On Free Vibrations of Orthotropic Plates in the Presence of Viscous Resistance

Ա ԱՍ Ա ՈՒԹ ՈՒ Ա Ա Ա Ա Ա Ղ Ա

64, №1, 2011

539.3

O ЫХ Х ЫХ

Ч

. ., . .

ч : , ,

, , .

Keywords: viscous resistance, frequency of free vibrations, symmetric and antisymmetric problems,

shear vibrations,longitudinal vibrations.

Ա .Ա.,Ս . .

,

: Ա

, 3 – 2 , 1

: Ո - ,

3 :

Aghalovyan L.A., Sargsyan .Z.

On Free Vibrations of Orthotropic Plates in the Presence of Viscous Resistance

The three-dimensional problem of elasticity theory of the free vibrationsof orthotropic platesin the presence of viscous resistance, on the facial plane of which mixed-boundary conditions of elasticity theory are given is considered. By the asymptotic method it is shown that 3 groups of free vibrations, 2 groups of shearing and 1 group of longitudinal free vibrations are appeared. The stress-deformed states, principal values of frequencies and the forms of natural vibrations of plates relevant to 3 groups of free vibrations are determined.

,

. , 3 –

2 , 1 .

-, ,

.

( ,

, )

- ё .

ё

- , [1-5].

, ё , ё

.

1.

( )

0

{( , , ), , , , }

D= x y z x y ∈D z≤h h<<l ( D0– , 2h – ,

l– ), ё

[6] :

0

zz xz yz

,

.

[5].

,

/ , = / , = /

x l y l z l

ξ = η ζ

/ , = / , = /

U=u l V v l W w l. h

l

ε = ,

:

( )

(

)

(

( ) ( ) ( )

)

(

)

1

2 2 2

* * *

, ,

, , ;

,

1, 2,3

0,

,

, ,

,

,

,

,

.

s s i t

jk

s s s

s i t s

s

e

x y z

j k

s

N

U V W

U

V

W

e

h

− + ω

αβ

ω

σ = ε

σ

α β =

=

=

= ε

ω = ε ω

ω = ρ ω

(1.3)

(1.3)

0 0 0 0

n

n n k n k

n n n k

a b a b

∞ ∞ ∞

−

= = = =

⎛ ⎞⎛ ⎞₌

⎜ ⎟⎜ ⎟

⎝

∑

⎠⎝

∑

⎠

∑∑

,

:

( ) ( ) ( )

(

)

( )

( )

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

( )

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

( )

( )

1 1

13 11 12

* * *

1 1 1

11 11 12 22 13 33 66 12

1 1

12 11 22 22 23 33 55 13

13 11 23

2

0, (1, 2, 3; , ,

),

,

,

s

s s

s m

m n n m

s s s

s s s s

s s s

s s s s

s

s

iK

U

U V W

U

U

V

a

a

a

a

V

W

U

a

a

a

a

W

a

a

− −

− −

− − −

− −

∂σ

∂σ

∂σ

+

+

+ ω

ω −

ω

=

∂ξ

∂η

∂ζ

∂

₌

_{σ +}

_{σ + σ}

∂

₊

∂

₌

_σ

∂ξ

∂η

∂ξ

∂

₌

_{σ +}

_{σ +}

_σ

∂

₊

∂

₌

_σ

∂η

∂ξ

∂ζ

∂

₌

_{σ +}

_σ

∂ζ

( ) ( ) ( )1 ( ) ( )

22 33 33 44 23

1

,

2

,

0, ,

0, ,

s s

s s

W

V

s

a

a

K

k h

n

m m

s

−

∂

∂

+

σ

+

=

σ

∂η

∂ζ

=

ρ

=

=

(1.4)

1

k – , ρ – .

