Dynamic behaviour of a coaxial cylindrical shells, with a gap partially filled with fluid

Ս Ի Ի Ո Թ Ո Ի Ի Ի Ի Ղ Ի

Ц Ь

64, №3, 2011

539.3

,

. ., . .

ч : , , ,

Key words: shell, fluid, stability, natural vibration

. ., Ս. .

,

,

և :

[1-4] և [5] :

[6] :

[7]

: ,

, [8-11]

, [8,11],

[9,10]: Ս [11]:

, ,

և ,

:

Ո և

:

:

Baghdasaryan G.Ye, Marukhyan S.A.

Dynamic behavior of a coaxial cylindrical shells, with a gap partially filled with fluid

There are numerous studies on the vibrations and dynamic stability of a cylindrical shell filled with fluid. Information about these studies can be found in monographs [1-4] and in a review article [5]. The problem of vibrations of coaxial cylindrical shells, filled with fluid of variable-depth, is considered in [6]. The problem of stability of cylindrical shells partially filled with fluid, with an external dynamic pressure is discussed in [7]. The problems of vibration of coaxial cylindrical shells are considered in [8-11], and besides [8, 11], in the remaining papers which deal with the same case, the vibrations of shells completely filled with fluid are researched in [9, 10]. The question of possible loss of stability is considered in [11].

In this paper, the problem of vibrations and stability of isotropic coaxial circular cylindrical shells of finite length in linear statement is considered, when the region between the shells (the gap) is partially filled with an incompressible fluid . The dependence of the vibration frequency on the depth of the filling and the thickness of the gap of the considered hydro-elastic system is studied. The possibility of loss of static stability of hydro-elastic system under the influence of hydrostatic pressure is shown.

,

, .

[1-4] [5]. ,

, [6]. [7]

, , .

[8-11], [8,11], , [9,10].

[11].

,

( ) .

.

.

1. .

l

,

( )

b

(

b

≤

l

)

.

( , , )

α θ

r

,

α

. ,

,

α =

0

, ( ё

).

:

) – ;

)

;

) ;

)

. [12]:

2 2

2

2 2

Φ

1

ρ ρ

α

i i

i i

i

i i i i i i

i

w w

D w h h Z

R t t

∂ ∂ ∂

Δ − + =

∂ ∂

+

ε

∂ ,

i

=

(1, 2)

, 2

2 2

1 1

Φ 0

α

i

i

i i i

w

E h R

∂

Δ + =

∂ ,

(

)

3 2 12 1

i i i

i

E h D =

− ν , (1.1)

1

i

=

,

i

=

2

– . (1.1) w_i– , Φ_i– , R_i– , h_i– , E_i– , ν_i– , ε_i

−

,ρ_i–

i

- , Z_i

−

,

Δ

– .

i

Z [6-8, 13]: 2

0 2

( ) 0

( 1) ρ ( ) 0

0 .

1

i i i

i i

w

g b b

Z R

b l

Z

∂

− + − α < α <

= ∂θ

< α <

⎧

⎡

⎤

⎪

_⎢

_⎥

⎨

_⎣

_⎦

⎪

⎩

(1.2)

0

Z – ё ,

g

– , ρ₀–

. – ( )

0 0

φ

ρ

t

i

i

r R

Z

=

∂

= −

∂

, (1.3)

2 2 2

2 2 2 2

φ

1

φ

1

φ

φ

0

θ

α

r

r

r

r

∂

₊

∂

₊

∂

₊

∂

₌

∂

∂

∂

∂

, (1.4)

, ,

:

φ

w

i

r

t

∂

∂

₌

∂

∂

r

=

R

i,

(

i

=

1, 2

)

, (1.5)

φ

0

α

∂

₌

∂

α =

0

, (1.6)

, [5-8, 13]:

φ

0

t

∂

₌

∂

α =

b

. (1.7)

φ

, (1.5),

. ,

.

, ё .

(1.1), ,

( )

( )

)

0 ( ) 0

(

cos

θ

t sin

λ α

,

Φ

cos

θ

t sin

λ α

,

i s

i s

i s

s

i s

s

W

w

n

n

∞

= ∞

=

=

=

Φ

∑

∑

(1.8)

λ

_s

=

(

m

+ π

s

) /

l

,

m

–

,

n

– ,

W

_s( )i

( )

t

Φ

( )_si

( )

t

– .

