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

11

## Texto

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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.

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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 yD zh h<<l ( D0– , 2h – ,

l– ), ё

[6] :

### 0

zz xz yz

(2)

,

.

[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

* * *

s s i t

jk

s s s

s i t s

s

− + ω

αβ

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

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

− −

− −

− − −

− −

### ∂ζ

( ) ( ) ( )1 ( ) ( )

22 33 33 44 23

1

s s

s s

s

### =

(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

s s s s s

s s

s s s s s

s s

s s s s

s s

− − − −

− − −

− −

( )1 13

s

### ∂η

(1.5)

2 2

2

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

11

22 33 12

### −

(3)

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 aa aa aa a

Δ Δ

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

( )

( ) ( )

2

55 * * *

2

55 44 11

s

s m s

m n n m U

− −

### ∂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

s s s s

s s

s s

U V

s s

s s

s W

− − − −

− −

− −

− −

(1.7)

,

(1.6) s=0:

( )

( )

0 2

0 2

55 *0 *0

2

55

44

11

(1.8)

(1.8) :

( ) ( )

( )

### )

0 0 2 0 2

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

55 44 11

U U

(1.9)

(1.9), , ё (1.5)

(1.1) (1.2) σxz

( )0

### 1

, (1.10)

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

( ) ( ) ( ) ( )

0 0

1 0 2 0

0 0

1 0 2 0

U U U U

U U U U

(1.11)

2

0 55 *0 2 *0

U a i K

γ = ω − ω .

:

0 0

U

U

, (1.12)

2 :

0 0

U

U

, (1.13)

0 0

U U

. (1.14)

(1.13) ,

:

2 2 2 *0

55

n

(4)

1)

55 n K

a π

> :

2 2 2 *0

55

n

### ∈

. (1.16)

(1.3), (1.16) , ,

2 2 2

55

2 2 2

55

.

2)

55 n K

a π

< :

2 2 2 *0

55

n

. (1.17)

(1.3) , .

55 13

1a =G – ,

. .

,

( “I”). , 2 2 2

* h

ω = ρ ω ,

2 2

I 2

0

55

n

### ⎠

. (1.18)

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

( )0 ( )0

( )0

I 1

0 1 I

n U U Un

,

(1.19)

(1.14) ( ,

“II”)

2

2

II 2

0

55

n

(1.20)

:

( )0 ( )0

II 2 II

n Un

### ∈ Ν

. (1.21)

, (1.1) (1.2) σyz,

:

2 2

III 2

0

44

n

( ) (1.22)

2

2

IV 2

0

44

n

( ) (1.23)

( )0 ( )0

III 1 III

n Vn

(5)

( )0 ( )0

IV 2 IV

n Vn

( ) (1.25)

(1.1) σzz,w.

( )0

;

( )0

### 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

W W W W

W W W W

(1.27)

:

0 0

W W

, (1.28)

2

2

V 2

0 11

n

### ⎠

. (1.29)

, (1.26)

. (1.27) C2( )W0 =C1( )W0tgγW0, , (1.9)

(1.28) :

( )0 ( )0

V 1 V

n Wn

, ( )

( )0

0 1

1

0

W W

W

(1.30)

(1.2) w

( )0

. (1.31)

(1.31), (1.9),

:

2

VI 2 2

0 11

n

( )(1.32)

VII 2 2 2

0 11

n

( ) (1.33)

( )0 ( )0

VI 1 VI

n Wn

( ) (1.34)

( )0 ( )0

VII 2 VII

n Wn

### ∈ Ν

( ) (1.35)

, , (1.19), (1.21),

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

,

(6)

I 0n

ω II

0n ω

( )0

yz

### σ

, W( )0 , , . .

, :

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

I I II II

n n n n

,

### .

(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

n n n n

### =

, (1.37)

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

n n n n n n

,

### .

(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

jkn jkn

n Un n Un

(1.39)

( ) ( )

( ) ( ) ( ) ( )

### )

0 0

III IV

0 0 0 0

23 III 1 III 23 IV 2 IV

44 44

jkn jkn

n Vn n Vn

(1.40)

, V VI

0n 0n, 0n

ω = ω ω (1.1), :

( )

( ) ( )

### )

0

0 0

11 V 23 1 V 23 13 11

jkn

n Wn

(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

jkn jkn

n Wn

n Wn

### ∈ 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

n m

,

,

(2.1)

(2.1), (1.8)

(7)

( )

( )

0 2

2 0

I I

I

55 *0 *0 I

2

m

m m m

. (2.2)

(2.2) Un( )0I ζ

−1, 1

, :

( )

( )

## )

( ) ( )

1 2 0 1

2

0 I I 0 0

I

I 55 *0 *0 I I

2

1 1

m

n m m n m

− −

### ∫

. (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

− −

(2.3) :

