Viscoelastic Substrates Effects on the Elimination or Reduction of the Sandwich Structures Oscillations Based on the Kelvin-Voigt Model

(1)

Abstract

Effects of viscoelastic substrates on the sandwich structures oscilla-tions are examined in this paper. In this regard, dynamic response of sandwich annular panels with FG polar orthotropic face sheets resting on viscoelastic substrate is presented. Young’s modulus in the radial and circumferential direction, shear modulus and density of each face sheet may be varied continuously in the radial direc-tion. The viscoelastic substrate is modeled as Kelvin-Voigt founda-tion. To describe more accurately response of sandwich structures, the governing dynamical equations are derived based on the layer-wise theory and five systems of second order coupled partial differ-ential equations are obtained. The effects of the stiffness and damp-ing coefficients of the foundation on the dynamic behavior of sand-wich plate are investigated for various transient loads and bounda-ry condition. Since no available results may be found in literature to demonstrate the efficiency and accuracy of the obtained results, the obtained results are verified by comparison with finite element results based on the three dimensional theory of elasticity for some special cases.

Keywords

Dynamic response; FG polar orthotropic; viscoelasticfoundation; transient loads, Kelvin-Voigt.

Viscoelastic Substrates Effects on the Elimination

or Reduction of the Sandwich Structures Oscillations

Based on the Kelvin-Voigt Model

1 INTRODUCTION

Sandwich structures are widely used in the industries. In practical applications, sandwich structures are subjected to various transient dynamic loads where reduce vibration of these structures under-dynamic loads is important in many engineering fields and viscoelastic substrate can be applied to reducethe vibration of structures. In addition, structures resting on viscous, elastic or viscoelastic foundationare extensively used in many engineering fields. However, most of the performed studies

M.M. Alipour a, * I. Rajabi b

a _{Assistant Professor, Department of} Mechanical Engineering, University of Mazandaran, Babolsar 47416-13534, IranE-mail: [email protected] [email protected]

b _{Assistant Professor, Department of} Mechanical Engineering, Malek Ashtar University, Shiraz, Iran.

E-mail: [email protected]

* Corresponding author

http://dx.doi.org/10.1590/1679-78254096

(2)

arelimited to the static and free vibration analyses of sandwich plates and rare researches are avail-able onthe dynamic analysis especially for plates under viscoelastic foundation.

(3)

beam under aharmonic line load. The differential governing equations were converted into algebraic equations byassuming the deflection and rotation of the beam in harmonic forms with respect to time and space.Some researchers analyzed beams resting on viscoelastic foundations subjected to moving loads. Thevibration instability analysis of an oscillator moving along a Timoshenko beam was performed byMetrikine and Verichev (2001) and Mazilu et al. (2012). The inHluence of a non-linear foundation on thedynamic response of a periodically supported beam was investigated by Hoang et al. (2016), based on the Euler–Bernoulli beam theory. Ding et al. (2013) investigated the dynamic response of infinitebeams supported by nonlinear viscoelastic foundations. The differential equations were obtained byemploying the Timoshenko beam theory and were solved based on the Adomian decomposition methodand a perturbation method in conjunction with complex Fourier transformation. The dynamic response of finite Timoshenko beamsresting on a six parameter foun-dation was studied by Yang et al. (2013).

Most of the existing researches were performed based on the equivalent single layer theories, whereusing the equivalent single-layer theories for analysis of sandwich plates may be inaccurate or erroneousin most circumstances. Various theories are presented for analysis of sandwich structure Carrera and Brischetto (2008).Many researchers investigated the laminated composite and sandwich plates based on the sandwichand multilayered structures theories. Thermoelastic bending of func-tionally graded sandwich plates were analyzed by Houari et al (2013) based on a new higher order shear and normal deformation theory and by Tounsi et al. (2013) based on A refined trigonometric shear deformation theory.Based on layerwise formulation, Alipour (2016a) presented anovel eco-nomical analytical method for bending and stress analysis of functionally graded sandwichcircular plates subjected to various loads with general elastic edge conditions. Alipour (2016b) analyzed the effects of elastically restrained edges on FG sandwich annular plates by using a novel solutionproce-dure and layerwise method. Alipour (2018) examined the dynamic response of sandwich plate with viscoelastic boundary support. Bending analysis of laminated compositeplates was performed based on a predictor-corrector approach and the zig-zag theory by Lee andCao (1996). Alipour and Shari-yat (2015) employed a zigzag-elasticity plate theory for bending andstress analysis of circu-lar/annular sandwich plates with orthotropic composite face sheets and auxeticcores. Free vibration analysis of circular and annular composite sandwich plates with auxetic cores wasperformed by Shariyat and Alipour (2017a).

(4)

founda-tion. To describe more accuratelyresponse of sandwich plates, the governing differential equations of motion are derived based on theminimum total potential energy principle by using the layerwise theory. The dynamic responses ofsandwich plate are examined for various stiffness and damping coefficients of the foundation, transientloads and boundary conditions. Since no existing work has been performed on the dynamic analysis ofFG polar orthotropic sandwich plates, accuracy and effi-ciency of the presented analysis are verified bycomparing the obtained results with results of the three-dimensional theory of elasticity extracted fromthe ABAQUS software for some special cases. The comparisons show that there is a very good agreementbetween present results and results of the three-dimensional theory of elasticity.