(1.4) U( )s ,V( )s ,W( )s

:

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

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

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

1 1 1 1

12 11 23 22 12

66

1 1 1

13 22 13 12 33

55

1 1

23 33 11 23

44

1

,

,

1

,

,

1

,

s s s s s

s s

s s s s s

s s

s s s s

s s

V

U

W

U

V

A

A

A

a

U

W

W

U

V

A

A

A

a

V

W

W

U

A

A

a

− − − −

− − −

− −

⎛

∂

∂

⎞

∂

∂

∂

σ =

⎜

_⎜

+

⎟

_⎟

σ = −

+

−

∂ξ

∂η

∂ζ

∂ξ

∂η

⎝

⎠

⎛

∂

∂

⎞

∂

∂

∂

σ =

⎜

_⎜

+

⎟

_⎟

σ = −

−

+

∂ζ

∂ξ

∂ζ

∂ξ

∂η

⎝

⎠

⎛

∂

∂

⎞

∂

∂

σ =

⎜

_⎜

+

⎟

_⎟

σ =

−

∂ζ

∂η

∂ζ

∂

⎝

⎠

( )1 13

,

s

V

A

−

∂

−

ξ

∂η

(1.5)

2 2

2

22 33 23 11 33 13 12 33 13 23 11 22 12

11

, , ,

22 33 12

,

a a

a

a a

a

a a

a a

a a

a

A

=

−

A

=

−

A

=

−

A

=

−

2 2 2 11 23 12 13 22 13 12 23

13 , 23 , 11 22 33 2 12 13 23 11 23 22 13 33 12.

a a a a a a a a

A = − A = − Δ =a a a + a a a −a a −a a −a a

Δ Δ

U( )s ,V( )s ,W( )s (1.4) :

( )

(

)

( ) ( )

(

)

2

55 * * *

2

55 44 11

2

,

0, ,

0, ,

, ,

;

,

,1

.

s

s m s

m n n m U

U

a

iK

U

R

n

m m

s

U V W a

a

A

− −

∂

₊

_ω

_{ω −}

_ω

₌

₌

₌

∂ζ

(1.6)

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

( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1

2 2

11 12 12 22

55 44

1 1

1 1

2 2

23 13 13 23

11 11 11

,

1

.

s s s s

s s

s s

U V

s s

s s

s W

W

W

R

a

R

a

A

U

A

V

R

A

A

A

− − − −

− −

− −

− −

⎛

∂σ

∂σ

⎞

⎛

∂σ

∂σ

⎞

∂

∂

= −

−

⎜

_⎜

+

⎟

_⎟

= −

−

⎜

_⎜

+

⎟

_⎟

∂ζ∂ξ

_⎝

∂ξ

∂η

_⎠

∂η∂ξ

_⎝

∂ξ

∂η

_⎠

⎛

∂σ

∂σ

⎞

∂

∂

=

+

−

⎜

_⎜

+

⎟

_⎟

∂ζ∂ξ

∂η∂ξ

_⎝

∂ξ

∂η

_⎠

(1.7)

,

(1.6) s=0:

( )

(

)

( )

0 2

0 2

55 *0 *0

2

2

0

U

a

i K

U

∂

+

ω −

ω

=

∂ζ

(

U V W a

, ,

;

55

,

a

44

,1

A

11

)

.

(1.8)

(1.8) :

( ) ( )

(

)

( )

(

)

(

)

0 0 2 0 2

1 55 *0 *0 2 55 *0 *0

55 44 11

cos

2

sin

2

, ,

;

,

,1

.

U U

U

C

a

i K

C

a

i K

U V W a

a

A

=

ω −

ω ζ +

ω −

ω ζ

(1.9)

(1.9), , ё (1.5)

(1.1) (1.2) σxz

( )0

0

U

∂

₌

∂ζ

ζ = ±

1

, (1.10)

( )0 ( )0 1U , 2U C C :

( ) ( ) ( ) ( )

0 0

1 0 2 0

0 0

1 0 2 0

sin

cos

0,

sin

cos

0,

U U U U

U U U U

C

C

C

C

⎧−

γ +

γ =

⎪

⎨

γ +

γ =

⎪⎩

(1.11)

(

2

)

0 55 *0 2 *0

U a i K

γ = ω − ω .

