Vibrations of an Elastic Orthotropic Unmoment Open Cylindrical Shell with Variable Curvature and Free End, when Other Edges are Rigid–Clamped

Ա Ա Ա Ի Ի ՈՒԹ ՈՒ ԻԱ Ա Ի Ա Ա ԻԱ Ի Ղ Ա Ի

64, №1, 2011 539.3:534.1

Ы

Ы Ё Щ Ё Ы

. ., . .

ч ы : , , , , ,

.

Keywords: vibrations, unmoment, classical, cylindrical, continuous spectrum, characteristics.

Ղ . . , Խ Ա.Ա.

Ա և ,

և ,

:

, և (

).

: Պ և

: Ghulghazaryan G.R., Khachanyan A.A.

Vibrations of an Elastic Orthotropic Unmoment Open Cylindrical Shell with Variable Curvature and Free End, when Other Edges are Rigid–Clamped

The problem of existence of free vibrations of an elastic orthotropic open cylindrical shell (with arbitrary directional curve) with free end, when other edges are rigid – clamped is studied. The investigation is carried out for elastic orthotropic shell when bending rigidity is vanishingly small (the moment free shell). The dispersion equations for finding the natural frequencies of vibrations are derived. The calculations were carried out for the shells with directing curve in form of a parabola with different values of curvature and lengths.

, ё ё . ,

ё ё (

).

.

.

.

.

, ,

, . .

[1],

,

.

,

-.

[2-7].

[8-11]. .

-, .

-, .

-.

,

-. .

-– [12-15]. :

( ) 1 cos

,

2 / , 0

,

1,

m

w

β = −

km

β

k

= π

s

≤ β ≤

s m

= +∞

. (1)

s

,

– . ,

(0)

( )

0,

(0)

( )

0

m m m m

w

=

w s

=

w

′

=

w s

′

=

. (2)

1. ч ы ч .

. -

.

.1.

( , )

α β

, α(0≤ α ≤l) β ≤ β ≤(0 s)

( . 1). l–

, s– . ,

2 2

0 _{1 1}

( / 2

_m m

cos

),

2 / , 0

,

_m m

R

−

=

k r

+

∑

∞₌

r

km

β

k

= π

s

≤ β ≤

s

∑

∞₌

r

< +∞

. (1.1)

, ,

-,

-,

α β

,

[16]

2 2 2

3

1 1 2

11 2 66 2

(

12 66

)

12 1

u

u

u

u

B

B

B

B

B

u

R

∂

∂

∂

∂ ⎛ ⎞

−

−

−

+

+

_⎜

_⎟

= λ

∂α

∂β

∂α∂β

∂α ⎝ ⎠

,

2 2 2

3

1 2 2

12 66 66 2 22 2 22 2

(

B

B

)

u

B

u

B

u

B

u

u

R

∂

∂

∂

∂ ⎛ ⎞

−

+

−

−

+

_⎜

_⎟

= λ

∂α∂β

∂α

∂β

∂α ⎝ ⎠

, (1.2)

O

y

•

β

α

M(α,β)

12 1 22 2 22

3 3

2

B

u

B

u

B

u

u

R

R

R

∂

∂

−

−

+

= λ

∂α

∂β

.

3 2 1

,

u

,

u

u

– ,

β

α

,

,

B

_ik– , 1 1

( )

R− =R− β

– .

λ

=

ω

2

ρ

,

ω

–

,

ρ

– .

[16]:

3

1 12 2 1 2

0

11 0

0

u

u

B

u

u

u

B

R

_{α= α=}

⎛

⎞

∂

₊

∂

₋

₌

∂

₊

∂

₌

⎜

⎟

∂α

_⎝

∂β

_⎠

∂β

∂α

, (1.3)

1 _l 2 _l

0

u

_α=

=

u

_α=

=

, (1.4)

1 0,s 2 0,s

0

u

_β=

=

u

_β=

=

, (1.5)

(1.3)

α =

0

, a

(1.4), (1.5) – α= β= β=l, 0, s

. - ( ) ( ) ( ) ( )

1 2 3

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

f α β =u u u j=

3

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

0 0

(

,

)

l s _{j j}

j

f

f

=

∫ ∫

∑

₌

u u

d d

β α

. (1.6)

- (1.6)

) ( 2 G

L , G=[0,l]x[0,s]. L2(G), , )

, (f f

f = . (1.2)-(1.5) ё

( . [9], . 84, [17]).

(c ) 0

L

.