2 DESCRIPTION OF THE MATERIAL PROPERTIES, DISPLACEMENT, STRAIN AND STRESS

FIELDS

As shown in figure 1, a three-layer sandwich annular plate with functionally graded polar ortho-tropic face sheets resting on the viscoelastic substrate which is modeled as Kelvin-Voigt foundation is considered.

Figure 1: Schematic of the sandwich annular plate on viscoelastic substrate.

Viscoelastic substrate is modeled as a continuously distributed medium with stiffness Kw and

damping coefficient Ct.

Face sheets may be fabricated from functionally graded polar orthotropic materials. Young’s modulus in the radial and circumferential direction, shear modulus and density of each face sheet may be varied continuously in the radial direction according to a power-law fraction, as follows:

(

)

(

)

( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( )

( ) 1 ,

( )

( ) 1 1, 3

k

k k

r r

k k k

o

k k

rz rz

k k k

o

E r E

E r E r r

G r G

r r r k

b

q q

h a

r r g

ì ü ì ü

ï ï ï ï

ï ï ï _{ï é} _ù

ï ï ï ï

ï ï₌ï ï_ê ₊ _- _ú

í ý í ý

ï ï ï ï êë úû

ï ï ï ï

î þ î þ

é ù

= ê_ê + - ú_ú =

ë û

(1)

r r

r

v v

E E

q q

q

(5)

where _a( )k _, _b( )k _, _g( )k _and _h( )k ₍_k ₌_{1, 3)}_{are inhomogeneity parameters for top (k=1) and bottom} (k=3) face sheets.

The three transverse local coordinates_x(1)_,_x(2)_and_x(3)_{are represented for top, core, and bottom}

layers, respectively. The local coordinates are measured from the mid plane of the corresponding layer and are positive upward.

Based on the layerwise theory with the piecewise linear local components, after some manipula-tions and imposing the continuity condimanipula-tions of the displacement components at the interfaces be-tween face sheets and core, the displacement field of the layers may be written as:

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

(2) (2) 2 0

(3) 3 (3) 2 (2) ( )

3 0

2 2

, 1, 2, 3

2 2 2 2

r r

r

i

i i

r r

h h

u u

h h h h

u u i

x y y

x y

x y y x

æ _ö÷

ç _÷

= +ç_ç + _÷_÷ +

çè ø

= +

æ _ö÷

ç _÷

= +ç_ç - _÷_÷ - - £ £ =

çè ø

(3)

whereu₀is the radial displacement component of the mid plane of the core and ( )i

r

y are the local

rotation of the layers of the plate.

Cauchy’s strain-displacement relations are:

, , ,

r r rz z r

u

u u w

r

q

e = e = e = + (4)

where the symbol “,” stands for the partial derivative. Based on Hooke's generalized stress-strain law:

( )

( ) ( )

( )

( ) ( )

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

( ) ( ) ( )

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

( ) ( )

1 1

( ) ( )

1 1

( )

k

k k

k

k r k r r

r _k _k r _k _k

r r r r

k k k

k r r k k

r

k k k k

r r r r

k k k

rz rz rz

E r v E r

v v v v

v E r E r

v v v v

G r

q

q q q q

q q

q q q q

s e e

t g

= +

-

-= +

-

-=

(5)

where the E, G andvsymbols denote Young’s modulus, shear modulus, and Poisson’s ratio,

respec-tively.

Based on Eqs.(1) to (5), the stress–displacement relations of face sheets and core may be writ-ten as:

(

)

(

)

(1)

(1) (1) (1)

(1) (1) 1 (1) (1) 2 (2) (2)

0, 0 , ,

(1) (1) (1) (1) (1)

1

2 2

1

r o

r r r

r r r r r r r r

r r o

E r r

v h v h v

u u

r r r

v v

E r r

v

b

q q q

q q b q

q

a

s x y y y y

a s

é ù

+

-ê ú é _æ _öæ ö_÷ æ öù_÷

ê ú ÷ç ç

ë û ê ç ÷ç ÷ ç ÷ú

= _ê + +_çç_ç + _÷_÷_ç_ç + ÷_÷+ _ç_ç + _÷÷_ú

÷ ÷

ç ç

è ø

- ê_ë è ø è øú_û

é ù

+

-ê ú

ê ú

ë û

=

-(1) (1) 1 (1) (1) (1) 2 (1) (2) (2)

0, 0 , ,

(1) (1)

1 1 1

2 2

r r r r r r r r r r

r r

h h

v u u v v

r r r

v q q q

q q

x y y y y

é æ_ç ö_÷æ_ç ö_÷ æ_ç ö_÷ù

ê + +çç + ÷÷ç + ÷÷+ ç + ÷÷ú

ê çè ÷øçè ÷ø çè ÷øú

ë û

(6)

(

)

(1)

₍

₎

(1) (1) (1) (1)

(2) (2) (2)

(2) (2) (2) (2)

0, 0 ,

(2)2 (2)

(2) (2) (2) (2) (2) (2)

