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Вывод частотного уравнения собственных колебаний упругой трехслойной пластины два противоположных края которой шарнирно закреплены а два других жестко закреплены.

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4/2010
ВЕСТНИК
МГСУ
ȼɕȼɈȾ ɑȺɋɌɈɌɇɈȽɈ ɍɊȺȼɇȿɇɂə ɋɈȻɋɌȼȿɇɇɕɏ
ɄɈɅȿȻȺɇɂɃ ɍɉɊɍȽɈɃ ɌɊȿɏɋɅɈɃɇɈɃ ɉɅȺɋɌɂɇɕ, ȾȼȺ
ɉɊɈɌɂȼɈɉɈɅɈɀɇɕɏ ɄɊȺə ɄɈɌɈɊɈɃ ɒȺɊɇɂɊɇɈ
ɁȺɄɊȿɉɅȿɇɕ, Ⱥ ȾȼȺ ȾɊɍȽɂɏ ɀȿɋɌɄɈ ɁȺɄɊȿɉɅȿɇɕ
DERIVATION OF THE FREQUENCY EQUATION OF
NATURAL VIBRATIONS OF AN ELASTIC THREE-LAYERED
PLATE, TWO OPPOSITE EDGES OF WHICH ARE HINGED
AND TWO ARE RIGIDLY FIXED
Ⱥ.ȼ. Ȼɨɝɞɚɧɨɜ, Ɉ.ɂ. ɉɨɞɞɚɟɜɚ
A.V. Bogdanov, O.I. Poddaeva
ȽɈɍ ȼɉɈ ɆȽɋɍ
ȼ ɷɬɨɣ ɫɬɚɬɶɟ ɪɚɫɫɦɨɬɪɟɧ ɜɵɜɨɞ ɱɚɫɬɨɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫɨɛɫɬɜɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ
ɭɩɪɭɝɨɣ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɬɪɟɯɫɥɨɣɧɨɣ ɩɥɚɫɬɢɧɵ, ɞɜɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɯ ɤɪɚɹ ɤɨɬɨɪɨɣ
ɲɚɪɧɢɪɧɨ ɡɚɤɪɟɩɥɟɧɵ, ɚ ɞɜɚ ɞɪɭɝɢɯ ɠɟɫɬɤɨ ɡɚɤɪɟɩɥɟɧɵ.
This article discusses the derivation of the frequency equation of natural vibrations of
an elastic rectangular three-layered plate, two opposite edges of which are hinged and two
are rigidly fixed.
Ɋɚɫɫɦɨɬɪɢɦ ɩɥɨɫɤɢɣ ɷɥɟɦɟɧɬ ɤɚɤ ɢɡɨɬɪɨɩɧɭɸ ɨɞɧɨɪɨɞɧɭɸ ɭɩɪɭɝɭɸ ɬɪɟɯɫɥɨɣɧɭɸ
ɩɥɚɫɬɢɧɭ.
Ȼɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ ɩɥɨɫɤɨɫɬɢ ɪɚɡɞɟɥɚ ɫɥɨɟɜ ɧɚɯɨɞɹɬɫɹ ɜ ɠɟɫɬɤɨɦ ɤɨɧɬɚɤɬɟ,
ɩɪɢ ɷɬɨɦ ɜɧɟɲɧɢɟ ɞɜɚ ɫɥɨɹ ɫɨɫɬɨɹɬ ɢɡ ɨɞɧɨɝɨ ɦɚɬɟɪɢɚɥɚ ɢ ɢɦɟɸɬ ɨɞɢɧɚɤɨɜɭɸ ɬɨɥɳɢɧɭ h2 h1 , ɚ ɜɧɭɬɪɟɧɧɢɣ ɫɥɨɣ – ɢɡ ɞɪɭɝɨɝɨ ɦɚɬɟɪɢɚɥɚ ɢ ɟɝɨ ɬɨɥɳɢɧɚ h1 . ȼ ɩɥɨɫɤɨɫɬɢ
XOY ɩɥɚɫɬɢɧɚ ɡɚɧɢɦɚɟɬ ɨɛɥɚɫɬɶ ^0 d x d l ; 0 d y d l ` .
1
2
ȼ ɞɚɥɶɧɟɣɲɟɦ ɩɚɪɚɦɟɬɪɵ ɜɧɭɬɪɟɧɧɟɝɨ ɫɥɨɹ ɨɛɨɡɧɚɱɚɬɶ ɢɧɞɟɤɫɨɦ "1" , ɚ ɜɧɟɲɧɢɯ
ɫɥɨɟɜ – ɢɧɞɟɤɫɨɦ " 2 " .