L

(c )₀

. ,

, (1.2),

,

[

0

,

λ

₀

]

–

2 2 4

66 11 22 12

4 4 2 2 2

66 11 22 11 22 12 12 66

(

)

( )sin

( , )

(

sin

cos

) (

2

) cos

sin

0

, 0

2

B

B B

B R

B

B

B

B B

B

B B

s

−

−

β

θ

Ω β θ =

θ+

θ +

−

−

θ

θ

≤ β ≤

≤ θ ≤ π

(1.7)

,

(1.2) – .

-, ,

.

(1.2)-(1.5)

.

( – )

[18]. , (1.2)-(1.5) –

( . [8];[9], . 97; [11])

2 2 2

66 22 22 11 11 22 12

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

0

Q B B R B B B B B R

s

− −

λ β = λ − β + λ λ − − β ≠

≤ β ≤ . (1.8)

λ

, (1.8), . .

( , )

0, 0

γ

Ω

.

[

0

,

λ

₀

]

∪

Ω

_γ

L

(c )₀

(1.8) [8], [11].

: ɜɧɟ ɦɧɨɠɟɫɬɜɚ [0,λ0]∪Ω_γ ɫɩɟɤɬɪ

ɨɩɟɪɚɬɨɪɚ (c ) 0

L

ɫɨɫɬɨɢɬ ɢɡ ɢɡɨɥɢɪɨɜɚɧɧɵɯ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɤɨɧɟɱɧɨɣ

ɤɪɚɬɧɨɫɬɢ.

[8].

. ,

λ

(c ) 0

L

, ,

λ

– (1.2)-(1.5),

, (1.2) –

(1.2)-(1.5).

,

L

(c )₀ e

-. .

2. ы

L

(c )₀ .

(1.2), (1.5),

(

)

(

)

(

)

1 ₁ 2 ₁

3 1

exp( ) sin , exp( ) (1 cos ) ,

exp( æ ) sin , / .

m m

m m

m m

u k u km u k v km

w k k w km w u R

∞ ∞

= =

∞ =

= χα β = χα − β

= α β =

∑

∑

∑

(2.1)

,

,

m m m

u v w

– ,

χ

–

. (2.1) (1.2).

sin

km

β

, cos

km

β

0

s

. :

2 2 2

11 66 12 66 12

2 2 2

12 66 66 22 22

(

/

)

(

)

,

(

)

(

/

)

,

m m m

m m m

B

B m

k u

B

B

mv

B

w

B

B

mu

B

B m

k v

B mw

⎧

χ −

+ λ

+

+

χ

=

χ

⎪

⎨

+

χ

−

χ −

+ λ

=

⎪⎩

(2.2)

:

,

m m m m m m m m

c u

= χ

a w

c v

= −

m b w

, (2.3)

2 2 2 2

12 22 12

2

11 11 11 66

,

m

B

B

B

a

m

B

B

B

k B

λ

=

χ +

+

η

η =

,

2

2 2 2

11 22 12 12 66 22 22

11 66 11 11

m

B B

B

B B

B

B

b

m

B B

B

B

−

−

=

χ −

+

η

, (2.4)

2

4 11 22 12 12 66 2 2 11 66 2 2 2 2 22 2 66 2

11 66 11 11 11

2

( )

m

B B B B B B B B B

c m m m

B B B B B

⎛ ⎞

− − +

= χ − χ + η χ + −η _⎜ − η _⎟

⎝ ⎠.

(1.2)

R

−1 (2.1)

. (1.1), (2.3),

([20], . 592), ,

sin

km

β

0

s

, ё

,

,

1

,

0

)

(

2

)

(

, 1 2 22 66 2

0

⎟⎟

+

−

=

=

∞

⎠

⎞

⎜⎜

⎝

⎛

−

−

∑

∞ ≠ = − +

m

w

A

r

r

w

B

B

A

r

r

m n n n n m n m n m m

m

η

(2.5)

2 2

12 22

,

,

1,

n n n n n n n

A

=

p c

p

= +

c

n b

−

B

B

χ

a

n

= +∞

. (2.6)

n

A

( )

2

1 n O

A_n = , ,

+∞ <

∑

∞ =1| |

n An . (1.1),

(

+

)

≤

(

+

∑

)(

∑

)

<

+∞

∑

∞ = ∞ = − + ∞

=1 0 1 1

,_m n m n m

3

/

2

m m n n

n

A

n

r

r

r

r

A

. (2.7)

, (2.5)

λ∉ λ ∪Ω

[0,

₀

]

_γ

χ

(2.6)

– [19]. (2.5)

, , ё

:

(

2 2

)

11 12 22 66 0 1

,

,

,

,

,

, , ,

,

_m

,

0

D

χ η

B

B

B

B

r r

…

r

…

=

. (2.8)

,

χ χ

1

,

2– (2.8)

-,

χ = −χ

3 1

χ = −χ

4 2

(2.8).

