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# Вывод частотного уравнения собственных колебаний упругой трехслойной пластины два противоположных края которой шарнирно закреплены а два других свободны от закрепления.

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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 ɋȼɈȻɈȾɇɕ ɈɌ ɁȺɄɊȿɉɅȿɇɂə
DERIVATION OF THE FREQUENCY EQUATION OF
NATURAL VIBRATIONS OF AN ELASTIC THREE-LAYERED
PLATE, TWO OPPOSITE EDGES OF WHICH ARE HINGED AND
TWO OTHERS ARE FREE FROM ATTACHMENT
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 ɫɜɨɛɨɞɧɵ ɨɬ ɡɚɤɪɟɩɥɟɧɢɹ.
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
others are free from attachment.
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
225
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
D § 1 · 1 D2 º ½°
2 § 2 · ª 2 D2 1 º 4 ª
¨1 ¸ « U
1» «1 U 4 22 ¨1 ¸ 2
»¾;
2 D2
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 ɫɜɨɛɨɞɧɵ ɨɬ ɡɚɤɪɟɩɥɟɧɢɹ.
Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɞɥɹ ɬɚɤɨɣ ɩɥɚɫɬɢɧɤɢ ɢɦɟɸɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ:
w 3W
wx3
0
x 0,l1
·
w 2W 3 2Q 1 § 2 w 2W
2 2
W
K
K
[
2
¨
¸
1
3
wx 2 7 4Q 1 ©
wy 2
¹
W
y 0, l2
ɝɞɟ K1
w 2W
wy 2
l1
, K3
l2
x 0,l1
0
y 0,l2
l1
S h1
Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (1) ɛɭɞɟɦ ɢɫɤɚɬɶ ɜ ɜɢɞɟ [2]:
226
½
°
°
°
°
0¾
°
°
°
°¿
(2)
4/2010
ВЕСТНИК
МГСУ
W x, y, t W x, y exp i[J t [
ɝɞɟ
(3)
- ɛɟɡɪɚɡɦɟɪɧɚɹ ɱɚɫɬɨɬɚ ɫɨɛɫɬɜɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɩɥɚɫɬɢɧɤɢ, J
Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ (11) ɞɥɹ
2
ª¬ ' B1' B2 º¼ W
W
0
b1
h1
ɩɪɢɦɟɬ ɜɢɞ:
(4)
ɝɞɟ
B1
A2 2 2
J [
A3
B2
1 2 2
J [ A1J 2[ 2 1
A3
(5)
ȼɜɟɞɟɦ ɧɨɜɵɟ ɛɟɡɪɚɡɦɟɪɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɢ ɮɭɧɤɰɢɸ ɩɪɨɝɢɛɚ
l1
x
S
l2
D; y
S
E W
l14
S4
V D , E (6)
ȼ ɧɨɜɵɯ ɤɨɨɪɞɢɧɚɬɚɯ ɭɪɚɜɧɟɧɢɟ (4) ɡɚɩɢɲɟɬɫɹ ɜ ɜɢɞɟ:
4
2
ª w4
·
w4
l12 § w 2
2
4 w
2 w
K
K
B
« 4 2K1
¨
¸
1
1
1
wD 2wE 2
wE 4
wE 2 ¹
¬ wD
B2
l14 º
V D , E 0
S 4 »¼
K1
ɝɞɟ
(7)
l1
l2
Ⱦɥɹ ɜɵɜɨɞɚ ɱɚɫɬɨɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɭɪɚɜɧɟɧɢɟɦ (7) ɢ ɝɪɚɧɢɱɧɵɦɢ
ɭɫɥɨɜɢɹɦɢ (2), ɡɚɩɢɫɚɧɵɦɢ ɜ ɛɟɡɪɚɡɦɟɪɧɵɯ ɤɨɨɪɞɢɧɚɬɚɯ (6).
