On the problem solution of piezolaminated plate dynamic deformation based on a hybrid WKB-Galerkin method

This is a repository copy of On the problem solution of piezolaminated plate dynamic deformation based on a hybrid WKB-Galerkin method.

White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/100641/

Article:

Ganilova, OA and Gristchak, VZ (2008) On the problem solution of piezolaminated plate dynamic deformation based on a hybrid WKB-Galerkin method. Reports of Ukrainian Academy of Science (5). pp. 12-20.

[email protected] https://eprints.whiterose.ac.uk/ Reuse

Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website.

Takedown

If you consider content in White Rose Research Online to be in breach of UK law, please notify us by

!"# 539.3

$%&'() Ç*+,%-+,(&+È

! "#$#%&' (")*+#,- .&%/,&0#12)3) .#4)",&")5/%&6 (7#8)9+#2:"&0#12&; ,%)3)1+)<%-; (+/1:&% %/ )1%)5# 3&*"&.%)3) =!>-?/+#"2&% ,#:)./

./(0+& 1. 2., .+3(456+ 7. 8.

1 96):( 9 ;(/5&(- (9<54=:56+3(%- <=%:5>4%&,/(?%9&(@ -+,%/(+456 6 956/%-%33A@

&5-<5:(,3A@ 9,/B&,B/+@ [1, 2, 3], 595CAD (3,%/%9 </%E9,+64)%, /+99-5,/%3(%

,/%@945D35D <=%:5>4%&,/(?%9&5D <4+9,(3A, 3+@5E)0%D9) 6 >4%&,/(?%9&5- <54% <5E

E%D9,6(%- E(3+-(?%9&5D 63%;3%D 3+F/B:&(. $4%EB%, 5,-%,(,=, ?,5 <5E5C35F5 /5E+ :+E+?+

CA4+ /+99-5,/%3+ 6 /+C5,% [4] E4) 6):&5B</BF(@ <4+9,(3, 95F4+935 -5E%4( 945(9,A@

5C545?%& (: 6):&5B</BF(@ <=%:5-+,%/(+456, 5<(9+335D 6 [5]. G%;%3(% 5935635F5

B/+63%3() :+E+?( </565E(459= -%,5E+-( B9/%E3%3() ( &53%?3A@ >4%-%3,56.

7C/+0+%, 3+ 9%C) 63(-+3(% ,5, H+&,, ?,5 <56%E%3(% 3+</)I%335-E%H5/-(/B%-A@

<4+9,(3 9 (3,%F/(/56+33A-( 6 3(@ +&,(6+,5/+-( ( 9%395/+-(, 6 5935635-,

/+99-+,/(6+%,9) 3+ 59356% -%,5E+ &53%?3A@ >4%-%3,56 [4, 6]. J5>,5-B 595CAD (3,%/%9

</%E9,+64)K, (994%E56+3() 6 >,5D 5C4+9,( 3+ 59356% ,5?3A@ ( </(C4(I%33A@

+3+4(,(?%9&(@ -%,5E56, <5:654)K0(% </56%9,( &+?%9,6%33AD +3+4(: <54B?%33A@

/%;%3(D.

J/( +3+4(:% /%+4=3A@ -%@+3(?%9&(@ </5'%9956 65:3(&+%, 3%5C@5E(-59,= 6

(3,%F/(/56+3(( 9(3FB4)/3A@ E(HH%/%3'(+4=3A@ B/+63%3(D, 95E%/I+0(@ C54=;(% (

-+4A% <+/+-%,/A </( 9,+/;%D </5(:65E35D ( 3% E5<B9&+K0(@ 6 /)E% 94B?+%6 ,5?3A@

/%;%3(D. 1 >,5- 94B?+% </(C4(I%335% /%;%3(% <59,+64%335D :+E+?( BE5C35 (9&+,= 6

6(E% +9(-<,5,(?%9&5F5 /%;%3() <5 E+335-B <+/+-%,/B.