(

, (2), , (),...

)

, 1,4 )

(

1 w w j=

w mj

j

j _…

(2.5)

χ

_j

,

j

=

1, 4

. (1.2)-(1.5)

4 _{( )} 4 _{( )}

1

,

1, 2,

1

,

j j

i j i j

u

=

∑

₌

u

i

=

w

=

∑

₌

w

(2.9)

4 , 1 , 2 , 1 , , ( ) ) ( _{= =} j i w

u_ij j – (1.2), (2.1)

j

χ = χ . (2.9) (1.3), (1.4).

(1.3) (1.4)

sin

km

β

,

cos

km

β

,

0

s

, :

( ) ( )

4 4

1 ( ) 2 ( )

( ) ( ) 1 1 ( ) ( ) 4 4 ( ) ( ) ( ) ( ) 1 1

0 ,

0,

exp( )

0,

exp( )

0

m m

j j j j j

m m

j j

j m j m

j j

j m j m j

j m j m

j j

j m j m

R

R

w

w

c

c

a

b

z w

z w

c

c

= = = =

χ

=

=

χ

=

=

∑

∑

∑

∑

,

m

=

1

,

+∞

(2.10)

( ) 2 ( ) 12 2 ( ) 12 ( ) ( ) ( ) ( )

1 2

11 11

,

(

),

,

m j j j m j j

j j m m m j m m j j

B

B

R

a

m b

c

R

a

b

z

k

l

B

B

= χ

−

−

=

+

= χ

(2.11)

) ( ) ( ) (

,

,

_mj _mj

j

m

b

c

a

–

a

_m

,

b

_m

,

c

_m (2.4) χ = χ_j .

(2.10) , ,

( ) ( ) ( ) ( )

11 12 11 1 12 2

( ) ( ) ( ) ( )

1 21 1 21 1 21 1 1 21 2

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

1 1 2 2 1 2

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

1 2

exp( ) exp( ) exp( ) exp( )

Det 0 , 1,

exp( ) exp( ) exp( ) exp( )

m m m m

m m m m

m m m m

m m m m

R R R z R z

R R R z R z

m

a z a z a a

b z b z b b

χ χ − χ − χ

= = +∞

χ χ − χ − χ

(2.12)

γ

λ

]

∪

Ω

,

0

[

0

λ

- . (2.12)

:

(

) (

)

(

)

(

) (

)

(

)

2 2 2

2 1 2 0 1 2 0 1 1 2

2

1 2 0 1 1 2

2 2 2

1 2 5 0 1 5 0 1 1 2

( ) , , ,... ,... , , ,... ,... (1 exp(2( ))) 4 , , ,... ,... exp( )

( ) , , ,... ,... , , ,... ,... exp(2 ) exp(2 ) 0, 1, .

m m m m

m

m m m m

x x K r r r Q r r r z z

x x R r r r z z

x x K r r r Q r r r z z m

− η η + + +

+ η + −

− + η η + = = ∞

(2.13)

(

2

)

2 2

(

2 2

)

0 1 1 1 2 2 1 2 3 1 2 4

, , ,... ,...

( 1)

i

,

2,5

im m m

K

η

r r

r

= δ

x x

+ − δ

x x

+ δ

x

+

x

+ δ

i

=

,

(

2

)

2 2

(

2 2

)

0 1 1 1 2 2 1 2 3 1 2 4

, , ,... ,...

( 1)

i

,

2,5

im m m

Q

η

r r

r

= γ

x x

+ − γ

x x

+ γ

x

+

x

+ γ

i

=

,

(

2

)

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

0 1 12 22 11 21

, , ,... ,...

m m m m

m m m m m m

R

η

r r

r

=

R

R

a b

+

R

R

a b

,

2 2

2 11 22 12 11 22 12 12

1 2

11 11 66 11

,

,

m m

B B

B

B B

B

B

B

B B

B

m

⎛

⎞

−

−

η

δ =

_⎜

−

η

_⎟

η =

⎝

⎠

2 2

2 22 11 22 12 12 11 22 12 22 66 12 66 2

2 3 3

11 11

(

( )

,

m m

B B B B B B B B

B B B B

B B

⎛ − − − − ⎞

δ = −η _⎜ + η _⎟

⎝ ⎠

(

)

(

)

2

2 2 12 66 12 22 4 2

12 11 22 12

3 3 4 3

11 11

(

)

(

)

1

,

1

,

m m m m

B B

B

B

B

B B

B

B

B

+

−

δ =

η

− η

δ =

η

− η

2

12 11 22 12 12 66

1 2

11 66

(

)

,

_j j

,

1, 2 ,

B

B B

B

B B

x

j

B B

m

χ

−

−

γ =

=

=

(2.14)

2 2

2 12 11 22 12 22 66 12 66 22 11 22 12

2 2 3

11 66 11

(

(

)

,

m

B

B B

B

B B

B B

B

B B

B

B B

B

⎛

−

−

−

−

⎞

γ = −

_⎜

+

η

_⎟

⎝

⎠

(

2

)

2 2

12 22 22 22 12

3 2 4

11 11 11 11

1

_m

,

(1

_m

)

_m

.