( ) ( )

( ) ( )

1 0 0 1

2 0 0

I I

I I

55 *0 *0 I I

1 1

n m

m m n m

− −

. (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

nv m

m m n m

− −

. (2.5)

(2.4) (2.5), :

I I

I I

### )

1 ( ) ( )0 0

55 *0 *0 *0 *0 I I

1

n m n m n m

(2.6)

nm, I I

*0n *0m

ω ≠ ω (2.6) :

( ) ( ) 1

0 0 I I 1

n m

. (2.7)

,

( )

( )

0 I

I 0 I

1 I

n

n n

Un

,

(2.8)

1− ≤ ζ ≤1.

n

Ψ .

3.

### 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

n

n n n n n n Un

(3.1)

(3.1) Un( )1I

:

( )1 ( )0

I I

1

n nm m

m

### b U

=

(8)

(1.1) (1.2), σxz.

(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

n m n m nm m n n n Un

m

=

(3.3)

(3.3) Uk( )I0 ζ

−1, 1

,

( )

0 I n

U , :

I I

I I

I

I

( )0 ( )1

55 *0 *0 *0 *0

### = 2

55 *1 *0 I I

nk n k n k n n Unk nk Unk

(3.4)

nk

δ – ,

( ) ( )

( ) ( )

( ) ( ) ( ) 1

0

0 1 I 1 1 0

I 0 I 0 2 I I

1

1 I 1 I

Un

Unk Unk Un k

Uk Uk

,

(3.5)

n=k (3.4) :

( )

1

I I

*1 I

55 *0

Unn n

n

,

(3.6)

nk:

( )

1 I

I I I I

55 *0 *0 *0 *0

Unk nk

n k n k

,

### .

(3.7)

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

nk

,

,

*1I n

,

. (3.8)

nn

b Un:

( )

( ) ( )

1 2

0 1

I I

2 0

1 I

n n

n

, ( )

( )

1 2

2

0 0

I I

1

n n

, (3.9)

: ( ) ( )

1

0 1 I I 1

n n

,

### .

(3.10)

( )1 I n

U (3.10), (3.2), :

( ) ( ) ( ) ( )

1 1

1

0 0 0 0

I I I I

0 1 1

n

nk n k nn n n

k

=

.

nk, ( ) ( )

1 0 0 I I 1

0 nn n n

b U U d

ζ =

,

nn

,

(3.11)

, ω*0n = ω*0In :

( )1 I

I

*1

n n

(9)

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

n

n n n n n n Un

(3.13)

:

( )2 ( )0

I I

1

n nr r

r

=

,

(3.14)

, :

( )

2

I I

*2 I

55 *0

Unn n

n

,

, (3.15)

( )

2 I

I I I I

55 *0 *0 *0 *0

Unk nk

n k n k

,

, (3.16)

( )

( )

( ) ( ) ( ) ( )

1 1

2 2 0 2

I 0 2 I I 0 I I

1 1 I 1

1 I

Unk Un k Un k

Uk Uk

− −

### 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

n n n

Un

. (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

,

### ,

(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

n

Wn Un

ξ

. (3.20)

(3.19), :

( )1 ( )1 ( )1

0I 1 2

55 11 55 11

n W W

### ζ

.

(3.19), (3.20) (3.19):

( )

( )

1 0 55 23

I 1 I

55 11

n Un

τ ξ

.

, Wn( )I1

( ) ( ) ( )

( )

### )

1 1 1 0 55 23

I 1 2 1 I

55 11 55 11 55 11

n W W Un

ξ

(3.21)

( )1 ( )1 1, 2

W W

(10)

( )1

( )I1 ( )0I

I 11 23

1

n n

n

ζ=

### ⎠

. (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

n W Un n W Un

+ ξ + ξ

### 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

n Un Un Un Un

+ ξξ

ξξ ηη

### ⎠

(3.24)

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

I 0 I

1 1 I

Unk Un Uk

. (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

Un n k Unk Uk

+ + ξξ

### −

. (3.26)

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

*2n ω cnk:

( )

( )

### )

0 1 I 55 11 55 23 11 23

I

*2 0 2 I

1 I 55 11 *

2 0 55 11

Un n Un n

ξξ

,

(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

n k Un nk

Uk n k n k

+ + ξξ

(3.28)

(11)

nn

,

(3.29)

, Un( )2I I

*2n

ω , . , :

( ) ( ) I I 2 I

* *0 *2

0 2 2

I I I

n n n

n n n

### ⎪⎩

(3.30)

(1.2) .

( )1 I 0 n

W ≠ , ,

, . .

. , Wn( )I0 =0,Wn( )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