*5I35 3+:6+,= /)E <5E@5E56, )64)K0(@9) F(C/(E3A-(, &5,5/A% 59356A6+K,9) 3+

(E%% <59,/5%3() ÇB,5?3)K0%F5È /%;%3(), <59,/5%335F5 3+ 59356% <594%E56+,%4=359,(

&55/E(3+,3A@ HB3&'(D +9(-<,5,(?%9&5F5 /+:45I%3() [7, 8, 9, 10]. L94( 9<%'(+4=3A-

5C/+:5- 5</%E%4(,= 3%(:6%9,3A% <+/+-%,/A, ,5 ÇB,5?3)K0%%È F(C/(E35% /%;%3(%

(9@5E35F5 B/+63%3() @5/5;5 95F4+9B%,9) 9 ,5?3A- /%;%3(%- 3+ C54=;5- (3,%/6+4%

(:-%3%3() &5>HH('(%3,56 B/+63%3(). .(C/(E3AD 1#M-.+4%/&(3 -%,5E, <5E/5C35%

5<(9+3(% &5,5/5F5 </%E9,+64%35 6 /+C5,+@ [7, 8] E%-539,/(/B%, 595C%335 @5/5;(%

/%:B4=,+,A </( 5,A9&+3(( </(C4(I%335F5 /%;%3() 9(3FB4)/3A@ E(HH%/%3'(+4=3A@

B/+63%3(D, 6 ?+9,359,( 9 <%/%-%33A-( &5>HH('(%3,+-(, 95E%/I+0(@ <+/+-%,/ </(

9,+/;%D </5(:65E35D, 3% :+6(9(-5 5, %F5 6%4(?(3A.

1 E+335D /+C5,% /+99-+,/(6+%,9) ,/%@945D3+) <=%:5>4%&,/(?%9&+) 5/,5,/5<3+)

<4+9,(3+, 3+@5E)0+)9) 6 >4%&,/(?%9&5- <54% <5E E%D9,6(%- E(3+-(?%9&5D 63%;3%D

3+F/B:&(, 9 B?%,5- <%/%-%335F5 65 6/%-%3( &5>HH('(%3,+ E%-<H(/56+3(). G%;%3(%

<59,+64%335D :+E+?( 59356+35 3+ (9<54=:56+3(( F(C/(E35F5 1#M-.+4%/&(3 -%,5E+.

G+99-5,/(- </)-5BF54=3BK <4+9,(3&B (a_!b_{) ;+/3(/35 5<%/,BK <5 69%-B &53,B/B.}

7/,5,/5<3+) <4+9,(3&+ 9(--%,/(?35 95C/+3+, 5,359(,%4=35 &55/E(3+,35D <459&59,(

xOy, (: <=%:5>4%&,/(?%9&5F5 -+,%/(+4+ ( (-%%, ,540(3B h_{. J/(45I%3+) 3+F/B:&+}

(3('((/B%,9) >4%&,/(?%9&5D 959,+64)K0%D V(x,y,t) ( -%@+3(?%9&5D E(3+-(?%9&5D

959,+64)K0%D 6 H5/-%:

2 2 )

( 2 sin sin ) (

t w h t w h t b

y a

x t

q Z_m

! ! " ! ! "

= % % #$ # , (1)

FE%

!

= =

3

1

k k k

h h

"

G(9. 1. 7/,5,/5<3+) 9(--%,/(?35 95C/+33+) <=%:5>4%&,/(?%9&+) <4+9,(3&+

"4) /+99-+,/(6+%-5D <4+9,(3&( (/(9. 1), 95F4+935 [11], E4) I%9,&59,%D <54B?(- 94%EBK0(% 6A/+I%3():

(

)

[

]

[

(

)

]

, 3 2 , 2 ,

0 ik ₀ ik0 ₀ ik ik ₀3 ik0 3 ₀3 ik

ik C B B D B B

K _{= = ! ! "! = ! + ! "! (3)}

B?(,A6+) ?,5

!

!

!