B B

B

B

B

B

B

B

B

⎛

⎞

γ = −

− η

γ = −

− η

_⎜

+

η

_⎟

⎝

⎠

(

)

2

( ) 11 22 12 12 66 2 2 12 66 2 2

1 2 2 2

11 11 11

1

,

m

j m j m m

B B

B B

B B

B

R

x

B

B

B

⎛

−

⎞

=

_⎜

−

η

_⎟

+

− η η

⎝

⎠

2

( ) 11 22 12 12 66 2 22 2

11 66 11

(1

),

j

m j m

B B

B

B B

B

b

x

B B

B

−

−

=

−

− η

2

( ) 11 22 12 2 12 22 2 ( ) 12 2 22 12 2

2 2

11 11 11 11 11

,

,

1, 2.

m j

j j m m j m

B B

B

B

B

B

B

B

R

x

a

x

j

B

B

B

B

B

−

+

=

+

η

=

+

+

η

=

, (2.13)

, (2.8)

.

. , 2

1

2 )

(x −x ,

, .

(

2

) (

2

)

2 0 1 2 0 1 1 2

2 11 21 32 42 12 22 31 41 1 2 11 21 41 31 1 2

, , ,... ,... , , ,... ,... (1 exp (2 ( )))

4( ) exp ( ) 4 [ ]

m m m m

K r r r Q r r r z z

m m m m m m m m z z m m m m z z

η η + + +

+ + + − − (2.15)

(

)

(

)

(

)

11 21 31 42 32 41 31 41 11 22 12 21 1 2 2 1

11 22 12 21 31 42 32 41 1 2

2[ ( ) ( )][ ] exp( ) exp( )

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

m m m m m m m m m m m m z z z z

m m m m m m m m z z m

− + + + − −

− + + + = = +∞

2

( ) 11 22 12 12 66 2

11 11 12 2 2 1 2

11 11

,

(

),

m

m

B B

B B

B

m

R

m

x

x

B

B

⎛

−

⎞

=

=

_⎜

−

η

_⎟

+

⎝

⎠

2

( ) 11 22 12 2 2 12 22 2

21 1 21 22 2 1 2 1 2

11 11

(1) 12 2 2 22 12 2

31 1 32 1 2 1 2

11 11 11

2

(1) 11 22 12 12 66

41 42 1 2

11 66

,

(

)

,

,

(

)

,

,

(

).

m

m

m m

m

B B

B

B

B

m

x R

m

x

x

x x

B

B

B

B

B

m

x a

m

x

x

x x

B

B

B

B B

B

B B

m

b

m

x

x

B B

−

+

=

=

+

+

+

η

=

=

+

+

+

+

η

−

−

=

=

+

(2.16)

, : ɟɫɥɢ −2(_β)

R ɦɨɠɧɨ

ɩɪɟɞɫɬɚɜɢɬɶɜɜɢɞɟ (1.1) ɢ

λ ∉

[0,

λ ∪ Ω

₀

]

_γ, ɬɨɭɪɚɜɧɟɧɢɹ (2.13) ɢ (2.15) ɹɜɥɹɸɬɫɹ

ɞɢɫɩɟɪɫɢɨɧɧɵɦɢ ɭɪɚɜɧɟɧɢɹɦɢ ɨɩɟɪɚɬɨɪɚ (c ) 0

L

, ɝɞɟ æ2=mx2 ɢ æ2=mx2? –

ɪɚɡɥɢɱɧɵɟɤɨɪɧɢɭɪɚɜɧɟɧɢɹ (2.8) ɫɧɟɩɨɥɨɠɢɬɟɥɶɧɵɦɢɞɟɣɫɬɜɢɬɟɥɶɧɵɦɢɱɚɫɬɹɦɢ.

,

χ =

₁

mx

₁

χ =

₂

mx

₂

,

ml

→

∞

(2.13) (2.16) :

(

2

)

2 2

(

2 2

)

2m m

, , ,... ,...