(1.8),

φ

( )

( )

s

0

φ

cos

θ

_s

( )

_n

λ

_{s s}

( )

_n

λ

sin

λ α

_s

s

n

A t I

r

B t K

r

∞

=

⎧

=

_{⎨ ⎣}

⎡

+

⎤

_⎦

+

⎩

∑

}

1

( ) sh

α α

( ) ch

α α

(

α

)

j nj j nj n nj

j

C t

D t

r

∞

=

⎡

⎤

+

∑

_⎣

+

_⎦

Ψ

, (1.9)

n

I

,

K

_n – ,

( )

1 1

(

α

)

(

α

)

α

(

α

)

(

α

)

n nj n nj

n nj

n nj n nj

J

r

Y

r

r

J

R

Y

R

Ψ

=

−

′

′

, (1.10)

n

J

,

Y

_n–

n

;

A

_s,

B

_s,

C

_j,

D

_j – ,

(1.5)–(1.7);

α

_nj– :

(

α 2

) (

α 1

)

(

α 1

) (

2

)

0

n nj n nj n nj n nj

J ′ R Y′ R −J ′ R Y′ α R = . (1.11)

( )

( )

.

z R

df z

f

R

dz

_=β

′ β =

(1.8) (1.9) (1.5) – (1.7) (1.11),

s

A

,

B

_s,

C

_j,

D

_j:

(

)

(

)

(

)

(

) (

)

(1) ( 2 2) 1

1 2 2 1

λ

λ

λ

λ

λ

)

(

(

)

λ

λ

s s

n s n s

s n s n s n s n s

s

W

W

K

R

K

R

t

t

I

R K

R

I

R K

A t

R

∂

∂

′

₋

′

∂

∂

⎡

′

′

₋

′

⎤

=

′

⎣

⎦

,

(

)

(

)

(

)

(

) (

)

(2) ( 1 1) 2

1 2 2 1

λ

λ

λ

λ

λ

)

(

(

)

λ

λ

s s

n s n s

s n s n s n s n s

s

W

W

I

R

I

R

t

t

I

R K

R

I

R K

B t

R

∂

∂

′

₋

′

∂

∂

⎡

′

′

₋

′

⎤

=

′

⎣

⎦

, (1.12)

(

)

(1)

(

)

(2)

1 1 2 2

0

λ

_α

_α

( , )

( )

N

s s s

n nj n j

j n

s

W

W

R

R

R

R

s

C t

j

t

t

=

⎛

∂

∂

⎞

=

_⎜

Ψ

− Ψ

_⎟

Δ

_⎝

∂

∂

_⎠

∑

,

(

)

₍

₎

(1)

₍

₎

(2)

s

1 1 2 2

0

λ

sh

α

α

sin

λ

α

α

,

( , ) ch

( )

α

N

nj nj s _{s s}

n nj n nj

s nj

j

b

b

_W

_W

R

R

R

R

s j

b

t

t

D t

=

−

_⎛

_∂

_∂

_⎞

=

_⎜

Ψ

− Ψ

_⎟

Δ

_⎝

∂

∂

_⎠

∑

(

)

2 2

(

)

2

2 2 2 2

2 2 2

2

( , )

α α

λ

1

2

nj nj s n nj

nj

R

n

s j

R

R

⎛

⎞

Δ

=

+

⎜

_⎜

−

⎟

_⎟

Ψ α

α

⎝

⎠

⎡

−

⎢

⎢⎣

(

)

2 2 2 1 1 2 2 1

1

2

_nj n nj

R

n

R

R

⎛

⎞

−

⎜

_⎜

−

⎟

_⎟

Ψ α

α

⎝

⎠

⎤

⎥

⎥⎦

.

