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Вывод частотного уравнения собственных поперечных колебаний предварительно напряженной пластины упруго закрепленной по одному краю и жестко закрепленной по другому.

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ВЕСТНИК
МГСУ
4/2010
ȼɕȼɈȾ ɑȺɋɌɈɌɇɈȽɈ ɍɊȺȼɇȿɇɂə ɋɈȻɋɌȼȿɇɇɕɏ
ɉɈɉȿɊȿɑɇɕɏ ɄɈɅȿȻȺɇɂɃ ɉɊȿȾȼȺɊɂɌȿɅɖɇɈ
ɇȺɉɊəɀȿɇɇɈɃ ɉɅȺɋɌɂɇɕ ɍɉɊɍȽɈ ɁȺɄɊȿɉɅȿɇɇɈɃ ɉɈ
ɈȾɇɈɆɍ ɄɊȺɘ ɂ ɀȿɋɌɄɈ ɁȺɄɊȿɉɅȿɇɇɈɃ ɉɈ ȾɊɍȽɈɆɍ
DERIVATION OF THE FREQUENCY EQUATION OF THE
NATURAL TRANSVERSE VIBRATIONS OF A PRESTRESSED
ELASTIC PLATE FIXED AT ONE END AND RIGIDLY FIXED ON
THE OTHER
Ɉ.Ⱥ. ȿɝɨɪɵɱɟɜ, Ɉ.Ɉ. ȿɝɨɪɵɱɟɜ, ȼ.ȼ. Ȼɪɟɧɞɷ
O.A. Egorychev, O.Ɉ. Egorychev, V.V. Brende
ȽɈɍ ȼɉɈ ɆȽɋɍ
ɇɟɢɡɜɟɫɬɧɚɹ ɜɟɥɢɱɢɧɚ ɩɪɨɝɢɛɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫ ɩɨɦɨɳɶɸ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨɣ
ɮɭɧɤɰɢɢ, ɜ ɚɪɝɭɦɟɧɬ ɤɨɬɨɪɨɣ ɜɯɨɞɹɬ ɝɟɨɦɟɬɪɢɱɟɫɤɢɟ ɢ ɮɢɡɢɱɟɫɤɢɟ ɩɚɪɚɦɟɬɪɵ, ɚ
ɧɟɢɡɜɟɫɬɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɡ ɛɢɤɜɚɞɪɚɬɧɨɝɨ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ
ɭɪɚɜɧɟɧɢɹ.
In solving the problem unknown amount of deflection is determined by the exponential
function, which argument consists of geometric and physical parameters, and the unknown
factor is determined by solving the biquadratic characteristic equation.
Ɋɚɫɫɦɨɬɪɢɦ ɨɪɬɨɬɪɨɩɧɭɸ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɧɚɩɪɹɠɟɧɧɭɸ ɩɥɚɫɬɢɧɭ-ɩɨɥɨɫɭ
ɫɟɪɟɞɢɧɧɚɹ ɩɥɨɫɤɨɫɬɶ, ɤɨɬɨɪɨɣ ɜ ɧɟɞɟɮɨɪɦɢɪɭɟɦɨɦ
ɫɨɫɬɨɹɧɢɢ ɫɨɜɩɚɞɚɟɬ ɫ
ɤɨɨɪɞɢɧɚɬɧɨɣ ɩɥɨɫɤɨɫɬɶɸ XOY, ɨɫɶ Z ɧɚɩɪɚɜɥɟɧɚ ɜɟɪɬɢɤɚɥɶɧɨ ɜɜɟɪɯ.
ɉɥɚɫɬɢɧɚ-ɩɨɥɨɫɚ ɡɚɧɢɦɚɟɬ ɫɥɟɞɭɸɳɭɸ ɨɛɥɚɫɬɶ:
^l d x d l; f d y d f; h d z d h` .
ɉɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɛɭɞɟɦ ɩɨɥɶɡɨɜɚɬɶɫɹ ɩɪɢɛɥɢɠɟɧɧɵɦ ɭɪɚɜɧɟɧɢɟɦ ɤɨɥɟɛɚɧɢɹ
ɩɥɚɫɬɢɧɤɢ-ɩɨɥɨɫɵ ɜ ɜɢɞɟ [3]:
w 2W
w 4W
w 4W
w 4W
A
A
A
0,
(1)
1
2
3
wt 2
wt 4
wx 2 wt 2
wx 4
ɝɞɟ W - ɮɭɧɤɰɢɹ ɩɪɨɝɢɛɚ.
h2 U ª
1
1
A1
1 c0 A331 3 1 a0 A551 º¼ ;
6 ¬
h2 ª
1
1
2 1 a0 1 c0 2 A13 A331 3 A33 A55 A132 A11 A33 º ;
¬
¼
6
2
h
2 1 c0 A331 A11 A13 A132 ,
A3
6U
ɝɞɟ U - ɩɥɨɬɧɨɫɬɶ, a0 , c0 - ɩɚɪɚɦɟɬɪɵ ɨɞɧɨɪɨɞɧɨɝɨ ɧɚɩɪɹɠɟɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ, Ai , j A2
ɭɩɪɭɝɢɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɨɩɪɟɞɟɥɹɸɳɢɟ ɨɪɬɨɬɪɨɩɢɸ.
Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (1) ɛɭɞɟɦ ɢɫɤɚɬɶ ɜ ɜɢɞɟ:
246
4/2010
ВЕСТНИК
МГСУ
§ b ·
(2)
W x, t W0 x exp ¨ i[ t ¸ ,
© h ¹
ɝɞɟ b - ɫɤɨɪɨɫɬɶ ɩɨɩɟɪɟɱɧɨɣ ɜɨɥɧɵ, [ - ɱɚɫɬɨɬɚ.
Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ (1) ɩɪɟɨɛɪɚɡɭɟɬɫɹ ɜ ɨɛɵɤɧɨɜɟɧɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ:
d 4W0
d 2W0
B2W0 0,
(3)
B
1
dx 4
dx 2
ɝɞɟ
2
2
2
º
A2 § b ·
1 § b· ª § b·
B
A
B1
[
;
[
[
(4)
2
¨
¸ « 1¨
¸ 1» .
¨
¸
A3 © h ¹ ¬« © h ¹
A3 © h ¹
¼»
Ⱦɥɹ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (3) ɡɚɩɢɲɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ
r 4 B1r 2 B2 0
(5)
ȿɝɨ ɪɟɲɟɧɢɟ ɢɦɟɟɬ ɜɢɞ:
1
2
ª B
º 2
§ B ·
r1,2,3,4 r « 1 r ¨ 1 ¸ B2 » .
« 2
»
© 2¹
¬
¼
ɂɫɩɨɥɶɡɭɹ ɫɨɨɬɧɨɲɟɧɢɟ (4), ɪɚɜɟɧɫɬɜɨ (6) ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ:
(6)
1
2 2
ª
b º
« A2 r A22 4 A1 A3 4 A3 §¨ [ ·¸ » .
(7)
r1,2,3,4
«
© h ¹ »¼
¬
Ɉɬɦɟɬɢɦ ɯɚɪɚɤɬɟɪɧɨɟ ɡɧɚɱɟɧɢɟ [ 0 , ɩɪɢ ɤɨɬɨɪɨɦ ɜɵɪɚɠɟɧɢɟ (7), ɫɬɨɹɳɟɟ ɩɨɞ
ɤɨɪɧɟɦ, ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɨɥɶ
§ b· 1
r ¨[ ¸
© h ¹ 2 A3
A3
A3
§h·
r2 ¨ ¸
;
! 0.
2
2
b
4
A
A
A
4
A
A
© ¹
1 3 A2
1 3
2
ɂɡ ɚɧɚɥɢɡɚ ɤɨɪɧɟɣ (7), ɜɨɡɦɨɠɧɵ ɞɜɚ ɜɚɪɢɚɧɬɚ.
ȼɚɪɢɚɧɬ I. [ [ 0 , ɬɨɝɞɚ h1 h.
[0
(8)
1
2
ª
ª § h ·2 º º
§ b· 1 «
2
»
r1,2,3,4 r ¨ [ ¸
A2 r A2 4 A1 A3 «1 ¨ ¸ » .