₁ = ₂ = , Bik! =Bik!! =Bik. (4) 1 >,5- 94B?+% 593563A% B/+63%3() E6(I%3() <4+9,(3&(, 6 /+-&+@ F(<5,%: 5<(9+33A@ 6 [11, 12, 13], -5I35 :+<(9+,= 6 6(E% [1, 14, 15]:

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

, , , 1 2 2 0 0 32 1 2 2 0 0 31 0 23 0 13 33 1 0 32 2 0 2 23 0 12 0 22 1 0 31 2 0 2 13 0 12 0 11

!

!

!

!

= = = = " " # " " # = # # " " + " " = # + " " + " " = # + N k k k N k k k m ik ik ik N k k ik ik ik N k k ik ik ik y V z e x V z e Z K L u K L w D L y V e t h w K L u C L C L x V e t u h w K L C L u C L k k k k $ $ % $ % $ (5) FE%

(

)

(

)

(

) (

)

(

)

(

)

(

)

(

2

)

,

(

)

2

(

2

)

.

, 2 , , , 2 2 4 66 12 4 4 22 4 4 11 33 2 3 66 12 3 3 22 23 2 3 66 12 3 3 11 13 2 66 12 12 2 2 66 2 2 22 22 2 2 66 2 2 11 11 y x D D y D x D D L x y K K y K K L y x K K x K K L y x C C C L x C y C C L y C x C C L ik ik ik ik ik ik ! ! ! + + ! ! + ! ! = ! ! ! + + ! ! = ! ! ! + + ! ! = ! ! ! + = ! ! + ! ! = ! ! + ! ! = (6)

#55/E(3+,+ 9/%E(335D <459&59,( k-F5 945), 95F4+935 [15], 5</%E%4%3+ &+&

(

1

)

0 2 1

! +

= k k

k

z z

z .

7,-%,(-, ?,5 (9@5E3+) 9(9,%-+ (5), 9 B?%,5- (3), /+9<+E+%,9) 3+ E6%

9+-59,5),%4=3A% <5E9(9,%-A: <%/6A% E6+ B/+63%3() </%E9,+64)K, 95C5D 3%&5,5/BK

<459&BK :+E+?B, + ,/%,=% Ð :+E+?B <5<%/%?35F5 (:F(C+ <4+9,(3&(, &5,5/+) 3+9 (3,%/%9B%,:

(

)

Z

y w D y x w D D x w D = ! ! + ! ! ! + + ! ! 4 4 22 2 2 4 66 12 4 4

11 2 2 , (7)

FE%

_!

_!

= = " " # " " # = N k k k N k k k m y V z e x V z e Z

Z k k

1 2 2 0 0 32 1 2 2 0 0 31 .

G+99-5,/(- /%;%3(% 5935635F5 B/+63%3(), 95F4+935 &4+99(?%9&5D ,%5/((

-35F5945D3A@ <4+9,(3, 6 94B?+% &5FE+ 3+ <4+9,(3&B E%D9,6B%, E(3+-(?%9&+) 3+F/B:&+

6(E+ (1):

(

_{12 66}

)

₂4 _{2 22} 4₄ 2₂

4 4

11 2 2 ( )sin sin 2 ( )

t w h t w h t b y a x t q y w D y x w D D x w D ! ! " ! ! " = ! ! + ! ! ! + + ! ! # #$ % %

. (8)

G%;%3(% CBE%- (9&+,= 6 6(E%:

b y a x t f t

w( )= ( )sin! sin! . (9)

N5FE+ 5C5:3+?(6 ! " # $ %

& + ₊

+

= 22₄

2 2 66 12 4 11 4 2 2 ~ b D b a D D a D

D ' ;

h h t t h t q t q h D D ! " ! # # ! ! 1 ; ) ( ) ( ; ) ( ) ( ~ ; ~ ~ ~ 2 2 2 2

2 = = =

B/+63%3(% (8) <%/%<(;%- 6 6(E% 94%EBK0%F5 3%5E35/5E35F5 E(HH%/%3'(+4=35F5 B/+63%3() 9 <%/%-%33A-( &5>HH('(%3,+-(:

) ( ~ ) ( ~ ~ ) ( ) ( 2 ) ( 2 t q t f D t f t t

f !! + " ! + =

# , (11) FE% _!2 _{- <+/+-%,/ </( 9,+/;%D </5(:65E35D.}

G%;%3(% B/+63%3() (11) (-%%, 6(E:

) ( ) ( )

(t f t f t

f = _c + _p , (12) FE% f_c(t) - /%;%3(% 5E35/5E35F5 B/+63%3() ( fp(t) - ?+9,35% /%;%3(%.