0, 0 ,

(2)2 (2) (2

1 ,

1

1 1

1

rz rz o r r

r r r r r

r r r r

rz

G r r w

E v v

u u

r r

v E

v u u v

r r

v G

b

q

t a y

s x y y

t

é ù

= ê_ê + - ú_ú +

ë û

é æ_ç öù_÷

ê ÷ú

= _ê + + ç_ç + _÷_÷÷_ú

çè ø

- _ë _û

é æ_ç öù_÷

ê ú

= _ê + + ç_ç + ÷_÷_÷_ú

è ø

- ë û

= )

(

(2) _,

)

r wr

y +

(

)

(

)

(3)

(3) (3)

(3) (3) (3)

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

0, 0 , ,

(3) (3) (3) (3) (3)

1

2 2

1 1

1

r o

r r r

r r r r r r r r

r r o

E r r

v h v h v

u u

r r r

v v

E r r

v

b

q q q

q q b q

q

a

s x y y y y

a s

é ₊ _- ù

ê ú é æ öæ ö_÷ æ öù_÷

ê ú ÷ç ç

ë û ê ç ÷ç ÷ ç ÷ú

= _- ê_ê + +_è_ççç - _ø÷_÷_ç_èç_ç + _÷_ø÷÷- _ç_è_çç + ÷÷_÷_ø_úú

ë û

é ₊ _- ù

ê ú

ë û

=

-(

)

(3)

(

)

(3) (3) 3 (3) (3) (3) 2 (3) (2) (2)

0, 0 , ,

(1) (1)

(3) (1) (3) (3)

1 1 1

2 2

1 ,

r r r r r r r r r r

r r

rz rz o r r

h h

v u u v v

r r r

v

G r r w

q q q

q q b

x y y y y

t a y

é æ_ç ö_÷æ_ç ö_÷ æ_ç ö_÷ù ê + +çç - ÷÷ç + ÷÷- ç + ÷÷ú ê çè ÷øçè ÷ø çè ÷øú

ë û

é ù

= ê_ê + - ú_ú +

ë û

3 THE GOVERNING EQUATIONS OF MOTION OF THE SANDWICH PLATE WITH FG POLAR

ORTHOTROPIC FACE SHEETS

The governing equations of motion of the sandwich annular plate with functionally graded polar orthotropic face sheets on viscoelastic substrate are derived based on using the minimum total po-tential energy principle:

0,

U K W

dP =d +d -d = ₍₇₎

where



U

,



K

and



W

are increments of the strain energy, kinetic energy and work done by

external applied loads, respectively:

(

)

(

)

r r rz rz

V

A

U dV

K u u w w dV

W P w dA

q q

d s de s de t dg

d r d d

d d

= + +

= +

=

ò



 ₍₈₎

For sandwich plate subjected to transient load (q(t)) resting on viscoelastic substrate based on the Kelvin–Voigt model, the distributed transverse load can be defined as follows.

( ) _w ( , ) _t ( , )

P =q t -K w r t -C w r t ₍₉₎

In which q(t) is the external dynamic loads, Kw and Ct are the spring and damper constant of

viscoelastic foundation.

(7)

(

)

(

)

( )

3 ( )

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

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

, 0 0 0 0 0 0 0 0

0 2 2 2

k k

k r

r r r r r

k

N N h h h

N I I I u I I I I

r

q _y _y _y

=

æ _- ö_÷

ç _÷

ç + ÷= + + + + -

-ç ÷

ç ÷

çè ø

å

_    ₍₁₀₎

(1) (1) (1)

(1) (1) (1) (2)

1 1 1 2

0 0 2

(1) (1) 1 (1) (1)

2 0

1 1

2 2 2 2

4

r r rz r

r r

h h h h

N M N M Q I u

r r r

h

I I

q q y

y y

æ ö æ ö æ ö

æ _¶ ö_÷ç _÷ _ç _÷ _ç _÷

ç ₊ _÷ç ₊ _÷_- _ç ₊ _÷_- ₌ _ç ₊ _÷₊

ç _÷ç ÷ _ç ÷ _ç ÷

ç ÷ç ÷ ç ÷ ç ÷

è ¶ èø ø è ø è ø

æ _ö÷

ç _÷

ç + ÷

ç _÷

ç ÷

è ø

 

  (11)

(

(1) (3)

)

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

2 2 2

,

2 2

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

2 1 2 3 2 2

0 0 0 0 0 2 0

1 1

2 2 2

2 2 2 2 4 4

r r r r rz

r r r

h h h

N M N N N Q

r r r

h h h h h h

I u I u I I I

q q

y y y

æ ö

æ _¶ ö_÷ç _÷

ç ₊ _÷ç ₊ _- _÷_- _- _- ₌

ç _÷ç _÷

ç ÷ç ÷

è ¶ èø 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

  

  (12)

(3) (3) (3)

(3) (3) (3) (3) (2) (3)

3 3 3 2 3

0 0 0 2

1

2 r r rz 2 2 r 4 r r

h h h h h

N M Q I u I I

r r y y y

æ ö æ ö

æ _¶ ö_÷ç _÷ _ç _÷

ç ₊ _÷ç_- ₊ _÷_- _{= -} _ç _- _÷₊ ₊

ç ÷ç ÷ ç ÷

è ¶ èø ø è  ø   (13)