»
h
© h1 ¹ 2 A3 ««
¬« © 1 ¹ ¼» »¼
¬
ª § h ·2 º
Ɂɞɟɫɶ «1 ¨ ¸ » 0 , ɫɥɟɞɨɜɚɬɟɥɶɧɨ
«¬ © h1 ¹ »¼
ª § h ·2 º
2
0 A2 4 A1 A3 «1 ¨ ¸ » A2 .
«¬ © h1 ¹ »¼
Ɍɨɝɞɚ
r1,2 ri E1 ; r3,4 ri E 2 ,
(9)
(10)
ɝɞɟ
E1
§ b · D2
;
¨[ ¸
© h1 ¹ 2 A3
E2
§ b · D3
¨[ ¸
© h1 ¹ 2 A3
247
ВЕСТНИК
МГСУ
4/2010
ª § h ·2 º
2
A
4
A
A
2 1 3 «1 ¨ h ¸ » ; D2 A2 D1 ; D3 A2 D1 .
«¬ © 1 ¹ »¼
Ⱦɥɹ ɞɚɧɧɨɝɨ ɜɚɪɢɚɧɬɚ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (3) ɢɦɟɟɬ ɜɢɞ:
W01 C1 cos E1 x C2 sin E1 x C3 cos E 2 x C4 sin E 2 x
D1
(11)
ȼɚɪɢɚɧɬ II. [ ! [ 0 , ɬɨɝɞɚ h2 ! h.
1
r1,2,3,4
ª
§ b· 1 «
r ¨[ ¸
A2 r
© h2 ¹ 2 A3 ««
¬
2
ª § h ·2 º º
2
»
A2 4 A1 A3 «1 ¨ h ¸ » » .
«¬ © 2 ¹ »¼
»¼
Ɂɞɟɫɶ
ª § h ·2 º
ª § h ·2 º
«1 ¨ ¸ » ! 0, A22 4 A1 A3 «1 ¨ ¸ » 0,
«¬ © h2 ¹ »¼
«¬ © h2 ¹ »¼
ɫɥɟɞɨɜɚɬɟɥɶɧɨ
1
§ b· 1
r1,2,3,4 r ¨ [ ¸
> A2 r iD1 @2 .
© h2 ¹ 2 A3
(12)
(13)
Ɉɛɨɡɧɚɱɢɦ
2
§ b · A2
§ b· D
;
E3 ¨ [ ¸ 1 .
¨[ ¸
© h2 ¹ 2 A3
© h2 ¹ 2 A3
Ɍɨɝɞɚ ɪɚɜɟɧɫɬɜɨ (13) ɩɪɢɦɟɬ ɜɢɞ:
1
1
M 2kS º
ª M 2kS
r i sin
,
r1,2,3,4 r D 3 r i E 3 2 r R 2 «cos
2
2 »¼
¬
ɝɞɟ
D3
R
D 32 E 32 ;
M
arctg
E3
.
D3
ȼɜɟɞɟɦ ɧɨɜɵɟ ɨɛɨɡɧɚɱɟɧɢɹ ɞɥɹ ɪɚɜɟɧɫɬɜɚ (14)
1
1
M 2kS
M 2kS
D 4 R 2 cos
; E 4 R 2 sin
.
2
2
Ɉɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɢɦ ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ (5) ɜ ɜɢɞɟ:
r1,2,3,4 r D 4 r i E 4 W02
(14)
Ⱦɥɹ ɞɚɧɧɨɝɨ ɜɚɪɢɚɧɬɚ II ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (3) ɩɪɟɞɫɬɚɜɢɦ ɜ ɜɢɞɟ:
eD 4 x C1 cos E 4 x C2 sin E 4 x e D 4 x C3 cos E 4 x C4 sin E 4 x (15)
(16)
Ɂɚɞɚɱɚ ɪɟɲɚɟɬɫɹ ɩɪɢ ɫɥɟɞɭɸɳɢɯ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɹɯ [2].