ɉɨɥɶɡɭɹɫɶ ɦɟɬɨɞɨɦ ɞɟɤɨɦɩɨɡɢɰɢɣ ɜɫɩɨɦɨɝɚɬɟɥɶɧɵɟ ɡɚɞɚɱɢ ɡɚɩɢɲɭɬɫɹ ɜ ɜɢɞɟ:
1.
w 4V1
w 3V1
f1 D , E ;
0
wD 4
wD 3
·
w 2V1 3 2v1 § 2 w 2V1
K32[ 2V1 ¸ 0 ɩɪɢ D
¨ 2K1
2
2
wD
wE
¹
4
2. K1
w 4V2
wE 4
f 2 D , E ; V2
w 2V2
wE 2
0 ɩɪɢ E
(8)
0, S
0, S
(9)
227
ВЕСТНИК
МГСУ
4/2010
2
ª 2 w4
·
l12 § w 2
l14 º
2 w
B1 2 ¨ 2 K1
« 2K1
¸ B2 4 » V3 S © wD
S ¼
wD 2wE 2
wE 2 ¹
3. ¬
f1 D , E f 2 D , E 0
(10)
ɋɥɟɞɭɹ ɦɟɬɨɞɭ ɞɟɤɨɦɩɨɡɢɰɢɣ, ɛɭɞɟɦ ɩɪɢɛɥɢɠɟɧɧɨ ɩɨɥɚɝɚɬɶ
V1 # V2 ; V3 #
1
V1 V2 2
(11)
ɜ ɡɚɞɚɧɧɵɯ ɬɨɱɤɚɯ ɩɥɚɫɬɢɧɤɢ.
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɪɨɢɡɜɨɥɶɧɵɯ ɮɭɧɤɰɢɣ
ɜɢɹɦɢ (8):
w 3V1
wD 3
Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ
\1 E \1 E ɩɪɢ
D
0, S
ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨ-
ɩɪɢɜɨɞɹɬ ɤ ɫɨɨɬɧɨɲɟɧɢɹɦ:
an ,m
¦ sin mE ;
n ,m 1 n
D 0
f
1
f
1
an ,m
n
1 sin mE 1 n
D
¦
n ,m
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
ɬ.ɟ. ɧɟɱɟɬɧɵɟ ɩɨ
0
\i
\1 E n an ,m
1
(12)
S
- ɟɫɬɶ ɱɟɬɧɚɹ ɮɭɧɤɰɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ
n n 2q ,
ɧɟɨɛɯɨɞɢɦɨ ɩɨɥɨɠɢɬɶ ɪɚɜɧɵɦɢ ɧɭɥɸ.
Ɍɨɝɞɚ
\1 E f
a2 q,m
q m 1
2q
¦
1
sin mE (13)
ɂɫɩɨɥɶɡɭɟɦ ɜɬɨɪɵɟ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ
·
w 2V1 3 2v1 § 2 w 2V1
K32[ 2V1 ¸ 0
¨ 2K1
2
2
wD
wE
¹
ɩɪɢ D =0, ɩɨɥɭɱɢɦ:
2
º
3 2v1 ª 2 w \ 4 E 2 2
0
\ 2 E K
[
\
E
« 2K1
»
3
4
7 4v1 ¬
wE 2
¼
ɚ ɩɪɢ D S , ɩɨɥɭɱɢɦ:
228
(14)
4/2010
ВЕСТНИК
МГСУ
º
3 2v1 ª 2 w 2\ 4
2 2
2
\ 2 E K
K
[
\
1
3
4» wE 2
7 4v1 «¬
¼
º
3 2v1 ª 2 w 2\ 3
K32[ 2\ 3 » S « 2K1
2
wE
7 4v1 ¬
¼
(15)
º
3 2v1 S 2 ª 2 w 2\ 2
K32[ 2\ 2 » 2K1
«
2
wE
7 4v1 2 ¬
¼
a2 q,m ª
3 2v1 S 3 2 2
2 2 º
¦
2
0
S
K
[
K
3