O+DE%- /%;%3(% 5E35/5E35F5 E(HH%/%3'(+4=35F5 B/+63%3() <54B?%335F5 (:

B/+63%3() (11). $5F4+935, 5<(9+335F5 6 [7, 8], F(C/(E35F5 1#M-.+4%/&(3 -%,5E+

1#M-</(C4(I%3(% (-%%, 6(E:

! " # $ % & ' ( ) * + , + =

-t a dt t f t f t

f( , ) exp 1 ₀( ) ₁( )

.

. . (13)

J5E9,+64)) (13) 6 5E35/5E35% B/+63%3(%, <54B?(-

0 ~ ~ 1 ) ( 2 1 2 1 1 1 0 1 0 2 1 2 0 2 1 0 2 = + !" # $% & + + !" # $% & + + + ' +

' _{f f f f f t f f} D

f ( ) ( ( (

( . (14)

J/(/+63(6+) &5>HH('(%3,A </( 5E(3+&56A@ 9,%<%3)@ <+/+-%,/+ !, </(@5E(- &

9(9,%-% B/+63%3(D: !" ! # $ = + % = + + , 0 2 , 0 ~ ~ ) ( 2 1 0 0 1 2 0 f f f D f t

f & ₍₄₍

! " ! # $ = % % & % = . 0 ~ ~ ) ( , ln 2 1 0 3 0 0 0 1 f D f f t f dt d f ' (15)

G%;%3(%, 6,5/5F5 B/+63%3() 9(9,%-A (15), 3+DE%35 (9<54=:B) <5E9,+356&B f₀ =U(t)V(t).

N+&(- 5C/+:5-, /%;%3(% 9(9,%-A (15) (-%%, 6(E:

! ! ! " ! ! ! # $ % & & & ' ( ) ) ) * + _% ± = , = ,

%

. ) ( 2 , ln 2 1 ) ( ~ ~ 2 / 1 ) ( ~ ~ 2 0 0 1 t dt D t dt D e dt t e i f f dt d f (16)

O+ 6,5/5- ;+F% /%;%3(), 95F4+935 F(C/(E35F5 1#M-.+4%/&(3 -%,5E+, (9@5E3+) HB3&'() </%E9,+64)%,9) 6 H5/-%:

( )

[

]

_! " # $ % & + =

_'

t a dt t f i t

f , exp ( ) ( ) ( )

~

0 02

01 ( ) (

)

( , (17)

FE% f0 - <54B?%33+) E4) 1#M Ð /%;%3() HB3&'() (16). N5FE+ 5E35/5E35% B/+63%3(%

<54B?%335% (: (11) </(-%, 6(E:

(

)

(

)

[

_{01 02 0 01}2 2 _{01 02 02}2 ₀2

]

2 ( )

[

_{01 02}

]

₀ ~~ 0

2 _{+ "+ + ! + + + =}

D f i t f i f

i# # # # # $ # #

#

% . (18)

$5F4+935 B9456() 5/,5F53+4=359,(

_"

(

₀,..., , ₀,..., , ₀!,..., ( #1), ,

)

( ) =0

b a i n N N

N u u u u t u t dt

R_{% % $}

E4) R ( N+1 &55/E(3+,35D HB3&'(( [7, 8], B?(,A6+) ?,5 f0 =±if0, <54B?(-:

(

)

(

)(

)

[

]

[

[

]

~~

(

)

0,

) ( 2 2 0 2 0 02 01 3 0 2 02 02 01 2 01 0 0 02 01 2 = !" # ± + $ $ + + ± + + $ + % $ $

&

dt f D i f i t f i i f f i b a ' ' ( ' ' ' ' ' ' ) (19)

! " #

= +

= + +

$

, 0 2

, 0

02 01 01

2 02 2

01 02

% % %

% %

%

B A

W B

B A

(20)

FE%

[

2 _{0 0} 2 ( ) ₀2

]

,

_!