:

0 0

sin

γ

_U

cos

γ =

_U

0

, (1.12)

2 :

0 0

sin

γ = ⇒ γ = π

_U

0

_U

n

,

n

∈ Ν

, (1.13)

(

)

0 0

cos

0

2

1 ,

2

U U

n

n

π

γ = ⇒ γ =

+

∈ Ν

. (1.14)

(1.13) ,

:

2 2 2 *0

55

,

n

n

iK

K

n

N

a

π

1)

55 n K

a π

> :

2 2 2 *0

55

,

n

n

iK

i K

n

N

a

π

ω =

±

−

∈

. (1.16)

(1.3), (1.16) , ,

2 2 2

55

1

exp

K

K

n

t

a

h

⎛

_π

⎞

− −

−

⎜

⎟

⎜

⎟

ρ ⎝

⎠

2 2 2

55

1

exp

K

K

n

t

a

h

⎛

_π

⎞

− +

−

⎜

⎟

⎜

⎟

ρ ⎝

⎠

.

2)

55 n K

a π

< :

2 2 2 *0

55

,

n

n

iK

K

n

a

π

ω =

±

−

∈ Ν

. (1.17)

(1.3) , .

55 13

1a =G – ,

. .

,

( “I”). , 2 2 2

* h

ω = ρ ω ,

2 2

I 2

0

55

1

,

n

n

iK

K

n

a

h

⎛

_π

⎞

ω =

⎜

_⎜

±

−

⎟

_⎟

∈ Ν

ρ ⎝

⎠

. (1.18)

(1.11) (1.13) C2( )U0 =0, , (1.9) :

( )0 ( )0

_{( )}

( )0

_{( )}

I 1

,

cos

0 1 I

,

cos

n U U Un

U

=

C

ξ η

γ ζ =

C

ξ η

π ζ

n

,

n

∈ Ν

.

(1.19)

(1.14) ( ,

“II”)

(

)

2

2

II 2

0

55

2

1

1

,

4

n

n

iK

K

n

a

h

⎛

_π

₊

⎞

⎜

⎟

ω =

±

−

∈ Ν

⎜

⎟

ρ

_⎝

_⎠

(1.20)

:

( )0 ( )0

_{( ) (}

)

II 2 II

2

1

,

sin

,

2

n Un

n

U

=

C

ξ η

+ π

ζ

n

∈ Ν

. (1.21)

, (1.1) (1.2) σ_yz,

:

2 2

III 2

0

44

1

,

n

n

iK

K

n

a

h

⎛

_π

⎞

ω =

⎜

_⎜

±

−

⎟

_⎟

∈ Ν

ρ ⎝

⎠

( ) (1.22)

(

)

2

2

IV 2

0

44

2

1

1

,

4

n

n

iK

K

n

a

h

⎛

_π

₊

⎞

⎜

⎟

ω =

±

−

∈ Ν

⎜

⎟

ρ

_⎝

_⎠

( ) (1.23)

( )0 ( )0

_{( )}

III 1 III

,

cos

,

n Vn

( )0 ( )0

_{( ) (}

)

IV 2 IV

2

1

,

sin

,

2

n Vn

n

V

=

C

ξ η

+ π

ζ

n

∈ Ν

( ) (1.25)

(1.1) σzz,w.

( )0

0

W

∂

₌

∂ζ

ζ =

1

;

( )0

0

W

=

ζ = −

1

. (1.26)

(1.26), (1.9),

( )0 ( )0 1W, 2W C C :

( ) ( ) ( ) ( )

0 0

1 0 2 0

0 0

1 0 2 0

sin

cos

0,

cos

sin

0.