ɉɥɚɫɬɢɧɚ ɧɚɯɨɞɢɬɫɹ ɜ ɭɩɪɭɝɨɦ ɤɨɧɬɚɤɬɟ ɫ ɜɟɪɬɢɤɚɥɶɧɨɣ ɞɟɮɨɪɦɢɪɭɟɦɨɣ ɩɥɚɫɬɢɧɨɣ.
ɉɚɪɚɦɟɬɪɵ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɥɚɫɬɢɧɵ ɨɛɨɡɧɚɱɢɦ ɢɧɞɟɤɫɨɦ (1), ɚ ɜɟɪɬɢɤɚɥɶɧɨɣ (2).
ɩɪɢ x l
d 2W0
dW0
K3
K 2W0 0;
2
dx
dx
d 3W0
d 2W0
dW0
K
K4
K 6W0
5
3
2
dx
dx
dx
248
½
°°
¾
0.°
°¿
(17)
ВЕСТНИК
4/2010
МГСУ
Ɂɞɟɫɶ
K2
§[ b · ;
2 A33 A13 3
U 2 h2 A551
1
1
2
7 A33 4 A13 2
1
1
3 A33 2 A13 § b ·
[ ¸ ;
U1 A55
1
1 ¨
7 A33 4 A13 © h1 ¹
1
K3
1
1
2
¨
¸
© h1 ¹
2
2
2
1
§ b·
2h1 A11 2 § A55 ·
2h1 U1 A11 2 1
1
b·
1 §
K 4 U 2 A551 ¨ [ ¸ ; K 5
;
K
A33 A13 ¨ [ ¸ .
¨
¸
6
2
2
1 ¨
2 ¸
1
2
3h2 A11 © A55 ¹
3h2 A33
© h1 ¹
© h1 ¹
ɩɪɢ x l
dW0
W0
0.
(18)
dx
Ɋɚɫɫɦɨɬɪɢɦ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ, ɢɫɩɨɥɶɡɭɹ ɫɨɨɬɧɨɲɟɧɢɟ (11), ɬ.ɟ. ɜɚɪɢɚɧɬ I.
ɉɨɞɫɬɚɜɢɦ ɜɵɪɚɠɟɧɢɟ (11) ɜ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ (17) ɢ (18), ɩɨɥɭɱɢɦ ɨɞɧɨɪɨɞɧɭɸ
ɫɢɫɬɟɦɭ, ɪɟɲɟɧɢɟ ɤɨɬɨɪɨɣ ɢ ɨɩɪɟɞɟɥɢɬ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ.
ɂɦɟɟɦ:
C1 cos E1l1 C2 sin E1l1 C3 cos E 2 l1 C4 sin E 2 l1 0
C1 E12 cos E1l1 C2 E12 sin E1l1 C3 E 22 cos E 2 l1 C4 E 22 sin E 2 l1
0
C1 K 2 cos E1l1 K 3 E1 sin E1l1 E cos E1l1 C2 K 2 sin E1l1 K 3 E1 cos E1l1 2
1
E 22 sin E1l1 C3 K 2 cos E 2 l1 K 3 E 2 sin E 2 l1 E 22 cos E 2 l1 C4 K 2 sin E 2 l1 K 3 E 2 cos E 2 l1 E 22 sin E 2 l1 (19)
0
C1 K 6 cos E1l1 K 4 E1 sin E1l1 K 5 E12 cos E1l1 E13 sin E1l1 C2 K 6 sin E1l1 K 4 E1 cos E1l1 K 5 E12 sin E1l1 E13 cos E1l1 C3 K 6 cos E 2 l1 K 4 E 2 sin E 2 l1 K 5 E 22 cos E 2 l1 E 23 sin E 2 l1 C4 K 6 sin E 2 l1 K 4 E 2 cos E 2 l1 K 5 E 22 sin E 2 l1 E 23 cos E 2 l1 0
l
.
h
ɇɟ ɬɪɢɜɢɚɥɶɧɨɟ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɨɞɧɨɪɨɞɧɵɯ ɭɪɚɜɧɟɧɢɣ (19) ɩɪɢɜɨɞɢɬ ɤ
ɬɪɚɧɫɰɟɧɞɟɧɬɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɜɢɞɚ:
E 22 E12 K 4 E1 E1l12 ª«¬ 12 sin 2E 2l1 K 2 E 22 K3 E 2l1 cos2 E 2l1 1
sin 2E 2 l1 » K 2 E12 K 4 E1 K 3 K 6 E1 K 5 E12 E 2 K 4 E1 E 22 (20)
2
ɝɞɟ l1
^
E13 E 22 º¼ K 3 E1 E 2 E 22 E12 sin 2 E 2 l1
`
0.