2 ₀3 ,

_!

~~ ₀ .

!

" # " = =

=

b

a b

a b

a

dt f D W dt f

B dt f t f

f

A $ % $ (21)

G%;%3(% 9(9,%-A (20) (-%%, 6(E:

! ! " !! # $

+ =

%

=

. 2

, 2

4

02

2

01

B A

B A BW

& &

(22)

N+&(- 5C/+:5-, F(C/(E35% /%;%3(% 5E35/5E35F5 B/+63%3() 5</%E%4)%,9) H5/-B45D (17), 9 B?%,5- (22).

"4) <54B?%3() ?+9,35F5 /%;%3() B/+63%3() (11), 6 &+?%9,6% </(-%/+ /+99-5,/(-

E6+ 94B?+) &5>HH('(%3,+ :+,B@+3() </( :+E+335- 63%;3%- 65:E%D9,6(( q(t):

_!(t₎₌Ct ₍ !₍t₎₌MeKt, (23) FE% C, M, K Ð </5(:654=3A% &539,+3,A. N5FE+ F(C/(E35% /%;%3(% (17) </(-%, 6(E

! " # $

% &

+

=e' c Dt c Dt

t

f( , ) Dt sin ~~ cos ~~

~

01 2

01 1 ~ ~

02 ( (

) (

, (24)

FE% c1, c2 - </5(:654=3A% &539,+3,A. P+9,35% /%;%3(% 5</%E%4%35 <5 -%,5EB 6+/(+'((

</5(:654=3A@ <59,5)33A@.

$5F4+935 (12) 5C0%% /%;%3(% (-%%, 6(E:

! " # $

% &

+ + +

=e' c c Dt c c Dt

t

f( ) Dt ( _{1 1})sin ₀₁ ~~ ( _{2 2})cos ₀₁ ~~

~ ~

02 ( (

(

, (25)

FE% =

_!

dt

D t D e

t q c

t D

~ ~

~ ~ cos )

( ~

01 2

01 ~

~

1

02

" #

"

"

, ="

_!

dt

D t D e

t q c

t D

~ ~

~ ~ sin )

( ~

01 2

01 ~

~

2

02

# $

#

#

.

"4) 95<59,+64%3() <54B?%335F5 </(C4(I%335F5 +3+4(,(?%9&5F5 /%;%3() 9

/%:B4=,+,+-( </)-5F5 ?(94%335F5 +3+4(:+ /+99-5,/(- &6+E/+,3BK <4+9,(3&B

(a=b=2_{-), 959,+64%33BK (: ,/+396%/9+4=35 (:5,/5<3A@ <=%:5>4%&,/(?%9&(@ 945%6,}

<459&59,( (:5,/5<((, &5,5/A@ <+/+44%4=3A &55/E(3+,35D <459&59,( xOy.

J=%:5>4%&,/(?%9&(- -+,%/(+45- 945%6 <4+9,(3A )64)%,9) BaTiO3, ,5FE+ 95F4+935 [1, 11]:

, / 10

120 9 2

22

11 Q " !

Q = = ! _{Q12 =}36,2_!109"/!2, _{Q66 =}42_!109"/!2_,

_!

"

=

2 /

2 /

2

h

h ij ij Q z dz

D ,

, /

3

,

12 2

0

0

! "#

e

e k ₌ k _{=! " =5,7!10}3"#_/!3_{, (26)}

10 ,

1 =

= _R

! , 8

0 1

=

q , =0,1, h=0,2

a h

, 0,1

2 2 ₀ = = = ! h

! ,

FE% h Ð <543+) ,540(3+ <4+9,(3&(.