W W W W

W W W W

C

C

C

C

⎧−

γ +

γ

=

⎪

⎨

γ −

γ

=

⎪⎩

(1.27)

:

(

)

0 0

2

1

cos 2

0

,

4

W W

n

n

+ π

γ

= ⇒ γ

=

∈ Ν

, (1.28)

(

)

2

2

V 2

0 11

1

2

1

,

16

n

iK

A

n

K

n

h

⎛

_π

⎞

ω =

⎜

_⎜

±

+

−

⎟

_⎟

∈ Ν

ρ ⎝

⎠

. (1.29)

, (1.26)

. (1.27) C₂( )_W0 =C₁( )_W0tgγ_W0, , (1.9)

(1.28) :

( )0 ( )0

_{( )}

(

) ( )

V 1 V

2

1

,

cos

1

,

4

n Wn

n

W

=

C

ξ η

+ π

− ζ

n

∈ Ν

, ( )

( )0

0 1

1

0

.

cos

W W

W

C

C

=

γ

(1.30)

(1.2) w

( )0

0

W

=

ζ = ±

1

. (1.31)

(1.31), (1.9),

:

(

)

2

VI 2 2

0 11

2

1

1

,

4

n

n

iK

A

K

n

h

⎛

₊

⎞

⎜

⎟

ω =

±

π

−

∈ Ν

⎜

⎟

ρ

_⎝

_⎠

( )(1.32)

(

)

VII 2 2 2

0 11

1

,

n

iK

A

n

K

n

h

ω =

±

π

−

∈ Ν

ρ

( ) (1.33)

( )0 ( )0

_{( )}

(

)

VI 1 VI

2

1

,

cos

,

2

n Wn

n

W

=

C

ξ η

+

πζ

n

∈ Ν

( ) (1.34)

( )0 ( )0

_{( )}

VII 2 VII

,

sin

,

n Wn

W

=

C

ξ η

π ζ

n

n

∈ Ν

( ) (1.35)

, , (1.19), (1.21),

(1.24), (1.25), (1.30), (1.34), (1.35),

,

I 0n

ω II

0n ω

( )0

yz

σ

, W( )0 , , . .

, :

( )0 ( )0 ( )0 ( )0

I I II II

0

n n n n

V

=

W

=

V

=

W

=

,

n

∈ Ν

.

(1.36)

III IV 0n 0n, 0n

ω = ω ω V VI VII

0n 0n, 0n, 0n ω = ω ω ω

( )0 ( )0 ( )0 ( )0 III III IV IV

0

n n n n

U

=

W

=

U

=

W

=

, (1.37)

( )0 ( )0 ( )0 ( )0 ( )0 ( )0 V V VI VI VII VII

0

n n n n n n

U

=

V

=

U

=

V

=

U

=

V

=

,

n

∈ Ν

.

(1.38)

, , I II

0n 0n, 0n ω = ω ω III IV

0n 0n, 0n

ω = ω ω :

( ) ( )

( ) ( ) ( ) ( )

(

)

(

)

0 0

I II

0 0 0 0

13 I 1 I 13 II 2 II

55 55

0, ,

1, 2,3,

13,

2

1

2

1

sin

,

cos

,

2

2

jkn jkn

n Un n Un

j k

jk

n

n

n

C

n

C

n

a

a

σ

= σ

=

=

≠

+ π

+ π

π

σ

= −

π ζ σ

=

ζ

∈ Ν

(1.39)

( ) ( )

( ) ( ) ( ) ( )

(

)

(

)

0 0

III IV

0 0 0 0

23 III 1 III 23 IV 2 IV

44 44

0, ,

1, 2,3,

23,

2

1

2

1

sin

,

cos

,

2

2

jkn jkn

n Vn n Vn

j k

jk

n

n

n

C

n

C

n

a

a

σ

= σ

=

=

≠

+ π

+ π

π

σ

= −

π ζ σ

=

ζ ∈Ν

(1.40)

, V VI

0n 0n, 0n

ω = ω ω (1.1), :

( )

( ) ( )

_{( )(}

)

(

) ( ) (

₎

0

0 0

11 V 23 1 V 23 13 11

0, ,

1,2,3,

,

,

2

1

2

1

,

sin

1

, 11,22,33;

,

,

,

4

4

jkn

n Wn

j k

j

k n

n

n

A C

A A

A

σ =

=

≠

∈Ν

+ π

+ π

σ

= −

ξ η

−ζ

−

(1.41)