0 1 m 1 1 2 2 1 2 3 1 2 4

0,

K

η

r r

r

= δ

x x

+δ

x x

+δ

x

+

x

+δ =

m

=

1

,

+∞

(2.17)

(

2

)

2 2

(

2 2

)

2m m

, , ,... ,...

0 1 m 1 1 2 2 1 2 3 1 2 4

0,

1,

Q

η

r r

r

= γ

x x

+ γ

x x

+ γ

x

+

x

+ γ =

m

= +∞

(2.18)

(2.17)

, ё

( . [11]). (2.18) ,

ё ё .

3. ы ч . (2.8)

. .

ч a)

R

−2

=

k

2

r

₀

/

2

(

r

_m

=

0

,

m

=

1

,

+∞

)

, . .

ё . (2.5)

2 66 0 2

22

(

m

)

m

2

m

0,

1,

B

r

r

A

w

m

B

⎛

⎞

−

−

η

=

= ∞

⎜

⎟

⎝

⎠

, (3.1)

, (2.8)

2 66 0

22

2 0, 1,

mm m m

B

r r p c m

B

= − η = = +∞, (3.2)

2 2

2 11 22 12 0 4 2 11 22 12 12 66 2 11 66 2

11 66 11 66 11

2 ( )

2

r B B B B B B B

B B B

m

B B B B B

⎛ − ⎞ ⎛ − − +

η − χ − η − η +

⎜ ⎟ ⎜

2

2 2 2 2 2 2

11 22 12 66 22 0 22 66 22 0

11 66 11 11 11

( ) 0, 1,

2 2

B B B B B r B B B r

m m m

B B B B B

⎞ ⎛ ⎞

− +

+ _⎟χ + + η −η _⎜ − η + _⎟= = +∞

⎝ ⎠

⎠

(3.3)

0

[0, ]

γ

λ∉ λ ∪Ω

χ χ

₁

,

₂ (3.3)

-. (3.3) ,

2 2 1 2

χ χ

,

χ +χ

₁2 2₂ .

(2.13), ё

ё .

(

) (

)

(

)

(

)

(

) (

)

(

)

2 2 2

2 1 2 0 2 0 1 2

2 2 2 2

1 2 0 1 2 2 1 5 0 5 0

1 2

(

)

,

,

1 exp(2(

))

4

,

exp(

) (

)

,

,

exp(2 ) exp(2 )

0,

1,

m m m m

m m m m m

x

x

K

r Q

r

z

z

x x R

r

z

z

x

x

K

r Q

r

z

z

m

−

η

η

+

+

−

−

η

+

−

+

η

η

×

×

+

=

= ∞

(3.4)

(

)

(

)

2 2

2 2 11 22 12 2 1 2 11 22 12

0 1 2

11 66 11 66

, (1 ) 1 ( 1)i , 2,5

im m m m m m m

B B B B B B

K r x x i

B B B B

−

⎡ − ⎤ ⎛ − ⎞

η = −η _⎢ +ε −η + −_{⎥ ⎜}η − ε _⎟ =

⎣ ⎦ ⎝ ⎠ ,

(

)

2

2 2 66 22 2 11 22 12

0 1 2

11 11 11 66

,

(1

)

( 1)

i

,

2,5;

im m m m m m

B

B

B B

B

Q

r

x x

i

B

B

B B

⎛

⎞

⎛

⎞

−

η

= − η

_⎜

−

ε + −

_⎟

_⎜

η −

ε

_⎟

=

⎝

⎠

⎝

⎠

( )

(

)

2

2 2 12 66 2 22 11 22 12 11 12 2 0

0 2

11 11 11 66 11

, 2(1 ) 1 , .

2

m m m m m m m

B B B B B B B B r

R r

B B B B B m

⎡ ⎤

⎛ + ⎞ − +

η = −η _⎜ η − ε _⎟⎢ +ε − η _⎥ ε =

⎝ ⎠⎣ ⎦ (3.5)

,

χ =

₁

mx

₁

χ =

₂

mx

₂

,

l

→

∞

(3.4) :

(

2

)

2

(

)

11 22 122 2 2 11 22 122

2 0 1 2

11 66 11 66

, (1 ) 1 0

m m m m m m m

B B B B B B

K r x x

B B B B

⎡ − ⎤ ⎛ − ⎞

η = − η _⎢ + ε − η −_{⎥ ⎜}η − ε _⎟=

⎣ ⎦ ⎝ ⎠ , (3.6)

(

)

2

2 2 66 2 22 2 11 22 12

2 0 1 2

11 11 11 66

, (1 ) 0,

1, .

m m m m m m m

B B B B B

Q r x x

B B B B

m

⎛ ⎞

⎛ ⎞ −

η = −η _⎜ η − ε +_{⎟ ⎜}η − ε =_⎟

⎝ ⎠ ⎝ ⎠

= +∞

(3.7)

(3.6)

.