J4+9,(3&+ <5E6%/F+%,9) -%@+3(?%9&5D 3+F/B:&% 6 H5/-% (1) ( >4%&,/(?%9&5D 6 H5/-%:

b y a

x t

t y x

Vk_{( , , )="}k_{( )sin}! _sin! _{, (27)}

E%D9,6(% &5,5/5D 5E(3+&565 65 69%@ 945)@, ,.%. _!k(t)_=!(t).

(

)

. sin sin ) ( sin sin ) ( ) ( 2 sin sin ) ( 2 2 1 2 0 0 32 1 2 0 0 31 2 2 4 4 22 2 2 4 66 12 4 4 11

!

!

= = " # $ % & ' + " # $ % & ' + + ( ( ) ( ( ) = ( ( + ( ( ( + + ( ( N k k N k k b y a x b t z e b y a x a t z e t w h t w h t b y a x t q y w D y x w D D x w D k

k * * *

+ * * * + , ,-* * (28)

1 &+?%9,6% </(-%/+ <545I(-, ?,5 q(t)=R!(t) ( _{"(t) q cos!} t

0

= .

N5FE+ B/+63%3(% (28) </(-%, 6(E:

(

)

b y a x t z e b z e a R t w h t w h t y w D y x w D D x w D N k k N k k k

k ! !

" ! ! # #$ sin sin ) ( ) ( 2 2 2 1 0 0 32 2 1 0 0 31 2 2 2 4 4 22 2 2 4 66 12 4 4 11 % % & ' ( ( ) * + , -. / 0 + + , -. / 0 + + + 1 1 2 1 1 2 = 1 1 + 1 1 1 + + 1 1

3

3

= = (29)

G%;%3(% >,5F5 B/+63%3() +3+45F(?35 /%;%3(K B/+63%3() (8).

"4) <54B?%3() F/+H(?%9&5D (3,%/</%,+'(( <54B?%335F5 /%;%3() (25)

659<54=:B%-9) </5F/+--3A- &5-<4%&95- MAPLE. "4) </)-5F5 ?(94%335F5 /%;%3()

(9@5E35F5 B/+63%3() (29) (9<54=:B%- -%,5E GB3F%-#B,,+.

G%:B4=,+,A ?(94%335F5 +3+4(:+ 3+ 59356% </%E45I%335F5 F(C/(E35F5

+9(-<,5,(?%9&5F5 -%,5E+ ( ?(94%335F5 -%,5E+ GB3F%-#B,,+ </%E9,+64%3A 3+ /(9.2 ( 6

,+C4('% 1.

G(9. 2. J56%E%3(% @+/+&,%/(9,(?%9&5D HB3&'(( _f(t) </( &5>HH('(%3,% E%-<H(/56+3()

t t₎₌

(

! (Ð Ð Ð ), _!(t₎₌₅t (---), t e t₎ 0.09

( =

! (Ð).

N+C4('+ 1 t

t) 1

( = !

t !"#$"%&'($()(&"( _f(t) *"+,(&&'($()(&"(-'

.(/'%010&2(-30//4

0.03 -0.1071743990 -0.105764863015825688 0.035 -0.1124892722 -0.133609903608793212 0.04 -0.1154273314 -0.159883542812304684 2 0.04565994967 0.0422282881712131159 3 0.1086357085 0.107532028638544108 4 0.07173229785 0.0716948593241360234 5 -0.03112145658 -0.0311270814887880908

$5<59,+64%3(% ?(94%33A@ /%:B4=,+,56 E%-539,/(/B%, E59,+,5?35 6A95&BK

>HH%&,(6359,=, <5 6%4(?(3% <+/+-%,/+, </(C4(I%335F5 +3+4(,(?%9&5F5 /%;%3(),

65:/+9,+K0BK 9 ,%?%3(%- 6/%-%3( 3% :+6(9(-5 5, @+/+&,%/+ (:-%3%3() &5>HH('(%3,+

E%-<H(/56+3().