(1.2) :

( ) ( )

( ) ( )

(

)

(

)

₍

₎

( ) ( )

₍

₎

0 0

VI VII

0 0

11 VI 23 1 VI 23 13 11

0 0

11 VII 23 2 VII 23 13 11

0, ,

1, 2, 3,

,

2

1

2

1

sin

,

11, 22, 33;

,

,

,

2

2

cos

,

11, 22, 33;

,

,

,

.

jkn jkn

n Wn

n Wn

j k

j

k

n

n

A C

A

A

A

A C

n

n

A

A

A

n

σ

= σ

=

=

≠

+ π

+ π

σ

=

ζ

−

σ

= −

π

π ζ

−

∈ Ν

(1.42)

, 3 , –

ω ωI0n, II0n, III IV 0n, 0n

ω ω –

ω ω0Vn, VI0n.

2. . ,

( )

{ }

0

{ }

( )0

{ }

( )0

{ }

( )0

{ }

( )0

{ }

( )0

{ }

( )0 I II , III , IV , V VI VII

n n n n n n n

U U V V W W W

1− ≤ ζ ≤1, . ., , ( ) ( )

1

0 0 I I 1

0

n m

U U

d

−

ζ =

∫

,

n

≠

m

(

U V W

, ,

)

,

n m

,

∈ Ν

.

(2.1)

(2.1), (1.8)

( )

( )

(

)

( )

0 2

2 ₀

I I

I

55 *0 *0 I

2

2

0

m

m m m

U

a

i K

U

∂

+

ω

−

ω

=

∂ζ

. (2.2)

(2.2) U_n( )0_I ζ

[

−1, 1

]

, :

( )

( )

(

( )

)

( ) ( )

1 2 0 1

2

0 I I 0 0

I

I 55 *0 *0 I I

2

1 1

2

0

m

n m m n m

U

U

d

a

i K

U U

d

− −

∂

ζ +

ω

−

ω

ζ =

∂ζ

∫

∫

. (2.3)

(2.3) (1.10), :

( )

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

1 2 0 0 1 0 0 1 0 0

0 0

I I I I I I

I I

2

1 ₁ 1 1

n n n m n m

m m

U

U

U

U

U

U

U

d

U

d

d

− ₋ − −

∂

∂

∂

∂

∂

∂

ζ =

−

ζ = −

ζ

∂ζ

∂ζ

∂ζ

∂ζ

∂ζ

∂ζ

∫

∫

∫

(2.3) :

( ) ( )

( )

(

)

( ) ( )

1 0 0 1

2 _{0 0}

I I

I I

55 *0 *0 I I

1 1

2

0

n m

m m n m

U

U

d

a

i K

U U

d

− −

∂

∂

−

ζ +

ω

−

ω

ζ =

∂ζ

∂ζ

∫

∫

. (2.4)

(1.8) Un( )0I ,

( )0 I m

U , :

( ) ( )

( )

(

)

( ) ( )

1 0 0 1

2 _{0 0}

I I

I

55 *0 *0 I I

1 1

2

0

nv m

m m n m

U

U

d

a

i K

U U

d

− −

∂

∂

−

ζ +

ω

−

ω

ζ =

∂ζ

∂ζ

∫

∫

. (2.5)

(2.4) (2.5), :

(

I I

)(

I I

)

1 ( ) ( )0 0

55 *0 *0 *0 *0 I I

1

2

0.

n m n m n m

a

i K

U U

d

−

ω − ω

ω + ω

−

∫

ζ =

(2.6)

n≠m, I I

*0n *0m

ω ≠ ω (2.6) :

( ) ( ) 1

0 0 I I 1

0

n m

U U

d

−

ζ =

∫

. (2.7)

,

( )

( )

_{( )}

{ } {

}

0 I

I 0 I

1 I

,

cos

,

n

n n

Un

U

n

C

Ψ =

Ψ

=

π ζ

η ξ

,

n

∈ Ν

(2.8)

1− ≤ ζ ≤1.

n

Ψ .