3

(1) (2) (3) ( )

0 0 0

0

1

( )

k

r w t

k

Q q K w C w I I I w

r r

=

æ _{¶ ÷}ö

ç + ÷ = - - + + +

ç _÷

ç ÷

è ¶ ø

å

  (14)

where the stress resultants M, N, Q and the higher-order inertias are defined as:

( ) 2

( ) ( ) ( ) ( )

2 2

( ) ( ) ( )

2

1

,

, 1,2, 3 ,

k

k k

k h k

i k k

i k

k

i h

h

k k k

r rz

h

N

d M

Q d k i r

s x

x

t x q

-ì ü _ì _ü

ï ï ï ï

ï ï _ï _ï

ï _{ï =}

í ý í ý

ï ï ï ï

ï ï ï_î ï_þ

ï ï

î þ

= = =

ò

(15)

(

)

( )

2

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

2 2

( ) ( ) ( ) ( )

2

( ) , 1 , 1, 3

1,2, 3 0,2

k

k i

i h

k k k j k k k k

j j o j

h h

i i i j i

j h

I r d I r r I k

I d i j

h

r x x g

r x x

-é ù

= = ê_ê + - ú_ú =

ë û

= = =

ò

(16)

(8)

(1) (1) 1 (1) 2 (2) (1) 1 (1) 2 (2)

0, , , 0

(2) (2) (2)

0, 0

(3) (3) 3 (3) 2 (2) (3) 3 (3) 2 (2)

0, , , 0

1

,

2 2 2 2

1 ,

1

2 2 2 2

i ri r r r r r i r r

i ri r i

i ri r r r r r i r r

h h h h

N A u A u

r

N A u A u

r

h h h h

N A u A u

r

q

y y y y

æ ö_÷ æ ö_÷

ç _÷ ç _÷

= ç_ç_ç + + _÷_÷+ ç_ç_ç + + _÷_÷

è ø è ø

= +

æ ö_÷ æ

ç _÷

= ç_ç - - _÷_÷+ -

-çè ø , i r, q

ìïï ïï ïï ïïï íï ïï ö

ï ç ÷

ï ç ÷ =

ï _ç_ç ÷_÷

ï _è _ø

ïïî

(17)

( ) ( ) ( ) ( ) ( ) ,

1

, 1, 2, 3, ,

k k k k k

i ri r r i r

M D D k i r

r q

y y q

= + = = (18)

(

)

( )k ( )k ( )k _{, ,} _{1,2, 3}

r rz r r

Q = A y +w k = ₍₁₉₎

where

( ) 2 ( ) ( ) 2 ( ) ( )

( ) ( )

( )2 ( )2

( ) ( ) ( ) ( ) ( ) ( ) 2 2 ( ) ( ) 2 1 1 ( ) ( ) , , 1 1 ( ) k k k k k h h

k k k k k

ii i k r r r k

k k

k k k k k k

ii h r r r h r r

k k

rz rz

h

A E r A v E r

d d

D v v D v v

A G r

q q

q

q q q q

x x

-

-ì ü ì ü ì ü ì ü

ï ï ï ï ï ï ï ï

ï ï₌ ï ï₌

í ý í ý í ý í ý

ï ï _- ï ï ï ï _- ï ï

ï ï ï_î ï_þ ï ï ï_î ï_þ

ï ï ï ï

î þ î þ

=

ò

2

( )_, _{1,2, 3} _,

k h

k

dx k = i =r q

ò

(20)

(

)

( )

(

)

( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) 1 , 1 , , , 1, 3

k k

ij ij k k k k

o rz rz o

k k

ij ij

A A

r r A A r r i j r k

D D

b b

a a q

ì ü ì ü

ï ï ï _{ï é} _ù _é _ù

ï ï ï ï

ï ï₌ï ï_ê ₊ _- _ú ₌ _ê ₊ _- _ú ₌ ₌

í ý í ý

ï ï ï ï ê_ë ú_û ê_ë ú_û

ï ï ï ï

î þ î þ (21)

( ) 2 ( ) ( ) 2 ( ) ( )

( ) ( )

( )2 ( )2

( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 ( ) ( ) ( ) 2 1 1 , , 1 1 k k k k k k h h

k k k k k

ii i k r r r k

k k

k k k k k k

ii h r r r h r r

h

k k k

rz rz

h

A E A v E

d d

D v v D v v

A G d

q q

q

q q q q

x x

x

-

-ì ü ì ü ì ü ì ü

ï ï ï ï ï ï ï ï

ï ï₌ ï ï₌

í ý í ý í ý í ý

ï ï _- ï ï ï ï _- ï ï

ï ï ï_î ï_þ ï ï ï_î ï_þ

ï ï ï ï

î þ î þ

=

ò

, i =r,q k =1, 3

(22)

Based on Eqs. (15) to (22), the governing equations (10) to (14) may be rewritten as:

(

)

(

)

(

)

(

)

(

)