ɑɚɫɬɨɬɵ ɫɨɛɫɬɜɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɭɩɪɭɝɨɣ ɬɪɟɯɫɥɨɣɧɨɣ ɩɥɚɫɬɢɧɵ ɛɭɞɟɦ ɨɩɪɟɞɟɥɹɬɶ, ɨɫɧɨɜɵɜɚɹɫɶ ɧɚ ɩɪɢɛɥɢɠɟɧɧɨɦ ɭɪɚɜɧɟɧɢɢ ɩɨɩɟɪɟɱɧɵɯ ɤɨɥɟɛɚɧɢɣ ɱɟɬɜɟɪɬɨɝɨ
ɩɨɪɹɞɤɚ [1]:
w 2W
w 4W
w 2W
A
A
'
A3' 2W
1
2
2
4
2
wt
wt
wt
0
(1)
ɝɞɟ
219
ВЕСТНИК
4/2010
МГСУ
2
h2 h1 ­° 1 7 8Q 1 h3 b 2 7 8Q 2 7 8Q 1 h3 h 2 ;
®
b 2 1 U h / 2 °̄ b 2 9b 1 Q 1 2 1 Q 2 16 1 Q 1 ª
3 4Q 2 º h °½ ;
U 1 « 2 U 2b 2
» ¾
2 1 Q 2 ¼ 4 ¿°
¬
A1
2
A2
h3 h2 h1 ­ 1
®
8 1 U h / 2 ¯ b 2
ª § 2· 1
º 4
« D1 ¨1 h ¸ 3 2 D1 » h3
¹
¬ ©
¼
ª § 2· 2
º
« D2 ¨ 1 h ¸ 3 2 D2 » ¹
¬ ©
¼
2 § 2 · ª 2 D2 1 º 4 ª
D § 1 · 1 D2 º ½°
¨1 ¸ « U
1» «1 U 4 22 ¨1 ¸ 2
»¾;
h© h ¹¬
2 D2
b © U¹
D22 ¼ °¿
¼ h¬
2
h3 h2 h1 b 2 ª 4 U D1 § 3 ·
1 D1 40 D2 º
A3
2 3 »;
¨1 ¸ 2 U
«
8 1 U h / 2 ¬ 3 © h ¹
h
3b h ¼
2h1
U1
b1
1
h
; U
; b
; Di
; i 1, 2 U2
h2 h1
b2
2 1 Q i bi - ɫɤɨɪɨɫɬɶ ɩɨɩɟɪɟɱɧɨɣ ɜɨɥɧɵ, Q i - ɤɨɷɮɮɢɰɢɟɧɬ ɉɭɚɫɫɨɧɚ,
Ui
- ɩɥɨɬɧɨɫɬɶ.
ɉɭɫɬɶ ɞɜɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɯ ɤɪɚɹ ɩɥɚɫɬɢɧɵ ɩɪɢ
ɨɩɟɪɬɵ, ɚ ɞɜɚ ɞɪɭɝɢɯ ɩɪɢ
y 0
ɢ
y l2
ɲɚɪɧɢɪɧɨ
x 0 ɢ x l1 ɠɟɫɬɤɨ ɡɚɤɪɟɩɥɟɧɵ.
Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɞɥɹ ɬɚɤɨɣ ɩɥɚɫɬɢɧɤɢ ɢɦɟɸɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ:
W
W
x 0
y 0
W
W
x l1
y l2
wW
wx
w 2W
wy 2
x 0
y 0
wW
wx
w 2W
wy 2
x l1
x l2
½
0°
°
¾
°
0°
¿
(2)
Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (1) ɛɭɞɟɦ ɢɫɤɚɬɶ ɜ ɜɢɞɟ [2]:
§ b · f
§ kS y ·
W x, y, t exp ¨ i[ 1 t ¸ ¦Wk ( x)sin ¨
¸
© h1 ¹ k 1
© l2 ¹
ɝɞɟ
[
- ɛɟɡɪɚɡɦɟɪɧɚɹ ɱɚɫɬɨɬɚ ɫɨɛɫɬɜɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɩɥɚɫɬɢɧɵ
ɉɨɞɫɬɚɜɥɹɹ (3) ɜ ɭɪɚɜɧɟɧɢɟ (1), ɞɥɹ
220
Wk ( x) ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ:
(3)
4/2010
d 4Wk
d 2Wk
B
B2Wk
1
dx 4
dx 2
ВЕСТНИК
МГСУ
0,
(4)
ɝɞɟ
B1
A2 2 2
r1 [ 2r2 2 ;
A3
B2
ªA
A1 4 4
1º
r1 [ r12 « 2 r22 » [ 2 r2 4
A3
A3 ¼
¬ A3
(5)
ɝɞɟ
r1
b1
h1
kS
; r2
l2
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (4) ɡɚɩɢɲɟɦ ɜ ɜɢɞɟ:
ª cos D 0 x cos D1 x º
ª cos D 0 x cos D1 x º
Wk x C1 «
» C2 «
»
n
n
n
n
D
D
D
D
0
1
0
1
¬
¼
¬
¼
ª sin D 0 x sin D1 x º
ª sin D 0 x sin D1 x º
C3 «
» C4 «
»,
m
m
m
m
D
D
D
D
0
1
0
1
¬
¼
¬
¼
(6)
ɝɞɟ C j - ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ, ɚ iD j - ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ:
D 4 B1D 2 B2
0
(7)
ɢ ɪɚɜɧɵ:
D 0,1
B1
2
2
§ B1 ·
¸ B2
© 2¹
r ¨
(8)
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɧɚ ɤɪɚɹɯ ɩɥɚɫɬɢɧɵ ɢɦɟɟɦ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ:
Wk
dWk
dx
0
(9)
Ɍɨɝɞɚ ɢɡ ɨɛɳɟɝɨ ɪɟɲɟɧɢɹ (6) n=0, m=1, ɩɪɢ ɷɬɨɦ ɢɡ ɭɫɥɨɜɢɹ ɧɚ ɥɟɜɨɦ ɤɪɚɸ ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ C1 ɢ C3 ɪɚɜɧɵ ɧɭɥɸ, ɚ ɢɡ ɭɫɥɨɜɢɹ ɧɚ ɩɪɚɜɨɦ ɤɪɚɸ ɩɨɥɭɱɚɟɦ:
ª sin D 0l1 sin D1l1 º
C2 ¬ªcos D 0l1 cos D1l1 ¼º C4 «
» 0
D
D
0
1
¬
¼
(10)
C2 ª¬D 0 sin D 0l1 D1 sin D1l1 º¼ C4 ª¬cos D 0l1 cos D1l1 º¼
0
221
ВЕСТНИК
4/2010
МГСУ
ɨɬɤɭɞɚ, ɢɡ ɭɫɥɨɜɢɹ ɧɟɬɪɢɜɢɚɥɶɧɨɫɬɢ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ (10), ɩɨɥɭɱɚɟɦ
ɬɪɚɧɫɰɟɧɞɟɧɬɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɢɞɚ:
D 02 D12
2
sin D 0l1 sin D1l1 2cos D 0l1 cos D1l1 0
D 0D1
(11)
Ⱦɥɹ ɚɧɚɥɢɡɚ ɱɚɫɬɨɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ (11) ɩɪɟɨɛɪɚɡɭɟɦ ɟɝɨ. ɉɪɟɞɫɬɚɜɢɦ ɫɢɧɭɫɵ ɢ
ɤɨɫɢɧɭɫɵ ɜ ɜɢɞɟ ɪɹɞɨɜ:
f
¦ 1
sin z
z 2 i 1
i
f
2i 1 !
i 0
;
¦ 1
cos z
j
j 0
z2 j
2 j !
;
Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ (11) ɷɤɜɢɜɚɥɟɧɬɧɨ ɫɥɟɞɭɸɳɟɦɭ:
2i
D 0D1 2 i 1
2 D D ¦ 1
e 2i 1! 2i 1!
i 0
f
2
0
2i
2
1
f
¦
i 0
ɉɭɫɬɶ
D
2j
D 0D1 e4 j
2¦ 1
2 j ! 2 j !