3.

s

≥

1

. (1.6) s=1 I *0n *0n

ω = ω .

(1.6): ( )

(

)

( )

(

)

( ) ( )

1 2

1 0 1

I I I I

I

55 *0 *0 I 55 *1 *0 I I

2

2

2

2

n

n n n n n n Un

U

a

iK U

a

iK U

R

∂

₊

_ω

_{ω −}

₊

_ω

_{ω −}

₌

∂ζ

(3.1)

(3.1) U_n( )1_I

:

( )1 ( )0

I I

1

n nm m

m

U

b U

∞

=

(1.1) (1.2), σ_xz.

(3.2) (3.1) , (2.2)

( )

( )

(

)

( )

0 2

2 ₀

I I

I

55 *0 *0 I

2 2

m

m m m

U

a i K U

∂

= − −

∂ζ ω ω , :

(

I I

)(

I I

)

( )0 I

(

I

)

( )0 ( )1

*0 *0 *0 *0 55 I 55 *1 *0 I I

1

2

2

n m n m nm m n n n Un

m

i K b a U

a

iK U

R

∞

=

ω −ω

ω +ω +

= −

ω

ω −

+

∑

(3.3)

(3.3) U_k( )_I0 ζ

[

−1, 1

]

,

( )

{ }

0 I n

U , :

(

I I

)(

I I

)

I

(

I

)

( )0 ( )1

55 *0 *0 *0 *0

2

= 2

55 *1 *0 I I

nk n k n k n n Unk nk Unk

b a

ω − ω

ω + ω +

i K

−

a

ω

ω −

iK C

δ +

R

(3.4)

nk

δ – ,

( ) ( )

( ) ( )

( )

_{( )} ( ) ( ) 1

0

0 1 I 1 1 0

I 0 I ₀ 2 I I

1

1 I _{1 I}

1

,

Un

Unk Unk Un k

Uk _Uk

C

C

R

R U

d

C

_C

₋

=

=

∫

ζ

,

n

∈ Ν

.

(3.5)

n=k (3.4) :

( )

(

)

1

I I

*1 _I

55 *0

2

Unn n

n

R

a

iK

ω =

ω −

,

n

∈ Ν

,

(3.6)

n≠k:

( )

(

)(

)

1 I

I I I I

55 *0 *0 *0 *0

2

Unk nk

n k n k

R

b

a

i K

=

ω − ω

ω + ω +

,

n

∈ Ν

.

(3.7)

(1.5), (1.7) , (1.36) RUn( )1I =0, , (3.5) (3.7) :

0

nk

b

=

,

n

≠

k

,

ω =

_*1I _n

0

,

n

∈

N

. (3.8)

nn

b Un:

( )

( ) ( )

(

)

1 ₂

0 1

I I

2 0

1 I

1

1

n n

n

U

U

d

U

−

+ ε

ζ =

∫

, ( )

( )

( )

1 ₂

2

0 0

I I

1

n n

U

U

d

−

=

_∫

ζ

, (3.9)

: ( ) ( )

1

0 1 I I 1

0

n n

U U d

−

ζ =

∫

,

n

∈ Ν

.

(3.10)

( )1 I n

U (3.10), (3.2), :

( ) ( ) ( ) ( )

1 1

1

0 0 0 0

I I I I

0 _{1 1}

0

n

nk n k nn n n

k

b

U U

d

b

U U

d

−

= _{− −}

ζ +

ζ =

∑ ∫

∫

.

n≠k, ( ) ( )

1 0 0 I I 1

0 nn n n

b U U d

−

ζ =

∫

,

0

nn

b

=

,

n

∈ Ν

.

