 are as follows:

Error(%) ABAQUS Error(%) FSDT present Mode (m,n) 5.68 326.14 3.84 319.89 307.61 (1,2) 3.96 385.27 1.10 374.13 370.01 (1,3) 4.93 560.95 2.28 545.75 533.29 (1,4) 4.44 753.83 2.03 735.25 720.32 (1,5)

Table 4: The first four natural frequencies relevant to the cylindrical sandwich panel for the first model (Hz).

Similar to example 1, the results of the present analysis, FSDT, and ABAQUS are very close.

When the face sheets are thick, the results of the present theory are better than those of FSDT theory. To clarify this, the geometric specifications of the cylindrical sandwich panel are defined in Table 5. It can be seen that the face sheets in Table 5 are thicker than those in Table 3.

Geometry

,/ 0.8 , 0.3 , 3

12 , [0 90 0 / / 0 90 0]

m

c

t b

h h R L Rc c

h h mm core

= = =

= =

Table 5: Geometry of the cylindrical sandwich panel for the second model (thick).

The results of four natural frequencies for a cylindrical sandwich panel based on Table 5 are as follows:

Error(%) ABAQUS Error(%) FSDT present Mode (m,n) 16.32 238.87 26.17 270.73 199.88 (1,2) 16.64 324.64 39.04 443.98 270.64 (1,3) 13.71 461.39 37.43 636.29 398.11 (1,4) 10.12 601.38 34.27 822.25 540.49 (1,5)

Table 6: The first four natural frequencies relevant to the cylindrical sandwich panel for the second model (Hz).

It is shown in Table 6 that in the cylindrical sandwich panel, when the face sheets are thick, the

maximum difference of magnitude of natural frequency of the sandwich panel with FSDT model is related to mode (m,n)= (1,3) which is about 39%. Comparing Tables 4 and 6 shows that when the face sheets are thin, the results of the present theory and FSDT theory are close, but when the face sheets are thick, the results of the present theory and FSDT theory are far from each other. The first four vibration mode shapes of sandwich panel obtained from ABAQUS FE code are presented in Figure 2.



(m,n)=(1,2) (m,n)=(1,1)

(m,n)=(1,4) (m,n)=(1,3)

Figure 2: The first four vibration mode shapes of sandwich panel obtained by ABAQUS.

For analysis and parametric study cylindrical sandwich panel, the aforementioned Geometrical and mechanical properties of Table 1 were chosen. The results of the parametric study which indi-cate variation of the natural frequencies and non-dimensional natural frequencies versus circumfer-ential wave number (n) for different longitudinal wave lengths (m) are depicted in Figure 3 .



Figure 3: Variation of the natural frequencies and non-dimensional frequencies versus circumferential wave.

It can be seen in Figure 3 that by increasing the circumferential wave number, the natural fre-

quencies and non-dimensional frequencies of the sandwich panel initially decrease and then increase when reaching a specific magnitude. The 3D view and 2D front view of the mode shape relevant to the first natural frequency are shown in Figure 4.

Figure 4: 3D view and 2D front view of the vibration mode shape relevant

to the first natural frequency in mode shape (m,n)=(1,4).

The effect of length-to-radius core ratio on natural frequency and non-dimensional natural frequency

Another parametric study is related to the study of variation of the natural frequencys and non-dimensional natural frequency versus the length-to-radius core ratio, shown in Figure 5. It can be seen that by increasing the length-to-radius core ratio, the natural frequency of the sandwich panel



initially decreases. Contrary to the natural frequency, the non-dimensional natural frequency in-

creases when the c

L

R ratio increases.

Figure 5: Effect of c

L

R on natural frequency and non-dimensional frequency of cylindrical sandwich panel.

The effect of the ratio of core thickness to total thickness on natural frequency and non-dimensional natural

frequency

Variation of the natural frequency and non-dimensional frequency versus the increase in the ratio of

core thickness to total thickness ( ch

h) can be seen in Figure 6. Based on the increase in the ratio of



core thickness to total thickness ( ch

h), the first four natural frequencies (Hz) decrease. Contrary to

natural frequencies, the non-dimensional natural frequency increases.

Figure 6: Effect of ch

h on natural frequency and non-dimensional frequency of cylindrical sandwich panel.

The effect of fiber angles of composite face sheets on natural frequency and non-dimensional natural frequency

The geometrical and mechanical properties of sandwich are presumed as in Table 1, and stacking sequence is considered parametrically as [ / / / / / / / / ]core . The effect

of fiber angles of composite face sheets on natural frequency and non-dimensional natural frequency



is shown in Figure 7. According to Figure 7, the natural frequency of the sandwich panel initially rises by increasing the fiber angle, which stems from the enhancement of sandwich panel stiffness, but this trend reverses as the fiber angle reaches a critical value. This critical value of the fiber an-gle is dependent on the mode shape. The variation of non-dimensional natural frequency versus fiber angle is almost similar to that of the natural frequency.