j 0
f
2
0
i
0, j
D12 e2
2j
0 , ɜɵɩɢɲɟɦ ɩɟɪɜɵɟ ɫɥɚɝɚɟɦɵɟ ɜ ɭɪɚɜɧɟɧɢɢ (12):
0
(13)
2
ɂɫɩɨɥɶɡɭɹ (8)
(12)
0
B1
§B ·
¨ 1 ¸ B2 ; D1
2
© 2¹
D0
2
B1
§B ·
¨ 1 ¸ B2 ɢɦɟɟɦ
2
© 2¹
D 02 D12 B1
D 2D 2 B2
0
(14)
(15)
1
Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ (13) ɩɪɢɦɟɬ ɜɢɞ:
B1
0
(16)
ɉɨɞɫɬɚɜɥɹɹ B1 ɢɡ (5) ɩɨɥɭɱɚɟɦ:
A2 2 2
r1 [ 2r2 2
A3
0
(17)
Ɍɨɝɞɚ ɱɚɫɬɨɬɚ ɫɨɛɫɬɜɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɪɚɜɧɚ:
2
[
2
ɉɪɢ
§r · A
2¨ 2 ¸ 3
© r1 ¹ A2
i 0,1 , j 0,1 :
DD
D D 0 1 2
0
222
2
1
36
2
e
6
(18)
D D 2 0 1
4
2
e4 D 02 D12 e2
0
(19)
4/2010
ВЕСТНИК
МГСУ
ɉɨɞɫɬɚɜɥɹɹ B1 , B2 ɩɨɥɭɱɚɟɦ:
B2 6 B2 4
e e B1e2 0
36
2
B1 4 1 2 B1
0
e e 36
2
B2
ɉɪɢ i 0,1,2 , j 0,1,2 ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ:
B1
DD
D D 0 1 2
0
2
1
2
36
e
D D D D 0 1
2
0
2
1
5!5!
6
D D 2 0 1
4
4
10
e
2
e4 D 02 D12 e2 D D D D 0 1
2
0
(20)
2
1
4!4!
4
e8
0
(21)
ɉɨɞɫɬɚɜɥɹɹ B1 , B2 ɩɨɥɭɱɚɟɦ:
B22 10
B22 8
B2 6 B2 4
2
B1 e e B1e B1
e B1
e
36
2
5!5!
4!4!
ɝɞɟ e
0
(22)
l1 / h1 .
ɂɫɩɨɥɶɡɭɹ ɡɧɚɱɟɧɢɹ B1 , B2 ɢɡ (5) ɩɨɥɭɱɢɦ ɱɚɫɬɨɬɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɢɞɚ:
d1[ 10 d 2[ 8 d3[ 6 d 4[ 4 d5[ 2 d6
0
(23)
ɝɞɟ
d1
A12 A2 e8 r110
k1 ;
A33
2 A1e8 r18 § A2
A22 r22 ·
2
d2 k1 ¨ A1r2 ¸;
A32
A3 ¹
© A3
§
··
e 6 r16 § 1
1
2
4
2
2 2
3 4
d3
A
A
e
k
A
A
r
A
r
A
r
A
r
A
6
4
2
¨
1 2
1¨
1 2 2
1 2
2 2
2 2
2 ¸¸
¸;
A32 ¨© 36
A3
©
¹¹
e 4 r 4 § 1 A2 2
1
1 A1
d4 1 ¨
e 1 r22 A1e 2 ;
A3 © 36 A3
18
2 A3
§
··
1
e 2 r22 k1 ¨ 4 A1r24 4 A22 r24 6 A2 r22 2 ¸ ¸¸ ;
A3
©
¹¹
223
ВЕСТНИК
4/2010
МГСУ
d5
d6
k1
e 2 r12 § 6 6
1 4 2 1
1 2
4 4
2 ·
¨ e r2 k1 4 5 A2 A2 e r2 A2e r2 e 1 A2 r2 ¸ ;
A3 ©
18
12
2
¹
1
1
e6 r26 e 4 r24 2e 2 r22 2e8 r210 k1 ;
18
2
§ e2
1 ·
¨
¸.
© 5!5! 4!4! ¹
Ⱥɧɚɥɨɝɢɱɧɨ ɦɨɠɧɨ ɜɡɹɬɶ ɩɟɪɜɵɟ ɱɟɬɵɪɟ ɢ ɛɨɥɶɲɟ ɫɥɚɝɚɟɦɵɯ ɜ ɜɵɪɚɠɟɧɢɢ (12) ɢ
ɩɨɥɭɱɢɬɶ ɛɨɥɟɟ ɬɨɱɧɨɟ ɱɚɫɬɨɬɧɨɟ ɭɪɚɜɧɟɧɢɟ ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɟɦɭ ɱɚɫɬɨɬɵ.
Ʌɢɬɟɪɚɬɭɪɚ
1. Ɏɢɥɢɩɩɨɜ ɂ.Ƚ., Ɏɢɥɢɩɩɨɜ ɋ.ɂ. Ʉɨɥɟɛɚɬɟɥɶɧɵɟ ɢ ɜɨɥɧɨɜɵɟ ɩɪɨɰɟɫɫɵ ɜ ɫɩɥɨɲɧɵɯ
ɫɠɢɦɚɟɦɵɯ ɫɪɟɞɚɯ. Ɇ. 2007.