(3.11)

, ω*0n = ω*0In :

( )1 I

I

0,

*1

0,

n n

2

s= . (1.6) s=2

: ( )

(

)

( )

(

)

( ) ( )

2 2

2 0 2

I I I I

I

55 *0 *0 I 55 *2 *0 I I

2

2

2

2

n

n n n n n n Un

U

a

iK U

a

iK U

R

∂

+

ω

ω −

+

ω

ω −

=

∂ζ

(3.13)

:

( )2 ( )0

I I

1

n nr r

r

U

c U

∞

=

=

∑

,

n

∈ Ν

.

(3.14)

, :

( )

(

)

2

I I

*2 _I

55 *0

2

Unn n

n

R

a

iK

ω =

ω −

,

n

=

k

, (3.15)

( )

(

)(

)

2 I

I I I I

55 *0 *0 *0 *0

2

Unk nk

n k n k

R

c

a

i K

=

ω − ω

ω + ω +

,

n

≠

k

, (3.16)

( )

( )

( )

( ) ( ) ( ) ( )

1 1

2 2 0 2

I ₀ 2 I I 0 I I

1 1 I 1

1 I

1

1

Unk Un k Un k

Uk Uk

R

R U

d

R

d

C

C

− −

=

_∫

ζ =

_∫

Ψ ζ

. (3.17)

( )2 I Un

R , (1.5), (1.7), (1.36), :

( )2

₍

₎

2 ( )I1 2 ( )0I 2 ( )0I

I 55 23 55 22 2 2

66

1

1

n n n

Un

W

U

U

R

a A

a

A

a

⎛

⎞

∂

∂

∂

=

−

−

⎜

_⎜

+

⎟

_⎟

∂ζ∂ξ

_⎝

∂ξ

∂η

_⎠

. (3.18)

(3.18) Wn( )I1 ,

(1.6), (1.13) :

( )

( ) ( )

1

2 2 2

1 1

I

I I

2

55 11 n

n Wn

W

n

W

R

a A

∂

₊

π

₌

∂ζ

,

n

∈ Ν

,

(3.19)

( )1 I Wn

R , (1.5), (1.7) ,(1.19):

( )1 2 ( )0I

( )

( )0

I 23 1 I 23

11 55 11 55

1

1

1

1

sin

n

Wn Un

U

R

A

C

A

n

n

A

a

ξ

A

a

⎛

⎞

∂

⎛

⎞

=

_⎜

−

_⎟

= −

_⎜

−

_⎟

π

π ζ

∂ξ∂ζ

⎝

⎠

⎝

⎠

. (3.20)

(3.19), :

( )1 ( )1 ( )1

0I 1 2

55 11 55 11

sin

cos

n W W

n

n

W

D

D

a A

a A

π

π

=

ζ +

ζ

.

(3.19), (3.20) (3.19):

( )

( )

( )

(

)

1 0 55 23

I 1 I

55 11

1

sin

1

n Un

a A

W

C

n

n

a A

τ _ξ

−

=

π ζ

π

−

.

, W_n( )_I1

( ) ( ) ( )

( )

( )

(

)

1 1 1 0 55 23

I 1 2 1 I

55 11 55 11 55 11

1

sin

cos

sin

1

n W W Un

a A

n

n

W

D

D

C

n

n

a A

a A

a A

ξ

−

π

π

=

ζ +

ζ +

π ζ

π

−

(3.21)

( )1 ( )1 1, 2

W W

( )1

₍

₎

( )I1 ( )0I

I 11 23

1

1

0;

n n

0

n

W

U

W

A

A

ζ=

⎛

∂

∂

⎞

ζ = − =

⎜

_⎜

−

⎟

_⎟

=

∂ζ

∂ξ

⎝

⎠

. (3.22)

(3.22), (1.19) (3.21),

( )1 ( )1 1, 2

W W

D D ,

, , :

( )

_{( )}

( )

( )

( )

_{( )}

( )

( ) 1

1 55 11 0 11 23

1 1 I

55 11 55 11 11

55 11

1

1 55 11 0 11 23

2 1 I

55 11 55 11 11

55 11

1

cos

,

2

1

cos

1

sin

.