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

0,

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

2

(1) (1) (1

1

,

1 2

r

rr rr rr rr o rr o rr

o o

o rr r rr

u

A A A A r r A r r u

r u

A A A A r r A r r

r h

r r A

b b

qq qq qq qq qq

b

a a

a y

æ ö

é ₊ ₊ ₊ _- ₊ _- ùç_ç ₊ ÷_÷

ê ú_ç ÷_÷

ê ú ç ÷

ë û è ø

é ù

-ê_ê + + + - + - ú_ú

ë û

é ù

+ ê_ê + - ú_ú

ë û

(1) (1)

, (1) )

2

r r _A r

r qq r

y y

é æ_ç ö_÷ ù

ê ç_ç ₊ ÷_÷_- ú

ê _ç_ç ÷_÷÷ ú

ê _è _ø ú

ë û

(9)

(

)

(

)

(

)

(

)

(

)

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

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

2

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

2 2 (3) (3) 3 , 2 2 1 2 r r

rr rr rr o rr o r rr

r

o o

o rr r

h

A A A r r A r r

r

h

A A A r r A r r

r h

r r A

b b

qq qq qq qq

b

y

a a y

y

a a

a y

æ _ö÷

é ù ç_ç _÷

+ _ê_ëê - + - - - ú ç_ú_û_ç + ÷_÷_÷

÷

çè ø

é ù

- ê_ê - + - - - ú_ú

ë û

é ù

- ê_ê + - ú_ú

ë û

(

)

(

)

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

(1) (1) (1) 1 (1) 2 (2)

0 (3) 1

(3) (3) (3) 3

0

2 2

r r r

rr

r

o rr r r

r

o rr

A

r r

A h h

r r A u

r r

A h

r r A u

r r

qq

b _q

y y

a b y y

a b

-é æ_ç ö_÷ ù

ê ç_ç ₊ ÷_÷_- ú

ê _ç_ç ÷_÷÷ ú

ê è ø ú

ë û

éæ_ç _¶ ö_÷_æ _ö_÷ù

êç ÷ç ÷ú

+ - _êê_è_çç_ç _¶ + _÷_ø÷ç÷_çç_è + + ÷_ø_÷ú_ú

ë û

æ _¶ ö_÷

ç _÷

ç

+ - _ç_ç + ÷_÷

-÷ ¶ çè ø

(

)

(

)

(

)

(1) (3)

(3) 2 (2)

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

0 0 0 0 0 0 0

(3) (1) (

(3) 2 (2) 3 (3) 2

0 0 0 0

2 2

2 2 2

r r

o r r r

o r r

h

h h h

I I I u r r I u I

h h h

r r I u I I

h

y y

g y y y

g y y

é _æ _ö÷ù

ê ç_ç _- _÷_÷ú₌

ê ç_çè ÷_øú

ê ú

ë û

æ _ö÷

ç _÷

+ + + - ç_ç + + _÷_÷+ +

çè ø

æ _ö÷

ç _÷

- ç_ç - - _÷_÷+

-çè ø

  

 



(

3)

)

(2) 3 (3) (3)

0

2

r r

h I

y - y

(

)

(

)

(1) (1) 0, (1) (1) (1) 1 0 0, ₂ (1)

2 2 (1)

, (1) (1)

(1) 1 (1) (1) (1) 1

, 1 2 1 4 4 r

o rr rr

r r r

o rr rr r rr

u

h u

r r A u A

r r

h h

r r A D A D

r r b qq b qq qq a y y a y

é æ ö ù

é + - ùê çç + ÷÷- ú

ê úê ç ÷÷ ú

ê ú ç ÷

ë ûê_ë è ø ú_û

æ ö

æ ö _÷ æ ö

æ ö÷ç ÷÷çç ÷ ç ÷÷

ç ç ç

+èçç + - ÷ø÷èçç + ø÷÷÷çèçç + ÷÷ø÷÷-ççè + ÷÷ø÷

(

)

(

)

₍

₎

(

)

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

(1) (1) (2)

1 2

, ₂

(1) 1

(1) (1) (1) (1) (1) (1) (1) (1)

,

1 4

1 ,

r r r

o rr r rr

r

rz o r r o rr r r r

h h

r r A A

r r

D

A r r w r r D

r

h

b

qq

b b _q

y y

a y

a y a b - y y

é ù

ê ú

ë û

é æ ö_÷ ù

é ùê çç ÷ ú

+ _ëê_ê + - ú_ú_û_êê _çç_ç + _÷÷_÷÷- ú_ú

è ø

ë û

æ ö

é ù ç_ç ÷_÷

- ê_ê + - ú_ú + + - _ç_ç + ÷_÷

÷ ç

ë û _è _ø

+

(

)

(

)

(1) (1) (1) 1

(1) (1) (1) (1) (2) (1) (2)

1 1 2 1 2

0, , , 0

(1) (1) (2) (1) (1) (1)

1 1 2 1 1 2

0 0 0 0

2 2 2 2 2

r

o rr r r r r r r r

r r o r

h h A h h

r r A u u

r

h h h h h h

I u I r r u

b _q

h

a b y y y y

y y g y

- é_ê æ_ç ö_÷ æ_ç ö_÷ù_ú

÷ ÷

- _ê _ç_çç + + _÷_÷+ _ç_çç + + _÷_÷_ú

è ø è ø

ê ú

ë û

æ _ö÷

ç _÷

= ç_ç + + _÷_÷+ - + +

çè   ø  

(

)