Figure 7: The effect of fiber angles of composite face sheets on natural frequency and

non-dimensional natural frequency of cylindrical sandwich panel.

6 CONCLUSIONS

In this study, an exact free vibration soulution for a cylindrical sandwich panel was presented based on an improved higher-order sandwich panel theory. Based on compatibility conditions in the inter-face of the core and the top and bottom face sheets, the total number of unknowns in the core and



the face sheets was reduced from 33 to 27. The mathematical formulation used Hamilton’s principle to derive the equations of motion and boundary conditions, and then used Fourier series to solve them. In order to verify the accuracy of the present theory, a comparison was made with the results of FSDT model and the commercial finite-element software ABAQUS. Finally, a parametric study was conducted, and the effect of different parameters on natural frequency and non-dimensional natural frequency were investigated. The natural frequency relevant to the cylindrical sandwich panel in different shape modes were determined. The results of this study can be summarized as follows:

The results of the present theory and those of the first shear deformation theory have a good agreement when the face sheets of the cylindrical sandwich panel are thin. Nevertheless, when the face sheets are thick, the results of the present theory and those of the first shear defor-mation theory are far from each other.

By increasing the circumferential wave number, the natural frequency and non-dimensional frequency of the sandwich panel initially decrease and then increase when n reaches a specific magnitude.

By increasing the length-to-radius core ratio, the natural frequencies of the sandwich panel initially decrease, but the non-dimensional natural frequencies increase.

When the ratio of core thickness to total thickness increases, the natural frequencies decrease. Contrary to natural frequencies, the non-dimensional natural frequencies increase.

the natural frequency of the sandwich panel initially rises by increasing the fiber angle. Nev-ertheless, this trend reverses as the fiber angle reaches a critical value. The variation of non-dimensional natural frequency versus fiber angle is almost similar to that of natural frequen-cy.

References

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APPENDIX A

Exact integration method

In this method, the integral in Eq. (30) is calculated accurately. After taking the exact integration, the following results are obtained:

( )

( ) ( )

1

1

1

11

12 1

22 2 2 1

3 1 1

1ln ,

1

ln ,1

1ln

1 2

k

k

k

k

k

k

h k

hk

h kk kh

k

h kk k k kh

k

R hH dz R

z R R h

R hzH dz R h h R

z R R h

R hzH dz R h h R h h R

z R R h

+

+

+

+

++

++ +

é ùæ ö+ ÷çê ú÷ç= = ÷ê úç ÷ç ÷+ +çê úè øë ûé ùæ ö+ ÷çê ú÷ç= = - - ÷ê úç ÷ç ÷+ +çê úè øë ûé ùæ ö+ ÷çê ú÷ç= = - - - + ÷ê úç ÷ç ÷+ +çê úè øë û

ò

ò

ò

( ) ( ) ( )

( ) ( ) ( ) ( )

1

1

33 3 2 2 2 3 1

4 1 1 1

4 4 3 3 2 2 2 31 1 1 14

54 1

,

1 1ln ,

1 3 2

1 1 1

4 3 2

1 ln

k

k

k

k

h kk k k k k kh

k

k k k k k k k kh

h k

k

R hzH dz R h h R h h R h h R

z R R h

h h R h h R h h R h hz

H dz R R hz R RR h

+

+

++ + +

+ + + +

+

é ùæ ö+ ÷çê ú÷ç= = - - - + - - ÷ê úç ÷ç ÷+ +çê úè øë ûéê - - - + - - -êê= = æ ö+ê ÷ç+ ÷çê+ ÷ç ÷ê ç ÷+çè øë

ò

ò

( ) ( ) ( ) ( )( )

( ) ( ) ( )

1

1

5 5 4 4 2 3 3 3 2 21 1 1 15

64 5 1

1

6 6 5 5 2 4 41 1 16

7

,

1 1 1 1

5 4 3 2,

1 ln

1 1 1

6 5 4

1

k

k

k

k

k k k k k k k kh

h kk k

k

k k k k k kh

h

h h R h h R h h R h hz

H dz R R hz R R h h RR h

h h R h h R h hz

H dz Rz R

+

+

+ + + +

++

+ + +

ùúúúúúú

ê úûé ùê ú- - - + - - -ê úê ú= = æ ö+ê ú÷ç+ ÷çê ú+ - - ÷ç ÷ê úç ÷+çè øê úë û

- - - + - -= =

+

ò

ò( )

( ) ( )

3 3 31

4 2 2 5 6 11 1

1

31

ln2

k k

kk k k k

k

R h h

R hR h h R h h R

R h

+

++ +

é ùê ú-ê úê úæ ö+ê ú÷ç ÷çê ú+ - - - + ÷ç ÷ê úç ÷+çè øê úë û

vibration analysis of a cylindrical sandwich panel with … · frostig's second model. in this...

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