2

₁

cos

n W Un n W Un

a A

A

A

n

D

C

n

a A

a A

A

n

a A

a A

A

A

n

D

C

n

_{a A}

_{a A}

A

n

a A

+ ξ + ξ

−

π

=

_π

−

−

π

−

π

=

−

π

−

π

(3.23)

(3.18), (3.21), (3.23) (1.19), :

( )

( )

( )

(

) ( )

(

)

(

)

(

)

( )

( )

( )

( )

1

2 0 55 23 11 23

I 1 I

55 11 55 11

11

55 11 2

0 0

55 23 55

55 22 1 I 1 I

55 11 66

1

1

cos

1

2

1

cos

1

cos

,

1

n Un Un Un Un

a A

A

A

n

R

C

n

a A

_A

a A

a A

a A

a

a A

C

C

n

a A

a

+ ξξ

ξξ ηη

−

−

π

=

−

_π

+ ζ −

−

⎛

⎛

₋

⎞

⎞

⎜

⎟

−

⎜

_⎜

−

⎟

_⎟

+

π ζ

⎜

_⎝

−

_⎠

⎟

⎝

⎠

(3.24)

( )2 I Unk R : ( ) ( ) ( ) 1 2 2

I 0 I

1 1 I

1

sin

Unk Un Uk

R

R

k d

C

₋

=

∫

π ζ ζ

. (3.25)

(3.24) (3.25), :

( )

_{( )}

(

)(

)

(

)

(

)

( )

( )

( ) 0 1 I 2

2 55 55 23 11 23 55 11

I 2 2 0

55 11 55 11 1 I

55 2 11

2

1

1

2

1

_{c s}

i

o

s n

Un n k Unk Uk

n

C

a A

a k a A

A

A

R

n

a A

n

a A k

C

a A

+ + ξξ

π

−

−

= −

π

π −

−

. (3.26)

, (3.15), (3.16), (3.26), I

*2n ω c_nk:

( )

( )

( )

₍

(

₎

)(

(

)

)

0 1 I 55 11 55 23 11 23

I

*2 0 2 I

1 I 55 11 *

2 0 55 11

1

2

1

_cos

sin

Un n Un n

n

C

_{a A}

_A

_A

_{a A}

n

C

a A

iK

_n

a A

ξξ

π

−

−

ω =

π

−

ω −

_π

,

n

∈ Ν

.

(3.27)

( )

( )

( )

(

)(

)

( )

(

)(

)

₍

₎

(

)

2 0

1 I 55 23 11 23

55 11

0 I I I I 2 2

1 I *0 *0 *0 *0 55 11 55 1 2

1

55 11

2

1

1

2

2

1

co

i

s

s n

n k Un nk

Uk n k n k

n

C

k

a A

A

A

a A

c

n

C

i K

a A

n

a A k

a A

+ + ξξ

π

−

−

−

=

π

π ω −ω

ω +ω +

−

−

(3.28)

0

nn

c

=

,

n

∈ Ν

.

(3.29)

, U_n( )2_I I

*2n

ω , . , :

( ) ( ) I I 2 I

* *0 *2

0 2 2

I I I

,

,

.

n n n

n n n

U

U

U

n

⎧ω = ω + ε ω

⎪

⎨

=

+ ε

∈ Ν

⎪⎩

(3.30)

(1.2) .

( )1 I 0 n

W ≠ , ,

, . .

. , W_n( )_I0 =0,W_n( )_I1 ≠0 ,

.

,

.

, .

1. . .

.// . - ,

-. . . 2000. №3. .8-11.

2. . .

.// . :

-“ ” , 2002. .9-19.

3. . ., . .

.// . 2006.

.70. .1. .111-125.

4. . ., . .

. // . . . 2005. .58. №2.

.48-58.

5. . . .

// - . .

. 2009. . 5-35.

6. . . . .: , 1959. 439 .

:

ч, , . ,

.

: 0019 , . 24 ,

: (+37410) 52-58-35, -mail: aghal@mechins.sci.am

–

.: (093)069950, : (0224)22335 E-mail: messarg@gmail.com