(1)

(2)

(1) (1) (1) (1) (1)

2 2

2 r

r o r

I I r r h

y

y g y

æ _ö÷ ç _÷ ç _÷ ç _÷ çè ø + + -   (24)

(

)

(

)

(

)

(

)

(

)

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

2

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

2 0

2 ,

(1) (1) (1)

1 2 , 2 2 1 4 r

rr rr rr o rr o rr

o o

r r

rr o r rr

u h

A A A r r A r r u

r

h u

A A A r r A r r

r h h

A r r

b b

qq qq qq qq

b a a a a y a y æ ö

é _- ₊ _- _- _- ùç_ç ₊ ÷_÷

ê ú_ç ÷_÷

ê ú ç ÷

ë û è ø

é ù

- ê_ê - + - - - ú_ú

ë û

é ù

+ ê_ê + - ú_ú +

ë û

(

)

(

)

(

)

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

2 2 2 2

,

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

2 2 2 2

,

1 4

4 4 4 4

r o

r r

rr rr rr rr o rr o r rr

h h

A r r

r r

h h h h

A D A A r r A r r

r b qq b b y a y

a a y

æ _ö÷ _é _ù

ç _÷

ç _÷_- _ê ₊ _- _ú

ç _÷

ç ÷_÷ ê_ë ú_û

çè ø

æ ö

é ù_ç _÷

÷

ê ú ç

+_ê + + + - + - _{ú ç}_ç + ÷_÷

÷÷ ç

ê ú è ø

ë û

(10)

(

)

(

)

(

)

(

)

(1) (3)

(3) (3)

2 2 2 2 (2)

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

2 2 2 2

2

(3) (3)

, (3)

(3) (3) (3) (3)

2 3 2 3

, ₂

4 4 4 4

1 1

4 4

r

o o

r r r

rr o r rr o

h h h h

A D A A r r A r r

r

h h h h

A r r A r r

r r

b b

qq qq qq qq qq

b b

qq

y

a a

y y

a y a

é ù

ê ú

-_ê + + + - + - _ú

ê ú

ë û

æ _ö÷

é ùç_ç _÷ é ù

+ _ë_êê + - ú_û_ú_çç_çè + ÷÷_÷_÷- ê_ê_ë + - ú_ú_û

-ø

(

)

(

)

(

)

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

(1) (1) (1) (1) (2) (1) (2)

2 1 2 1 2

0, , , 0

1

(3) (3) (3) (3) (2)

2 3 2

0, , ,

,

2 2 2 2 2

2 2 2

rz r r

r

o rr r r r r r

A w

h h h A h h

r r A u u

r

h h h A

r r A u

b _q

b

y

a b y y y y

a b y y

-+

é _æ _ö_÷ _æ _ö_÷ù

ê ç _÷ ç _÷ú

+ - _ê _ç_çç + + _÷_÷+ ç_ç_ç + + _÷_÷_ú

è ø è ø

ê ú

ë û

æ _ö÷

ç _÷

- - ç_ç_çè - - _÷_÷+ ø

(

)

(1)

(3)

(3) (2)

3 2

0

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

2 1 2 2 2 3

0 0 2 0 0

(1) (1) (1) (2

2 1 2

0 0

2 2

2 2 2 2 2 2

2 2 2

r r

r r r r r

o r r

h h

u r

h h h h h h

I u I I u

h h h

I r r u

q

h

y y

y y y y y

g y y

é _æ _ö÷ù

ê ç_ç _- _- _÷_÷ú

ê ç_çè ÷_øú

ê ú

ë û

æ ö_÷ æ ö_÷

ç _÷ ç _÷

= _ç_çç + + _÷_÷+ - _çç_ç - - _÷_÷+

è ø è ø

- + +

    

 

 ) 2 (3) (3)

(

)

(3) 2 (2) 3 (3)

0 0

2 o 2 r 2 r

h h h

I g r r h u y y

æ ö_÷ æ ö_÷

ç _÷_- _- ç _- _- _÷

ç _÷ ç _÷

ç ÷ ç ÷

ç ç

è ø è   ø

(

)

(

)

(

)

(3) (3)

(3)

0, (3)

(3) (3) (3)

3 3 0

0, ₂

(2) (2) , (3)

(3) (3) (2)

2 3

, ₂

2 (3) 3 (3)

1 1 2 2 1 4 4 r

rr o rr o

r r r

o rr r rr

rr rr

u

h h u

A r r u A r r

r r

h h

r r A A

r r h D A b b qq b qq a a y y a y æ ö

é ùç_ç ÷_÷ é ù

- _êê + - _úú_ç_ç + _÷_÷÷+ ê_ê + - _úú

ë ûè ø ë û

é æ ö_÷ ù

é ù_ê ç_ç _÷ _ú

+ _ëê_ê + - ú_ú_û_êê _çç_ç + _÷÷_÷÷- ú_ú

è ø ë û + +

(

)

(

)

(

)

(

)

(

)

(3) (3) (3) (3)