(1.12) (1.9), (1.3)

ё :

( )

_{( )}

{

( )

0 0 1

0

ρ

cos

θ

i

sin

λ α

i

s s

Z

n

A

s

∞

=

⎡

= −

∑

_⎣

+

(1.13)

( )

_{( )}

( )

_{( )}

(

)

2 ( )1

1 1 2

j 1

,

sh

α α

,

ch

α α

i i s

nj nj

d W

B

s j

C

s j

dt

∞ =

⎤

+

+

_⎥

⎦

+

∑

( )

_{( )}

(

( )

_{( )}

( )

_{( )}

)

2 ( )2

2 2 2 2

j 1

sin

λ α

,

sh

α α

,

ch

α α

.

i i i s

s nj nj

d W

A

s

B

s j

C

s j

dt

∞ =

⎫

⎡

⎤

_⎪

+

_⎢

+

+

_⎥

_⎬

⎪

⎣

∑

⎦

⎭

( )

_{( )}

(

)

(

)

(

)

(

)

(

)

(

) (

)

( ) ( ) 1

1 2 2 1

λ

λ

λ

λ

( 1)

λ

λ

(

λ

)

λ

λ

p p

n s n s i n s n s i

i p r r

p

s n s n s n s n s

K

R

I

R

I

R

K

R

A

s

I

R K

R

I

R K

R

+

′

−

′

= −

⎡

′

′

₋

′

′

⎤

⎣

⎦

,

( )

_{( )}

1

λ

(

α

) (

α

)

,

( 1)

( , )

s p n nj p n nj i

i p

p

R

R

R

B

s j

s j

+

Ψ

Ψ

= −

( )

_{( )}

1

(

λ

s

sh

α

α

sin

λ

)

(

α

) (

α

)

,

( 1)

( , ) ch

α

nj nj s p n nj p n nj i

i p

p

nj

b

b R

R

R

C

s j

s j

b

+

−

Ψ

Ψ

= −

Δ

,

(

p

=

1, 2;

r

( )p

= +

1

δ

_1p

,

δ

_ij– ).

(1.8) (1.1),

Φ

( )_si

( )

t

( )

_{( )}

(

)

( )

( )

2

s 2 s

2 2 2

λ

.

λ

/

Φ

i

t

i i s i

t

i _{s i}

E h

W

R

_n

_R

=

+

(1.14)

,

W

_s( )1

( )

t

( )2

_{( )}

s

t

W

.

– , (1.2), (1.8), (1.13) (1.14), (1.1),

W

_s( )1

( )

t

W

_s( )2

( )

t

: ( ) ( )

( )

( ) 1 1 2 1 2 1 1 2

,

к к к

d W

dW

k n W

dt

+ ε

dt

+ Ω

+

( )1 2 ( )1

_{( )}

( )1

_{( )}

( )2

(1) (1)

1 2

2 0

,

,

0,

ks s s s

s

d

b W

m

k s W

m

k s W

dt

∞ =

⎛

_⎡

_⎤

⎞

+

_⎜

+

_⎣

+

_⎦

_⎟

=

⎝

⎠

∑

( ) ( )

( )

( ) 2 2 2 2 2 2 2 2

,

к к к

d W

dW

k n W

dt

+ ε

dt

+ Ω

+

( )2 2 ( )2

_{( )}

( )1

_{( )}

( )2

(2) (2)

1 2

2 0

,

,

0,

ks s s s

s

d

b W

m

k s W

m

k s W

dt

∞ =

⎛

_⎡

_⎤

⎞

+

_⎜

+

_⎣

+

_⎦

_⎟

=

⎝

⎠

∑

(

k

=

0,1, 2, 3,...)

. (1.15)

( )

(

)

2 4 2 2 2 2

2 2 2

2

2 2 2

12(1

,

λ

/

)

λ

ρ

_λ

i k i k i i i

i i i _{k i}

D

n

k n

h

R

R h

_n

_R

ν

⎡

_⎛

_⎞

₋

⎤

⎢

⎥

Ω

=

_⎜

+

_⎟

+

⎢

_⎝

_⎠

₊

⎥

⎣

⎦

,

(

)

(

)

(

(

)

)

2

( ) 0

2 2

1 cos

λ λ

1 cos

λ λ

ρ

( 1)

ρ

λ λ

λ λ

s k s k

i i

ks

i i i _{s k s k}

b

b

gn

b

lR h

⎡

₋

₋

₋

₊

⎤

= −

⎢

−

⎥

−

+

⎢

⎥

⎣

⎦

, (1.16)