ɂɫɩɨɥɶɡɭɹ ɪɚɜɟɧɫɬɜɨ (16) ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ (3), (17) ɢ (18), ɬ.ɟ. ɪɟɲɚɹ ɜɚɪɢɚɧɬ
II, ɩɨɥɭɱɢɦ ɫɢɫɬɟɦɭ ɨɞɧɨɪɨɞɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜɢɞɚ:
ɋ1 cos E 4 l1 C2 sin E 4 l1 C3 e 2D 4 l1 cos E 4 l1 C4 e 2D 4 l1 sin E 4 l1 0
ɋ1 D 4 cos E 4 l1 E 4 sin E 4 l1 ɋ2 D 4 sin E 4 l1 E 4 cos E 4 l1 e 2D 4l1 ª¬C3 D 4 cos E 4 l1 E 4 sin E 4 l1 ɋ4 D 4 sin E 4 l1 E 4 cos E 4 l1 º¼
0
249
ВЕСТНИК
МГСУ
4/2010
^
e 2D 4l1 ɋ1 ª¬D 42 cos E 4 l1 2D 4 E 4 sin E 4 l1 E 42 cos E 4 l1 K 3 D 4 cos E 4 l1 E 4 sin E 4 l1 K 2 cos E 4 l1 º¼ ɋ2 ª¬D 42 sin E 4 l1 2D 4 E 4 cos E 4 l1 `
E 42 sin E 4 l1 K 3 D 4 sin E 4 l1 E 4 cos E 4 l1 K 2 sin E 4 l1 º¼ (21)
ɋ3 ª¬D 42 cos E 4 l1 2D 4 E 4 sin E 4 l1 E 42 cos E 4 l1 K 3 D 4 cos E 4 l1 E 4 sin E 4 l1 K 2 cos E 4 l1 @ ɋ4 ª¬ D 42 sin E 4 l1 2D 4 E 4 cos E 4 l1 E 42 sin E 4 l1 K 3 D 4 sin E 4 l1 E 4 cos E 4 l1 K 2 sin E 4 l1 @ 0.
^
e 2D 4l1 ɋ1 ª¬D 43 cos E 4 l1 3D 42 E 4 3D 4 E 42 cos E 4 l1 E 43 sin E 4 l1 K 5 D 42 cos E 4 l1 2D 4 E 4 sin E 4 l1 E 42 cos E 4 l1 K 4 D 4 cos E 4 l1 E 4 sin E 4 l1 K 6 cos E 4 l1 º¼ ɋ2 ª¬D 43 sin E 4 l1 3D 42 E 4 cos E 4 l1 3D 4 E 42 sin E 4 l1 E 43 cos E 4 l1 K 5 D 42 sin E 4 l1 2D 4 E 4 cos E 4 l1 E 42 sin E 4 l1 K 4 D 4 sin E 4 l1 E 4 cos E 4 l1 K 6 sin E 4 l1 @` ɋ3 ª¬ D 43 cos E 4 l1 3D 42 E 4 sin E 4 l1 3D 4 E 42 cos E 4 l1 E 43 sin E 4 l1 K 5 D 42 cos E 4 l1 2D 4 E 4 sin E 4 l1 E 42 cos E 4 l1 K 4 D 4 cos E 4 l1 E 4 sin E 4 l1 K 6 cos E 4 l1 @ ɋ4 ª¬D 43 sin E 4 l1 3D 42 E 4 cos E 4 l1 3D 4 E 42 sin E 4 l1 E 43 cos E 4 l1 K 5 D 42 sin E 4 l1 2D 4 E 4 cos E 4 l1 E 42 sin E 4 l1 K 4 D 4 sin E 4 l1 E 4 cos E 4 l1 K 6 sin E 4l1 º¼
0.