7,-%,(-, ?,5 <54B?%335% </(C4(I%335% +3+4(,(?%9&5% /%;%3(% (25) B/+63%3() (8)

-5I%, CA,= (9<54=:56+35, 95F4+935 [3], 6 94B?+% &54%C+3(D 5E35945D35D

<=%:5>4%&,/(?%9&5D <4+9,(3A, >HH%&, 96):+3359,( >4%&,/(?%9&5F5 ( -%@+3(?%9&5F5

<54%D &5,5/5D 3+@5E(, 5,/+I%3(% 6 (:-%3%3(( I%9,&59,3A@ @+/+&,%/(9,(&

(&5>HH('(%3,+ JB+9953+ ( I%9,&59,( </( '(4(3E/(?%9&5- (:F(C%), + ,+& I% 6 94B?+%

&54%C+3(D C(-5/H35D <=%:5>4%&,/(?%9&5D <4+9,(3A. J54B?%335% /%;%3(% <5:654)%,

/%;+,= /)E </+&,(?%9&(@ :+E+?, 96):+33A@ 9 6A?(94%3(%- 6%4(?(3A ,5&+ </565E(-59,( 6

'%<( F%3%/+,5/+ ( <+/+-%,/56 >4%&,/5B</BF5F5 <54) (3+</)I%3359,= >4%&,/(?%9&5F5

( ,.E.), 9 +3+4(:5- /%:53+393A@ ( +3,(/%:53+393A@ ?+9,5, &54%C+3(D 6 <%/%-%335-

>4%&,/(?%9&5- <54% ( +3+4(:5- E/BF(@ E(3+-(?%9&(@ >HH%&,56, 65:3(&+K0(@ 6

96):+33A@ >4%&,/5-%@+3(?%9&(@ <54)@.

1. Ishihara M., Noda N. Nonlinear Dynamic Behaviour of a Piezothermoelastic Laminated Plate with Anisotropic Material Properties// Acta Mechanica. Ð 2003. Ð Q166. Ð P. 103 Ð 118. 2. Kasap S. O. Principles of Electrical Engineering Materials and Devices. Ð Canada:

McGraw-Hill, 2000. Ð 690 p.

3. ./(3?%3&5 1.N., !4(,&5 8.R., SB4=F+ O.8. T4%&,/5B</BF59,= Ð #.: O+B&. EB-&+, 1989.

Ð 280 9. Ð (*%@+3(&+ 96):+33A@ <54%D 6 >4%-%3,+@ &539,/B&'(D: 1 5,( ,.; N.5)

4. #+/3+B@56 1. .., #5:456 8. 1., J),%'&+) L. 1. "%-<H(/56+3(% &54%C+3(D 6):&5B</BF(@

<4+9,(3 9 <5-50=K /+9</%E%4%33A@ <=%:5>4%&,/(?%9&(@ 6&4K?%3(D // 8&B9,. 6U93.- 2002. Ð 5, - N 4. Ð C. 15-32.

5. #+/3+B@56 1. .., #(/(?5& V. R., #5:456 1. V. T4%&,/5-%@+3(?%9&(% &54%C+3() (

E(99(<+,(63AD /+:5F/%6 6):&5B</BF(@ ,53&59,%33A@ >4%-%3,56 9 <=%:5>HH%&,5- // J/(&4. -%@. Ð 2001. Ð 37, N 2. Ð $. 45-77.

6. Cristovao M. Mota Soares, Carlos A. Mota Soares, Victor M. Franco Correia. Optimal Design of Piezolaminated Structures// Composite Structures. Ð 1999. Ð Q47. Ð P. 625-634

7. Gristchak V. Z., Ganilova O. A. Application of a Hybrid WKB-Galerkin Method in Control of the Dynamic Instability of a Piezolaminated Imperfect Column//Technische Mechanik. Ð 2006. Ð Q26. Ð P. 106 Ð 116.