( )

_{( ) ( )}

( )

_{( )}

(

)

(

)

(

(

)

)

0

sin

λ λ

sin

λ

λ

2

ρ

,

1

ρ

2

λ λ

2

λ

λ

i

i i s k s k

p p

i i s k s k

b

b

m

k s

A

s

lh

⎧

⎡

−

+

⎤

⎪

= −

_⎨

_⎢

−

_⎥

+

−

+

⎪

⎣

⎦

⎩

( )

_{( )}

2 2 1

α

ch

α

sin

λ

λ

sh

α

cos

λ

,

α

λ

i nj nj k k nj k

k p

j nj

b

b

b

b

B

s j

∞ =

⎡

−

+

⎢

+

+

⎢⎣

∑

( )

_{( )}

2 2

λ

α

sh

α

sin

λ

λ

ch

α

cos

λ

,

α

λ

i k nj nj k k nj k

p

j k

n

b

b

b

b

C

s j

+

−

⎤

⎫

⎪

+

_⎥⎬

(

)

1

k n

,

Ω

Ω

₂

( )

k n

,

–

,

b

_ks( )i –

,

m

( )pi

( )

k s

,

– ё .

(

s

=

0)

. ,

λ

_s

s

=

0

m

m

π

/

l

.

( )

s i

W

,

b

_ks( )i ,

m

( )_pi

( )

k s

,

Ω

_i

(

k n

,

)

0

s

=

m

n

. (1.15) :

(

)

2 (1) (1) (1)

1 2 1

1

_M

_{m n}

,

d W

mn

dW

mn

+

dt

dt

⎡

+

⎤

+ ε

⎣

⎦

(

)

(

)

2 ( 2)

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

1 2 2

Ω

,

,

mn

0

mn mn mn

d W

m n W

M

m n

B W

dt

+

+

+

=

,

(

)

2 ( 2) ( 2) ( 2)

2 2 2

1

,

d W

mn

dW

mn

M

m n

dt

dt

⎡

+

⎤

+ ε

+

⎣

⎦

(1.17)

(

)

(

)

2 (1)

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

2 1 2

Ω

,

,

mn

0

mn mn mn

d W

m n W

M

m n

B W

dt

+

+

+

=

,

:

( )

( )

0

( , )

i i mn

W

=

W

m n

,

M

_q( )i

(

m n

,

)

=

m

_q( )i

( )

0, 0

,

(

q

=

1, 2

)

2 2

( ) ( ) 0

2 00

( , )

(

ρ

sin

1

2

ρ

1)

i

i i m

mn

i i i m

gn

b

B

b

l

b

m n

R h

b

⎡

_⎛

_μ

_⎞

⎤

⎢

⎥

=

_−⎜

_⎟

μ

⎢ ⎝

⎠

⎣

=

⎦

−

⎥

,

μ = λ = π

m 0

m

/

l

. (1.18) (1.17)

( )

, ё .

.

2. ч ч . (1.17)

(1) (1)

( )

t

mn mn

W

t

=

c e

ν , ( 2 )

( )

( 2 ) t mn mn

W

t

=

c e

ν . (2.1)

(2.1) (1.17)

(1)

mn

c

,

c

(2)_mn:

(

)

(

(1)

)

2

(

2

(

)

(1)

)

(1) (

(

)

2 (2)

2 1)

1 1 1

1

M

m n

,

Ω

m

,

n

B

_mn

c

_mn

M

m n

,

c

_mn

0

⎡

+

⎤

+

⎣

+

ν + ε ν +

⎦

ν

=

,

(2.2)

(

)

(

(

)

)

(

(

)

)

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

2 2

(1) (2)

1

,

mn

1

2

,

Ω

,

mn mn

0

M

m n

ν

c

+

_⎣

⎡

+

M

m n

ν + ε ν +

m

n

+

B

⎤

_⎦

c

=

.