2. ȿɝɨɪɵɱɟɜ Ɉ.Ɉ. Ʉɨɥɟɛɚɧɢɹ ɩɥɨɫɤɢɯ ɷɥɟɦɟɧɬɨɜ ɤɨɧɫɬɪɭɤɰɢɣ. Ɇ.: Ⱥɋȼ, 2005.
3. Isaev S.A., Leontev A.I., Frolov DP. «Identification of self-organizing by the numerical
simulation of laminar three-dimensional flow around a crater on a plane by a flow of viscous
incompressible fluid.» Technical physics letters. V.24, Issue 3, pp 209-211, Mart 1998.
4. ɂɫɚɟɜ C.A., ɋɭɞɚɤɨɜ Ⱥ.Ƚ., Ȼɚɪɚɧɨɜ ɉ.Ⱥ., ɍɫɚɱɨɜ Ⱥ.ȿ., ɋɬɪɢɠɚɤ ɋ.ȼ., Ʌɨɯɚɧɫɤɢɣ ə.Ʉ.,
Ƚɭɜɟɪɧɸɤ ɋ.ȼ. «Ɋɚɡɪɚɛɨɬɤɚ, ɜɟɪɢɮɢɤɚɰɢɹ ɢ ɩɪɢɦɟɧɟɧɢɟ ɨɫɧɨɜɚɧɧɨɝɨ ɧɚ ɦɧɨɝɨɛɥɨɱɧɵɯ ɜɵɱɢɫɥɢɬɟɥɶɧɵɯ ɬɟɯɧɨɥɨɝɢɹɯ ɪɚɫɩɚɪɚɥɥɟɥɟɧɧɨɝɨ ɩɚɤɟɬɚ ɨɬɤɪɵɬɨɝɨ ɬɢɩɚ VP2/3 ɞɥɹ ɪɟɲɟɧɢɹ ɮɭɧɞɚɦɟɧɬɚɥɶɧɵɯ, ɩɪɢɤɥɚɞɧɵɯ ɢ ɷɤɫɩɥɭɚɬɚɰɢɨɧɧɵɯ ɡɚɞɚɱ ɚɷɪɨɦɟɯɚɧɢɤɢ ɢ ɬɟɩɥɨɮɢɡɢɤɢ», ȼɟɫɬɧɢɤ
ɘɠɧɨ-ɍɪɚɥɶɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ. ɋɟɪɢɹ: Ɇɚɬɟɦɚɬɢɱɟɫɤɨɟ ɦɨɞɟɥɢɪɨɜɚɧɢɟ ɢ
ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɟ. 2009. Ɍ. 150. ʋ 17. ɋ. 59-72.
Literature
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M, 2007.
2. Egorychev O.O. Kolebania ploskih elementov konstruktsii. M.: ASV, 2005.
3. Isaev S.A., Leontev A.I., Frolov DP. «Identification of self-organizing by the numerical simulation of laminar three-dimensional flow around a crater on a plane by a flow of viscous incompressible
fluid.» Technical physics letters. V.24, Issue 3, pp 209-211, Mart 1998.
4. Isaev S.A., Sudakov A.G., Baranov P.A., Usachev A.E., Strizhak S.V., Lohanskii Ya.K., Guvernyuk S.V. «Development, Verification and application-based multiblock computational technologies parallelized package open-VP2 / 3 for basic, applied and operational objectives of Aeromechanics
and Thermophysics», Journal of South-Ural State University. Series: Mathematical Modeling and Programming. 2009. V. 150. ʋ 17. pp. 59-72.
Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɭɩɪɭɝɚɹ ɬɪɟɯɫɥɨɣɧɚɹ ɩɥɚɫɬɢɧɚ, ɤɨɥɟɛɚɧɢɟ, ɤɨɷɮɮɢɰɢɟɧɬ ɉɭɚɫɫɨɧɚ,
ɲɚɪɧɢɪɧɨɟ ɡɚɤɪɟɩɥɟɧɢɟ, ɠɟɫɬɤɨɟ ɡɚɤɪɟɩɥɟɧɢɟ, ɩɨɩɟɪɟɱɧɚɹ ɜɨɥɧɚ, ɩɥɨɬɧɨɫɬɶ
Key words: elastic three-layer plate, vibration, Poisson's ratio, pinned, stiffening, transverse
wave, density
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