8. Gristchak V., Dmitrieva Ye. A Hybrid WKB-Galerkin Method and Its Application// Technische Mechanik. Ð 1995. Ð Q15. Ð P. 281-294.

9. Geer J. F., Andersen C. M. Hybrid Perturbation Galerkin Tecnique with Application to Slender Body Theory// SIAM J. Appl. Math. Ð 1989. Ð Q49. Ð P. 344-361.

10. Geer J. F., Andersen C. M. Investigating A Hybrid Perturbation-Galerkin Technique Using Computer Algebra / NASI-18107. Ð Hampton, Virginia. Ð 1988. Ð 26 p.

11. 8-C+/'B-)3 $. 8. N%5/() +3(:5,/5<3A@ <4+9,(3. J/5?359,=, B9,5D?(659,= (

&54%C+3(). Ð *59&6+: O+B&+,1987. Ð 3609.

12. 8-C+/'B-)3 $.8. 7C0+) ,%5/() +3(:5,/5<3A@ 5C545?%&. Ð *59&6+: O+B&+,1974. Ð

4489.

13. W%@3('&(D $. .. 83(:5,/5<3A% <4+9,(3&(. Ð *59&6+: R(:-+,F(:, 1957. Ð 4639.

14. Lee C. K. Theory of Laminated Piezoelectric Plates for the Design of Distributed Sensors and Actuators. Part 1: Governing Equations and Reciprocal Relationships// Journal of the Acoustical Society of America. Ð 1990. Ð 87, Q3. Ð P. 1144 Ð 1158.

1 /+C5,% /+99-+,/(6+%,9) <56%E%3(% ,/%@945D35D 9(--%,/(?35 95C/+335D

5/,5,/5<35D <=%:5>4%&,/(?%9&5D <4+9,(3&(, 3+@5E)0%D9) <5E E%D9,6(%- 35/-+4=35

</(45I%335D E(3+-(?%9&5D 3+F/B:&( q(x,y,t), B?(,A6+) <%/%-%33AD 65 6/%-%3(

&5>HH('(%3, E%-<H(/56+3().

"4) /%;%3() <59,+64%335D :+E+?( CA4 </(-%3%3 F(C/(E3AD -%,5E

1#M-.+4%/&(3+. J54B?%33A% /%:B4=,+,A CA4( 95<59,+64%3A 9 /%:B4=,+,+-( </)-5F5

?(94%335F5 -%,5E+, + ,+& I% </5+3+4(:(/56+35 64()3(% &5>HH('(%3,+ E%-<H(/56+3() 3+ <54B?%33A% /%:B4=,+,A.

On the problem solution of piezolaminated plate dynamic deformation based on a hybrid WKB-Galerkin method

The paper deals with the problem of behavior of a symmetrical orthotropic piezoelectric sandwich plate subjected to the dynamic normal loading q(x,y,t), taking into consideration variable in time damping coefficient.

A Hybrid WKB-Galerkin method was applied to solve the problem. Obtained results are compared with numerical method results. The damping coefficient effect on the solution is studied.

./(0+& 1. 2.

2+<5/5I9&(D 3+'(53+4=3AD B3(6%/9(,%,

[email protected]

.+3(456+ 7. 8.

2+<5/5I9&(D 3+'(53+4=3AD B3(6%/9(,%,

[email protected]

2+<5/5I9&(D 3+'(53+4=3AD B3(6%/9(,%,

G(9B3&(

G(9. 1. 7/,5,/5<3+) 9(--%,/(?35 95C/+33+) <=%:5>4%&,/(?%9&+) <4+9,(3&+

0.02 0.04 0.06 0.08 0.1

t

-0.14 -0.13 -0.12 -0.11 -0.09 -0.08 f

G(9. 2. J56%E%3(% @+/+&,%/(9,(?%9&5D HB3&'(( _f(t) _{</( &5>HH('(%3,% E%-<H(/56+3()}

t t₎₌

(

! (Ð Ð Ð ), !₍t₎₌₅t (---), t e t₎ 0.09

( =

! (Ð).

h 2_!0

2 1

! !

x