(2.2) :

4 3 2

0 1 2 3 4

0

a

ν + ν + ν + ν +

a

a

a

a

=

, (2.3)

(

)

(

(1)

)

(

( 2)

(

)

)

( 2)

(

)

(1)

(

)

0

1

1

,

1

2

,

1

,

2

,

(

)

(

(2)

)

(

(1)

(

)

)

1 1

1

2

,

2

1

1

,

a

= ε

+

M

m n

+ ε

+

M

m n

, (2.4)

(

)

(

2 (1)

)

(

(2)

(

)

)

(

2

(

)

(2)

)

(

(1)

(

)

)

1 2 2 1 1 2

2

Ω

m n

,

B

mn

1

M

m n

,

Ω

,

mn

1

,

a

=

+

+

+

m n

+

B

+

M

m n

+ ε

ε

,

(

)

(

2 (1)

)

(

2

(

)

(2)

)

3 2

Ω

1

,

mn 1

Ω

2

,

mn

a

= ε

m n

+

B

+ ε

m n

+

B

,

(

)

(

2 (1)

)

(

2

(

)

( 2 )

)

4

Ω

1

,

mn

Ω

2

,

mn

a

=

m n

+

B

m n

+

B

.

, , (2.3)

0

0

a

>

, ,

,

, :

0 (

0, 4)

i

a

>

i

=

, (2.5)

(

)

(

)

(

(

)

)

(

(

)

)

(

(

)

)

2

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

1 2

⎡

Ω

1

m n

,

B

mn

1

M

2

m n

,

Ω

2

m n

,

B

mn

1

M

1

m n

,

⎤

ε ε

_⎣

+

+

−

+

+

_⎦

+

( )

(

)

(

( )

)

(

( )

)

(

( )

)

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

1 2

⎡

2

Ω

1

m n

,

B

mn

1

M

1

m n

,

1

Ω

2

m n

,

B

mn

1

M

2

m n

,

⎤

+ε ε ε

_⎣

+

+

+

ε

+

+

_⎦

+

(

)

(

)

(

(

)

)

(

(

)

)

(

(

)

)

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

1 2

⎡

Ω

1

m n

,

+

B

mn

1

M

2

m n

,

Ω

2

m n

,

+

B

mn

1

M

1

m n

,

⎤

+ε ε

_⎣

+

+

+

_⎦

+

(

)

(

)

(

(

)

)

(

(

)

)

2

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

1

,

2

,

2

Ω

1

,

mn 1

Ω

2

,

mn

0.

M

m n M

m n

⎡

m n

B

m n

B

⎤

+

_⎣

ε

+

+ ε

+

_⎦

>

(

i

=

0,

i

=

1)

(2.5) , [5]

M

_i( )i

(

m n

,

)

>

0,

,

(

)

(

)

(

)

(

)

(1) (2)

(2 1

)

1

2

) 2 (1

,

,

,

,

M

m n

M

m n

M

m n

M

m n

>

>

. (2.6)

(1.18) ,

B

mn( 2)

>

0,

(1)

0

mn

B

<

, , (1.16),

(

)

2 (2)

2

0

Ω

m n

,

+

B

_mn

>

,

Ω

₁2

(

m n

,

)

+

B

_mn(1) .

(

)

2 (1)

1

0

Ω

m n

,

+

B

_mn

>

, , (2.4) (2.5), ( (2.5)) ,

.

Ω

₁2

(

m n

,

)

+

B

_mn(1)

<

0

,

4

0,

a

<

,

.

(

)

2 (1)

1

Ω

m n

,

+

B

_mn

=

0

. (2.7)

(2.7), ,

b

=

l

,

, ,

2 2

2 4 2

2 1 1 2 1

2

π

_β

π

π

m R

m R

m R

T

gl

n

n

n

l

l

l

−

⎧

_⎡

_⎤

_⎡

_⎤

⎫

⎪

⎛

⎞

⎛

⎞

⎛

⎞

⎪

=

⎨

⎢

+

⎜

⎟

⎥

+

⎜

⎟

⎢

+

⎜

⎟

⎥

⎬

⎝

⎠

⎝

⎠

⎝

⎠

⎢

⎥

⎢

⎥

⎪

⎣

⎦

⎣

⎦

⎪

⎩

⎭

, (2.8)

2 3

1 1

0

3(1

1

)

1

t

h

T

c

R

ν

⎛

⎞

ρ

=

_⎜

_⎟

ρ

−

_⎝

_⎠

,

(

)

2 1 1 2 1

β

12 1

R

h

ν

⎛

⎞

=

−

_⎜

_⎟

⎝

⎠

,

1 1 2

1

2 (1

)

t

E

c

ν

=

t

c

– .

m

n

,

(2.8) . ,

m

m

=

1

.