(3.7) (1.2)-(1.5) ( . (1.8) (1.9)).

, , (3.7) 2

2

66 22 0 2

11 11

2

2 2

2

0 66 0

22 11 22 12

2 2 2

11 11 11 66

(1 )

2

1 1 0, 1,

2 2

B B r

m B B

r B r

B B B B

m

m B m B m B B

⎛ ⎞

η

− _⎜ η − _⎟+

⎝ ⎠

⎛ ⎞

⎛ ⎞

⎛ η ⎞ ⎛ ⎞ η −

+η _⎜ − _{⎟⎜ ⎜} + _⎟− _⎟⎜η − _⎟= = +∞

⎝ ⎠

⎝ ⎠⎝ ⎠⎝ ⎠

(3.8)

∞

→

m

(3.8)

2

2 2 2

66 22 0 22 11 22 12 0

11 11 11 11 66

0,

2

2

B

B

r

B

B B

B

r

B

B

B

B B

⎛

−

⎞

η −

+

η η −

_⎜

_⎟

=

oe (1.2) – (1.5) _R−2 =_k2_r0/2

.

ε →

_m

0

(3.3) 2

4 11 22 12 12 66 2 11 66 2 2

11 66 11

2 ( )

c_m B B B B B m B B

B B B

⎛ − − + ⎞

= χ −_⎜ − η χ +_⎟

⎝ ⎠

2 2 22 2 66 2

11 11

(m ) B m B 0, m 1, ,

B B

⎛ ⎞

+ − η _⎜ − η =_⎟ = +∞

⎝ ⎠ (3.10)

,

-. (3.4)

(

)

(

)

2 2 2

2 1 2 2 1 2

2 2 2 2

1 2 1 2 2 1 5 5

1 2

(

)

(

)

(

) 1 exp(2(

))

4

(

) exp(

)

(

)

(

)

(

)

exp(2 )

exp(2

)

0,

1, ;

m m m m

m m m m m

x

x

K

Q

z

z

x x R

z

z

x

x

K

Q

z

z

m

−

η

η

+

+

−

−

η

+

−

+

η

η ×

×

+

=

= ∞

(3.11)

2

2 2 11 22 12 2 1 2

1 2 11 66

(

)

(1

)

( 1)

i

,

2,5

im m m m m

B B

B

K

x x

i

B B

−

⎛

−

⎞

η = − η

_⎜

− η + −

_⎟

η

=

⎝

⎠

,

2 2 66

1 2 11

(

)

(1

)

( 1)

i

0,

1,

,

2, 5

im m m

B

Q

x x

m

i

B

η = − η

+ −

=

= +∞

=

, (3.12)

( )

2

2 12 66 2 11 22 12 11 12 2

11 11 66 11

2

(1

)

m m m

B

B

B B

B

B

B

R

B

B B

B

⎛

⎞

+

−

+

η =

− η

_⎜

−

η

_⎟

⎝

⎠

.

1

x m

1

,

2

x m

2

χ =

χ =

– (3.10)

-. (3.11)

ё ё .

1

mx

1

χ =

χ =

₂

mx

₂ ,

∞ →

l (3.11)

( )

2

2 2 11 22 12 2 2

2 1 2

11 66

(1

)

0,

1,

m m m m m

B B

B

K

x x

m

B B

⎛

−

⎞

η = − η

_⎜

− η − η

_⎟

=

= +∞

⎝

⎠

, (3.13)

2 2 66

2 1 2

11

(

)

(1

)

0,

1,

.

m m m

B

Q

x x

m

B

η = − η

+

=

= +∞

(3.14)

(3.13)

-– ё ё

[11].

ч ) −2

=

2

(

₀

/

2

+

₁

cos

)

(

=

0

,

=

2

,

+∞

m

r

k

r

r

k

R

β

_m .

(2.5) :

{

r p

1 m−1

ω

m−1

+

r

mm

ω +

m

r p

1 m+1

ω

m+1

=

0,

m

= +∞

1,

, (3.15) 2

0 66 22

/

,

2

/

,

m

w

m

c

m

r

mm

r p

m

B

c

m

B

ω =

=

−

η

(3.16)

m m

c

p

,

(2.6), (3.10).

(3.15) ,

(

2 2

)

11 12 22 66 0 1

,

,

,

,

,

, ,

0

D

χ η

B

B

B

B

r r

=

, (3.17)

2

χ

- (3.17) . ё ё

D

n

:

(

2 2

)

11 12 22 66 0 1

,

,

,

,

,

, ,

0

n

D

χ η

B

B

B

B

r r

=

. (3.18)

n

χ

- (3.18).