(3) 2 (3)

, (3) (3)

(3) (3) 3 (3)

, ₂

1

(3) (3) (3) (3) (3) (3) (3)

,

1 1

4

1 ,

r r r

o r rr o

r

rz o r r o rr r r

h

r r D A r r

r r

D

A r r w r r D

b b

qq qq

b b

y y

a y a

a y a b - y

æ ö

æ ö_÷_é _ù_ç _÷ æ ö_÷_é _ù

ç _÷ _ç _÷ ç _÷

ç ÷ê + - úç + ÷-ç + ÷ê + - ú

ç ÷ê úç ÷÷ ç ÷ê ú

ç ÷ë ûç ÷ ç ÷ë û

è ø è ø è ø

é ù

- ê_ê + - ú_ú + + - +

ë û

(

)

(3)

(3) (3) (3)

1

(3) (3) (3) (3) (2) (3) (2)

3 3 2 3 2

0, , , 0

(3) (2) (3) (3) (3

3 2 3 3

0 0 0

2 2 2 2 2

2 2 2 2

r

r r

r

h h h A h h

r r A u u

r

h h h h

I u I

q

b _q

y

a b y y y y

y y g

-æ _ö÷ ç _÷ ç _÷ ç ÷ ç ÷ çè ø

é æ ö_÷ æ ö_÷ù

ê ç _÷ ç _÷ú

- - _ê _ç_çç - - _÷_÷+ ç_ç_ç - - _÷_÷_ú

è ø è ø

ê ú

ë û

æ _ö÷

ç _÷

= - ç_ç - - _÷_÷

-çè   ø

(

)

(

)

(3) (3)

) 2 (2) 3 (3)

0 (3) (3) (3) (3) (3)

2 2

o r r

r o r

h h

r r u

I I r r

h

y y

y g y

æ _ö÷

ç _÷

- ç_ç - - _÷_÷

çè ø + + -     (26)

(

)

(

)

(

)

(

)

(1) (3) (1) (3)

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

(1) (2)

(2)

(1) (1) (1) (2)

, 44 ,

(3) (3) , , 1 1 r

rz rz rz rz o rz o rr

r r

rz o r r r r

rz o

w

A A A A r r A r r w

r

A r r A

r r

A r r

b b

b

a a

y y

a y y

a

æ ö

é ₊ ₊ ₊ _- ₊ _- ù_ç ₊ _÷_÷

ç

ê ú_ç_ç _÷_÷

ê ú è ø

ë û

æ ö æ ö

é ùç_ç ÷_÷ ç_ç ÷_÷

+ ê_ê + - ú_{ú ç}_ç + ÷_÷_÷+ _ç_ç + ÷_÷_÷

ë û è ø è ø

é

+ +

-ë

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(1) (3) (1) (3) (3) 1

(3) (1) (1) (1) (1)

,

1 (1) (2) (3)

(3) (3) (3) (3)

0 0 0

(1) (3) (1) (3) 0 0 , , r

r r rz o r r

rz o r r w t

o o

A r r w

r

A r r w q K w C w I I I w

r r I w r r I w

b

h h

y

y a b y

a b y

g g

-æ ö

ùç_ç ₊ ÷_÷₊ _- ₊

ê ú_ç ÷_÷

ê ú ç_{û è} ÷_ø

+ - + = - + + + + + +

- +

- 

 

(27)

For solution of the governing equations (23) to (27), various combinations of the edge condi-tions may be employed.

(11)

0 (1) (2) (3)

0

0 0

r r r

u

w

y y y ì ₌ ïï ïï ₌ ïï ïï ₌ íï ïï = ïï ï = ïïî

(28)

II. Simply-supported movable edge:

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

(1) (2) (3)

2 2

(3 (3) 3

0

0 2

0

2 2

0 2

0

r r r

r r

r r r

r r

N N N

h

N M

h h

N M N

h

N M

w

ìï + + =

ïï ïï

ï ₊ ₌

ïï ïï

ïï ₊ _- ₌

íï ïï

ïï- + =

ïï ïï = ïïïî

(29)

III. Free edge:

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

(1) (2) (3)

2 2

(3 (3) 3

(1) (2) (3)

0

0 2

0

2 2

0 2

0

r r r

r r

r r r

r r

r r r

N N N

h

N M

h h

N M N

h

N M

Q Q Q

ìï ₊ ₊ ₌

ïï ïï

ï ₊ ₌

ïï ïï

ïï ₊ _- ₌

íï ïï

ïï- + =

ïï

ïï ₊ ₊ ₌

ïïïî

(30)

Also, initial conditions of the sandwich plate are:

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

( , 0) ( , 0) 0,

( , 0) ( , 0) 0.

r r

u r u r

r r

w r w r

y y

= =

    

(31)

4 SEMI-ANALYTICAL SOLUTION FOR DYNAMIC ANALYSIS OF FG POLAR ORTHOTROPIC

SANDWICH PLATE

(12)

Based on Taylor’s expansion, the unknown displacement functions can be expressed by the fol-lowing power series:

(

)

(1) (1)

,

0 ,

(1) (1)

0 , ,

0 (1) (1)

0 , ,

( , )