, , (2.8)

n

,

*

n

(

)

2 2

4 4 4 0

0 0 _{2 2} 2

0

3

β

n

a

0

n

a

a

n

a

+

−

−

=

+

,

a

0

=

π

R

1

/

l

. (2.9)

1 R1

h1×103

0.5 1 2

2 2.6 5

1.6 10

0.95 22 3 4.3

3 2.6

7 1.56

15 5 8.1

2 4.9

5 2.93

10 7 12.4

2 7.4

4 4.44

8 10 19.3

1 11.5

3 6.09

5

(2.8) (2.9) *

n

*

l

. (

l

*),

« – »

(

c

_t=5100 / ,

ρ

₁

/

ρ

₀=2.7, ν₁

=

0.3

) 1

h

R

₁, .1,

*

n

. ,

,

l

*

, –

( .1).

3. . (2.3),

(

)

2 (1)

1

Ω

m n

,

+

B

mn

>

0

( )

(

ε << Ω

_{i j}

,

( ,

i j

=

1, 2)

)

. e (2.3) (

ν = ω ω −

i

,

):

(

)

(

(1)

)

(

(2)

(

)

)

(2)

(

)

(1)

(

)

4

1 2 1 2

1

M

m n

,

1

M

m n

,

M

m n M

,

m n

,

⎡

+

+

−

⎤

ω −

⎣

⎦

(

)

(

2 (1)

)

(

(2)

(

)

)

(

2

(

)

(2)

)

(

(1)

(

)

)

2

1 2 2 1

Ω

m n

,

B

_mn

1

M

m n

,

Ω

m n

,

B

_mn

1

M

m n

,

⎡

⎤

−

_⎣

+

+

+

+

+

_⎦

ω +

(

)

(

2 (1)

)

(

2

(

)

(2)

)

1 2

Ω

m n

,

+

B

_mn

Ω

m n

,

+

B

_mn

0

+

=

. (3.1)

ё , (3.1)

(

)

(

)

(

)

(

)

2 (2)

2

1 (2)

2 2

2 (1)

1

2 (1)

1

Ω

,

,

1

,

ω

Ω

,

.

1

,

mn

mn

mn

m n

B

D

M

m n

m n

B

D

M

m n

⎧

+

= ∞

⎪ +

⎪

= ⎨

+

⎪

_{= ∞}

⎪ +

⎩

(3.2)

(3.1)

Ω

₁2

(

m n

,

)

+

B

_mn(1)

>

0

, , ,

:

(

)

2

( 1)

i

/

i

A

B

C

ω =

+ −

,

(

i

=

1, 2)

, (3.3)

(

)

(

)

(

)

(

)

2 (1) 2 (2)

1 2

(1) (2)

1 2

Ω

,

Ω

,

1

,

1

,

mn mn

m n

B

m n

B

M

m n

M

m n

A

+

+

+

+

=

+

,

(

)

(

)

(

)

(

)

(

(

)

)

(2) (1)

1 2

(1) (2)

1 2

,

,

2

,

1

1

1

,

M

m n M

m n

M

n

C

m

M

m n

+

+

⎛

⎞

⎜

⎟

=

−

⎜

⎟

⎝

⎠

, (3.4)

(

)

(

)

(

)

(

)

2

2 (1) 2 (2)

1 2

(1) (2)

1 2

Ω , Ω ,

1 , 1 ,

mn mn

m n B m n B

B

M m n M m n

⎛ + + ⎞

=⎜_⎜ − ⎟_⎟ +

+ +

⎝ ⎠

(

)

(

)

(

)

(

)

(

(

)

)

(

)

(

(

)

)

(

(

)

)

(2) (1)

1 2 2 (1) 2 (2)

1 2

2

(1) (2

2 ) 1

4

,

,

Ω

,

Ω

,

1

,

1

,

.

mn mn

M

m n M

m n

m n

B

m n

B

M

m n

M

m n

+

+

+

+

+

(1.16) (2.6) ,

C

>

0

B

>

0

. , (3.3) (3.4) ,

ω < ω

1 2

.