(3.17)

χ

_n

n

→

∞

.

D

_m

, :

3

,

2 1 2 1 1 2 1 2 1 1 22 2 11 1

≥

−

=

−

=

=

− −

−

r

p

p

D

m

D

r

D

p

p

r

D

r

D

r

D

m m m m mm m (3.19)

[11]: ɩɪɢ ɮɢɤɫɢɪɨɜɚɧɧɨɦ

m

≥

2

ɢ ɩɪɢ

ɭɫɥɨɜɢɢ

λ ∉

[0,

λ ∪ Ω

0

]

γ ɭɪɚɜɧɟɧɢɹ (3.17) ɢɦɟɸɬ 2

χ

-ɮɨɪɦɚɥɶɧɵɟɪɟɲɟɧɢɹɜɢɞɚ

( ) ( )

2 2

( ) ( ) ( ) 2 ( ) 4

1 1

,

,

1, 2

i m m i

j j j

r

jm

r

i j

χ

= χ

+ α

+ β

+

_…

=

, (3.20)

ɝɞɟ ( )m j

χ

– ɤɨɪɧɢ ɭɪɚɜɧɟɧɢɹ

r

_mm

=

0

(ɢɥɢ ɭɪɚɜɧɟɧɢɹ (3.3)) ɫ ɧɟɩɨɥɨɠɢɬɟɥɶɧɵɦɢ

ɞɟɣɫɬɜɢɬɟɥɶɧɵɦɢɱɚɫɬɹɦɢɢ

( ) 1 1 1 1 1 1 ( )

1 1 1 1

(

)

,

1, 2

m j m m m m m m m m

j

m m m m mm

p

p

r

p

r

j

r

r

r

− + + + − −

− − + + _χ=χ

+

α

=

=

′

. (3.21)

mm

r

′

– χ2_{. ,}

2 , 1 , , ) ( ₌ j i kχ_ji

ё

( )

(

₂

)

1 2 ( ) ( ) ( ) 2

1

, ,

1, 2

i m m

j j j

r

i j

χ ≈ − χ

+ α

=

, (3.22)

/

m

η

– (2.13), (2.15). ч )

R

−2

=

k r

2₁

cos

k

β

(

r

₀

=

0

,

r

_m

=

0

,

m

=

2

,

+∞

)

.

,

0 0=

r ( . [21]).

mm

r

α

( )_jm

,

j

=

1, 2

(3.16), (3.21) :

1 1 1 1

2 ( )

66 22 2 2 ' ( )

66 22 1 1

( )

2 / , , 1, 2, 0,

4( / )

m m m m m

m

mm m j m

j

m m m

p p c p c

r B B c j m

B B c c c

− + + − χ=χ − +

+

= − η α = = = +∞

η . (3.23)

(3.23)

χ

( )jm

,

j

=

1,2

–

(3.10) .

. 1– 4, (2.15), (3.4), (3.11),

/

m

η

m

,

a

,

b

2

,

2

;

1

,

2

;

),

(

2 2

2

−

≤

=

=

=

=

=

−

b

a

b

a

a

x

x

a

ba

, : 0 / max{ Re 1/ , Re 2/ }

kχ m= k χ m k χ m (3.25)

. 1–4 1,2,3

;

0

2

=

−

R

R

−2

=

k

2

r

₀

/

2

;

R

−2

=

k r

2

( / 2

₀

+

r

₁

cos

k

β

)

(3.24),

[22]

3 2.4 10

ρ = ⋅ / 3;

Ε

₁

=

6

.

37

⋅

10

10 / 2;

Ε

₂

=

1

.

47

⋅

10

10

;

G

=

4

.

9

⋅

10

6;

v

₁=0.26;

2

v

=0.06 (3.26)

: ɚ=2,b=1,l=15,s=4.5912,

r

₀

=

0

.

205

,

r

₁

=

0

.

0306

;

1

,

2

=

=

b

ɚ

,

l

=

5

,

s

=

4

.

5912

,

r

₀

=

0

.

205

,

r

₁

=

0

.

0306

;

ɚ

=

2

,

b

=

2

,

l

=

15

,

s

=

5

.

9158

,

3021

.

0

,

8653

.

0

₁

0

=

r

=

r

;

ɚ

=

2

,

b

=

2

,

l

=

5

,

s

=

5

.

9158

,

r

₀

=

0

.

8653

,

r

₁

=

0

.

3021

.