( , ) , ( , )

( , ) ( , )

r x y

x y

x

x y o r x y

x

x y r x y

r t y t

u r t y t U

u r t y t U r r r t y t

u r t y t U r t y t

y y y ¥

=

ì ü ì

ì ü ì ü ï = D ï F

ï = D ï ï ï _ï _ï

ï ï ï ï _ï _ï

ï _{= D} ï₌ ï ï _- ï _{= D} ï₌ _F

í ý í ý í ý

ï ï ï ï ï ï

ï = D ï ï ï ï = D ï F

ï ï ï ï ï ï

î þ î þ ïî ïþ

å

     

(

)

(

)

0

(2) (2) (3)

,

(2) (2) (3)

, 0 (2) (2) , , ( , ) ( , ) ( , ) , ( , ) ( , ) x o x

r x y r

x

r x y o r

x

r x y r

r r

r t y t r t y t

r t y t r r r t y t

r t y t

y y y y y y ¥ = ¥ = ü ï ï ï ï ï ï ï ï

ï ï

-í ý ï ï ï ï ï ï ï ï ï ï î þ

ì ü ì ü

ï = D ï ïF ï = D

ï ï ï ï

ï _{= D} ï₌ ï_F ï _- _{= D}

í ý í ý

ï ï ï ï

ï = D ï ïF ï

ï ï ï ï

î þ î þ

å

    

(

)

(

)

(3) , (3) , 0 (3) (3) , , , 0 , , ( , ) ( , ) ( , ) ( , ) x y x

x y o

x

x y x y

x

x y o

x

x y

r r

r t y t

w r t y t W

w r t y t W r r

w r t y t W

¥ = ¥

=

ì ü ì ü

ï ï ïF ï

ï ï ï ï

ï ï₌ ï_F ï

-í ý í ý

ï ï ï ï

ï = D ï ïF ï

ï ï ï ï

î þ î þ

ì ü ì ü

ï = D ï ï ï

ï ï ï ï

ï _{= D} ï₌ ï ï

-í ý í ý

ï ï ï ï

ï = D ï ï ï

ï ï ï ï

î þ î þ

å

    (32)

whereDTis the time step, y is the time step counter and U_{x y}_, ,U_{x y}_, ,U_{x y}_, ,F(1)_{x y}_, ,F(1)_{x y}_, , F(1)_{x y}_, ,F_{x y}(2)_, ,

(2) (2) (3) (3) (3)

, , , , , , , , , , , , ,

x y x y x y x y x y Wx y Wx y

F F F F F and Wx y_, are the coefficients of series in each time step.

On the other hands, Taylor transform of the functions 1/r and 1/r2_{may be expressed by the}

following power series whose center is located at r=ro.

(

)

(

)

1 2

2

0 0

1 1 1 1

, ( 1)

s s

o o

s o s o

r r s r r

r r r r

+ +

¥ ¥

= =

æ ö_÷ æ ö_÷

ç _÷ ç _÷

= - _ç_çç- _÷_÷ - = + _çç_ç- _÷_÷

-è ø è ø

å

(33)

In practical applications, the transformed form of functions must be expressed by means of fi-nite series.

The transformed form of the governing equations may be obtained by substituting Eqs. (32) and (33) into the governing Eqs. (23) to (27) and performing some manipulations.

(

)

(

)(

)

(

)

(

)(

)

(1) (1) (1) (3)

(1) (2) (3) 1

2, 1,

0 0

(1) (1) (1) (1) 1 (1)

2, 1,

0 (3) (3) (3) (3)

( 2)( 1) ( 1)

2 1 1

2 1

X x

s

rr rr rr x y x s y

x s

rr x y x s y

s

rr x

A A A x x U x s U

A x x U x s U

A x x U

b

b b

b

c

a b b c b

a b b

+ + - + = = -+ - + - - + =

-é é ù

ê ₊ ₊ ê ₊ ₊ _- _{- +} ú

ê ê ú

ë ë û

é ù

ê _- ₊ _- ₊ _- _- _{- +} ú₊

ê ú ê ú ë û - + - +

å

(

)

(

)

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

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

, ,

0 0

(3) (3) 2 1 (1)

2 ,

0

1

( 1) ( 1)

( 1) ( 2)( 1)

2

x s

y x s y

s

x x

s s

x s y x s y

s s

x

s

rr x

x s y s

x s U

A A A s U A s U

h

A s U A x x

b

qq qq qq qq b

b

qq b

c b

c a c

a c -+ + - - + = -+ + - - -= = -+ + - -= é ù

ê _- _- _{- +} ú₊

ê ú

ë û

- + + + - +

- + + + + F

å

(

)(

)

(1) (1)

(

)

(1)

(1) 1 (1)

, 1,

0

(1) (1)

(1) (1) (1) (1) 1 (1)

1

2, 1,

0

( 1)

2 1 1

2

x s

y x s y

s x

s

rr _x _y _x _s _y

s

x s

h

A x x x s

b

b b

c

a b b c b

+ - + = -+ - + - - + = é ù

ê _- _{- + F} ú

ê ú

ë û

é ù

ê ú

+ _ê - + - + F - - - + F _ú

ê ú

ë û