,

i j

ε << Ω

( ,

i j

=

1, 2)

, (2.2) , 1

ω = ω

,

c

_mn(1)

c

_mn( 2 )

<

0

, ,

ω

1

( ),

2

ω = ω

,

c

_mn(1)

c

_mn( 2 )

>

0

,

ω

₂

( ).

.

3.1. ч . (3.1) (1.16) (1.18)

(

ω

₂) ,

« – » – ( )i

5100 /

t

c

=

,

ρ ρ

_i

/

₀

=

2 7

.

,

0.3

i

ν

=

,

i

=

1, 2

.

ω

₂

b l

/

.1

1

m

=

,

R

₁

=

2

,

l

=

2

R

₁,

h

i

10

3

−

=

n

λ

2

1

R

R

⎛

_{λ =}

⎞

⎜

⎟

. 1

.1 ,

b l

/

λ

.

b l

/

.

λ

b l

/

(

b l

/

)

.

- , , .2

ω

₂ n

λ =

1.1

b l

/

.

.2

.2 ,

n

,

.

b l

/ ,

,

ω

2 ,

2

ω

. , .1

n

, .

1

ω

. (3.1) – (3.4)

i

=

1

ω

1

- , .

. 3.

.3 .3 , :

)

λ

;

) , ,

;

) ,

ω

1 ё

b l

/

, .

, .1

b

=

0

,

, ,

ω = Ω

_{1 1},

R

₁

(

λ=

1.1,

λ=

1.5

)

– .

- , .2

1

ω

n

λ =

1.1

b l

/

.

2 n

b/l 0 1 2 3 4 5 6 7 8

0 2549 1814 972.6 548.5 340.7 229.2 164.2 123.8 98.05 0.1 383.5 1381 772.1 445.8 281.6 192.4 139.6 106.4 85.03 0.2 141.2 411.5 242.5 151.1 103.2 75.85 58.71 47.21 39.13 0.4 56.34 86.91 58.38 40.58 29.92 22.99 17.41 8.986 - 0.5 44.11 55.05 39.28 28.24 21.16 15.99 9.084 - - 0.6 37.45 39.26 29.55 21.81 16.46 11.54 - - - 0.7 33.81 30.51 24.08 18.16 13.69 7.519 - - - 0.8 31.98 25.33 20.88 16.05 11.95 - - - - 1 31.19 20.29 18.04 14.31 9.884 - - - -

.2 ,

a)

b l

/

(

b l

/

<

b l

₀

/

)

n

=

1

(

b

₀ к - м ч к

а ам а ач );

)

b l

/

>

b l

₀

/

,

ω

₁

( )

n

.

1. . . .

: , 1975. 192 .

2. . ., . . : .: ” ”, 1970.

356 .

3. . ., . . :

. .5. : , 1982. 400 .

4. . . . . .:

, 1979. 320 .

5. . ., . . ,

. // . 2000. .36. № 4.

6. . ., .Ц. ,

.// . . 1965. .XI. № 4.

7. . ., .Ц. ,

. // . . . 2007. .60. № 1. .25–32.

8. . . ,

. // . . . 1968. .XXI. № 4. .40–47.

9. . ., . .

, .

// . 1992. 28. № 4. .42-48.

10. Myung Jo Jhung, Yong Beum Kim, Kyeong Hoon Jeong, Suhn Choi Modal Analysis of Coaxial Shells with fluid-Filled Annulus. J. Of the Korean Nuclear Society. V.32, n.4, pp.328-341,2000.

11. . ., . .

, .// .

. . . . “ ”. 2010.

.1. .113-118.

12. . . ё . .: ,

1949.

13. Mixon John S., Herr Robert W. An investigation of the vibration characteristics of pressurized thin-walled circular cylinders partly filled with liquid. Techn. Rept. NASA, Nr. R – 145, 1962.

х:

ч – , , ,

, .: (374 10) 55 29 64 E-mail: Gevorgb@rau.am

х –

.: (374 93) 37 25 04

E-mail: satineh.marukhyan@gmail.com