1

1 2 3

m

0

/

/

k

χ

m

η

m

k

χ

₀

/

m

η

/

m

k

χ

₀

/

m

η

/

m

1 -0.0650 0.9900

2 -0.0856 0.9825 -0.0689 0.9850 -0.0689 0.9850 3 -0.0865 0.9822 -0.0733 0.9831 -0.0733 0.9831 4 -0.0866 0.9821 -0.0743 0.9826 -0.0743 0.9826 5 -0.0866 0.9821 -0.0747 0.9824 -0.0747 0.9824 10 -0.0866 0.9821 -0.0752 0.9822 -0.0752 0.9822 20 -0.0866 0.9821 -0.0754 0.9821 -0.0754 0.9821 100 -0.0866 0.9821 -0.0754 0.9821 -0.0754 0.9821

2

1 2 3

m

0

/

/

k

χ

m

η

m

k

χ

₀

/

m

η

/

m

k

χ

₀

/

m

η

/

m

3 -0.0650 0.9900 -0.0373 0.9956 -0.0373 0.9956 4 -0.0797 0.9849 -0.0634 0.9874 -0.0634 0.9874 5 -0.0840 0.9832 -0.0704 0.9844 -0.0704 0.9844 6 -0.0856 0.9826 -0.0731 0.9832 -0.0731 0.9832 7 -0.0862 0.9823 -0.0742 0.9827 -0.0742 0.9827 10 -0.0865 0.9821 -0.0752 0.9822 -0.0752 0.9822 20 -0.0866 0.9821 -0.0754 0.9821 -0.0754 0.9821 100 -0.0866 0.9821 -0.0754 0.9821 -0.0754 0.9821

3

1 2 3

m

0

/

/

k

χ

m

η

m

k

χ

₀

/

m

η

/

m

k

χ

₀

/

m

η

/

m

1 -0.0650 0.9900

4

1 2 3

m

0

/

/

k

χ

m

η

m

k

χ

₀

/

m

η

/

m

k

χ

₀

/

m

η

/

m

3 -0.0650 0.9900

4 -0.0797 0.9849 -0.0278 0.9960 -0.0277 0.9960 5 -0.0840 0.9832 -0.0460 0.9889 -0.0460 0.9890 6 -0.0856 0.9826 -0.0522 0.9858 -0.0522 0.9858 7 -0.0862 0.9823 -0.0550 0.9842 -0.0550 0.9842 10 -0.0865 0.9821 -0.0576 0.9827 -0.0576 0.9827 20 -0.0866 0.9821 -0.0584 0.9822 -0.0584 0.9822 100 -0.0866 0.9821 -0.0585 0.9821 -0.0585 0.9821

ч . ,

-,

(1.1),

-, ё .

(2.13) (2.15).

– (3.4) (2.15) – (3.11) (2.15). -,

-. , ,

,

.

m

-, . , ml→∞

–

, .

-, .

1. . ., . ., . .

// . .: . . 1992. . 87-91.

2. . .

( ) // .

. . . .1961. .1. № 3. . 542-545.

3. . ., . ., . ., . .

-. : ,

1986. 170 .

4. . .

// . . . 1979.

№ 2. .139-143.

5. . ., . .

// . . 1972. .8. .8. .113–116. 6. Grigorenko Ya.M., Rozhok L. S. Solving the Stress Problem for Hollow Cylinders

with Corrugated Elliptical Cross Section// Int. Appl. Mech. 2004, 40, № 2. p. 169-175. 7. Semenyuk N.P., Babich I.Yu., Zhukova N.B. Natural Vibrations of Corrugated

8. . ., . ., . .

// . .

1973. .4. № 5. .978-986.

9. . ., . ., . . . .: , 1979. 383 .

10. . ., . . . .: , 1974. 156 .

11. , . .

-// . . . 2007. №1. . 84–99.

12. . .

// . 1932. № 11. .33-38; № 12. .21-26.

13. . .

// . . . . . . 1933. № 5. .647-653.

14. . ., . ., . .

-. -.

// . . . 1982. .44. № 6. . 1007-1013.

15. . .

– // . . 2008. .44.

№ 11. .99-111.

16. . . . .: , 1974. 446 .

17. ., – . . .: ,

1979. 587 .

18. . . ,

. – . // . . . 1964. . 28. . 665-706. . - , 1970. . 110. №6. .233-297.

19. . ., . . ё . .- .:

, 1952. 695 .

20. . . . .3.

.: , 1963. 656 .

21. . ., . .

// . . 2006. .42. № 12. .97-114.

22. . ., . .

, . // . .

. 1982. № 3. .171-174. х:

ч – .- . ., .

. . . .: (+37410) 64-91-21; -mail:

ghulgr@yahoo.com

ч – . . .

.: (37410)63-86-83