1 m »
«
7 4v1 6
q ,m 1 2 q ¬
¼
ɂɦɟɟɦ ɞɜɚ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɬɪɟɯ ɧɟɢɡɜɟɫɬɧɵɯ \ 2 ,\ 3 ,\ 4 . Ȼɟɡ ɨɝɪɚɧɢɱɟɧɢɹ ɨɛɳɧɨɫɬɢ ɨɞɧɭ ɢɡ ɮɭɧɤɰɢɣ ɦɨɠɧɨ ɩɨɥɨɠɢɬɶ ɧɭɥɸ, ɩɭɫɬɶ \ 3 E 0. Ɋɟɲɚɹ ɫɨɜɦɟɫɬ1
f
ɧɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ (14) ɢ (15), ɩɨɥɭɱɢɦ:
1
1
2
f
a2 q , m § K 3 2
2·
¨ 2 [ m ¸ sin mE q ,m 1 2q © 2K1
¹
2
1
·
1 7 4v1 f a2 q ,m § K32 2
D ¨ 2 [ m 2 ¸ sin mE ¦
2
2K1 3 2v1 q ,m 1 2q © 2K1
¹
\ 2 E ¦
(16)
\ 4 E (17)
ɝɞɟ
1 ª 7 4v1 S 2 2 2
2 2 º
m »
K
[
2
K
3
1
«
SK12 ¬ 3 2v1 6
¼
Ɍɨɝɞɚ ɮɭɧɤɰɢɹ V1 D , E ɭɞɨɜɥɟɬɜɨɪɹɸɳɚɹ ɪɟɲɟɧɢɸ
D
ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɡɚɞɚ-
ɱɢ (8) ɩɪɢɦɟɬ ɜɢɞ:
V1 D , E f
¦
q ,m 1
1
a2 q,m
1
·
­D 3 D 2 § K32 2
sin mE ® D ¨ 2 [ m2 ¸ 2q
¯6
¹
1
1 7 4v1 § K
1
°½
2
2·
2
[ m ¸ sin 2qD ¾
D¨
3
2q ¹
°¿
2
3
2
1
(18)
ɝɞɟ
D
1 ª 7 4v1 S 2 2 2
2 2 º
K
[
2
K
1 m »
SK12 «¬ 3 2v1 6 3
¼
229
ВЕСТНИК
4/2010
МГСУ
ɂɡ ɭɫɥɨɜɢɹ
V2
w 2V2
wE 2
0 ɩɪɢ E
0
ɉɨɥɭɱɚɟɦ
\ 2 0;
(19)
\4 0
ɂɡ ɭɫɥɨɜɢɹ
V2
w 2V2
wE 2
0 ɩɪɢ E
S
ɉɨɥɭɱɚɟɦ
\ 1 0;
(20)
\3 0
Ɍɨɝɞɚ ɮɭɧɤɰɢɹ V2 D , E ɭɞɨɜɥɟɬɜɨɪɹɸɳɚɹ ɪɟɲɟɧɢɸ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɡɚɞɚɱɢ
(9) ɩɪɢɦɟɬ ɜɢɞ:
an ,m
(21)
V2 D , E ¦ 4 4 sin nD sin mE n , m 1 K1 m
Ɂɚɬɟɦ ɢɫɩɨɥɶɡɭɟɦ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɣ V1 D , E ɢ V2 D , E ɩɪɢ ɪɟɲɟɧɢɢ ɞɢɮɮɟf
2
ɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ (10) ɢ ɭɞɨɜɥɟɬɜɨɪɹɹ ɩɪɢɛɥɢɠɟɧɧɵɦ ɭɫɥɨɜɢɹɦ (11), ɩɨɥɭɱɢɦ
ɫɢɫɬɟɦɭ ɞɜɭɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ. ɇɟɬɪɢɜɢɚɥɶɧɨɟ ɪɟɲɟɧɢɟ ɷɬɨɣ ɫɢɫɬɟɦɵ ɩɪɢɜɨɞɢɬ
ɤ ɱɚɫɬɨɬɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɜɨɫɶɦɨɝɨ ɩɨɪɹɞɤɚ, , ɩɪɢ n
d1[ 8 d 2[ 6 d3[ 4 d 4[ 2 d5
m 1, D
E
S
2
0
ɝɞɟ
K 24K34 S 3
;
4K14 16
d1
A1
d2
A1
K22
k2 · 1 K 22K34 ª S 3
K 24K32 S § S 2
5S º
k
1 2K 22 1
¨
¸ A2
2
4 «
3¹ 2
4K1 ¬ 48
6 ¼»
l12 K34 S 3
;
S 2 4K14 48
§S3 S2
· 1 K 2K 4 ª 1
A1K24 ¨ k1 k1k2 ¸ A2 2 4 3 « S k1 K1 ¬ 2
¹ 2
§S2
l2 K2 § S 2
S ·
S · Sº
1 2K 22 ¨ k1 k2 ¸ » K 22 1 2 32 ¨ k1 k2 ¸ 6 ¹ 6¼
6 ¹
d3
A3
230
K34 ª 2 § S 3 S · S 3 º
«K1 ¨ ¸ » ;
4K14 ¬ © 48 6 ¹ 48 ¼
.
(22)
4/2010
d4
ª
§S3 S2
·
1
Sº
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Ɋɟɡɭɥɶɬɚɬɵ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (22) ɞɚɸɬ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɱɚɫɬɨɬ ɫɨɛɫɬɜɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɬɪɟɯɫɥɨɣɧɨɣ ɩɥɚɫɬɢɧɵ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɭɤɚɡɚɧɧɵɯ ɛɟɡɪɚɡɦɟɪɧɵɯ ɩɚɪɚɦɟɬɪɨɜ.
Ʌɢɬɟɪɚɬɭɪɚ
1.
2. Ɏɢɥɢɩɩɨɜ ɂ.Ƚ., Ɏɢɥɢɩɩɨɜ ɋ.ɂ. Ʉɨɥɟɛɚɬɟɥɶɧɵɟ ɢ ɜɨɥɧɨɜɵɟ ɩɪɨɰɟɫɫɵ ɜ ɫɩɥɨɲɧɵɯ ɫɠɢɦɚɟɦɵɯ
ɫɪɟɞɚɯ. Ɇ, 2007.
3. ȿɝɨɪɵɱɟɜ Ɉ.Ɉ. Ʉɨɥɟɛɚɧɢɹ ɩɥɨɫɤɢɯ ɷɥɟɦɟɧɬɨɜ ɤɨɧɫɬɪɭɤɰɢɣ. Ɇ.: Ⱥɋȼ, 2005.
4. 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.
5. 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
1. Filippov I.G., Filippov S.I. Kolebatelnie i volnovie protsessi v sploshnih szhimaemih sredah. 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
ɉɨɱɬɨɜɵɣ ɚɞɪɟɫ: 129337 ɝ. Ɇɨɫɤɜɚ əɪɨɫɥɚɜɫɤɨɟ ɲɨɫɫɟ ɞɨɦ 26
e-mail: misi@mgsu.ru, Ʉɨɧɬɚɤɬɧɵɟ ɞɚɧɧɵɟ: (495) 739-33-63, e-mail: misi@mgsu.ru
Ɋɟɰɟɧɡɟɧɬ: Ʉɢɫɟɥɟɜ Ⱥ.Ȼ., ɞ.ɮ.-ɦ.ɧ., ɩɪɨɮɟɫɫɨɪ ɆȽɍ ɢɦ. Ɇ.ȼ. Ʌɨɦɨɧɨɫɨɜɚ
231
```
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