Ɋɟɡɭɥɶɬɚɬɨɦ ɪɟɲɟɧɢɹ ɨɞɧɨɪɨɞɧɨɣ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ (21) ɩɨɥɭɱɢɦ ɬɪɚɧɫɰɟɧɞɟɧɬɧɨɟ
ɭɪɚɜɧɟɧɢɟ ɜɢɞɚ:
H
e8D 4 l1 1 e 4D 4l1 1 0,
(22)
H2
ɝɞɟ
H1
^sin 2E l ª¬D D
4 1
4
2
4
3E 42 K 5 E 42 D 42 K 4D 4 K 6 º¼ cos 2E 4 l1 ª¬ E 4 E 42 `^
3D 42 K 5 2D 4 K 4 º¼ sin 2 E 4 l1 ª¬D 42 E 42 D 4 K3 K 2D 4 @ ` ^
cos 2 E 4 l1 ¬ª E 4 D 42 E 42 K 2 ¼º sin 2E 4 l1 ª¬ E 43 K 3 K 4 K 2 º¼ `^
cos 2 E 4 l1 ª¬ E 4 2D 4 K 3 º¼ sin 2 E 4 l1 ª¬D 42 E 42 E 42 D 42 K 5D 4 K 4 K 6D 4 º¼ ` ^
cos 2 E 4 l1 ª¬D 42 E 42 2D 4 E 4 K 5 E 4 K 6 E 4 º¼ sin 2E 4 l1 ª¬D 42 E 42 K 3D 4 K 4 º¼ `^
cos 2 E 4 l1 ¬ª E 4 2D 4 K 3 ¼º sin 2E 4 l1 ¬ªD E
2
4
2
4
K
4
K 5D 4 D 42 E 42 K 6D 4 ¼º ` ^
cos 2 E 4 l1 ¬ªD 42 E 42 K 5 E 4 2D 4 E 4 K 6D 4 ¼º sin 2E 4 l1 ¬ªD 4 D 42 3E 42 K 5 D 42 E 42 K 4D 4 K 6 º¼ cos 2E 4 l1 ª¬ E 4 3D 42 E 42 K 5 2D 4 E 4 ^
`
K 4 E 4 @` sin 2 E 4 l1 ¬ªD 42 E 42 D 4 K 3 K 2D 4 ¼º cos 2E 4 l1 ¬ª E 4 K 2 D 42 E 42 ¼º ;
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МГСУ
E 4 ^D 42 E 4 D 42 2E 42 E 45 K 4 E 4 D 42 E 42 K 6 2D 4 E 4 `
K 3 ª¬ E 4 D 42 E 42 2D 4 K 5 K 6 E 4 º¼ K 2 E 4 ª¬3D 42 E 42 K 5 2D 4 K 4 º¼ .
ȼ ɜɵɪɚɠɟɧɢɹɯ (20) ɢ (22) ɩɪɟɞɫɬɚɜɢɦ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟ ɢ ɩɨɤɚɡɚɬɟɥɶɧɵɟ
ɮɭɧɤɰɢɢ ɜ ɜɢɞɟ ɫɬɟɩɟɧɧɨɝɨ ɪɚɡɥɨɠɟɧɢɹ, ɩɨɥɭɱɢɦ ɱɚɫɬɨɬɧɨɟ ɜɵɪɚɠɟɧɢɟ ɜ ɜɢɞɟ
ɚɥɝɟɛɪɚɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɥɸɛɨɣ ɧɟɨɛɯɨɞɢɦɨɣ ɫɬɟɩɟɧɢ.
Ʌɢɬɟɪɚɬɭɪɚ
1. ȿɝɨɪɵɱɟɜ Ɉ.Ⱥ., ȿɝɨɪɵɱɟɜ Ɉ.Ɉ. ȼɥɢɹɧɢɟ ɮɨɪɦɭɥɢɪɨɜɤɢ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɩɪɢ
ɨɩɪɟɞɟɥɟɧɢɢ ɫɨɛɫɬɜɟɧɧɵɯ ɱɚɫɬɨɬ ɤɨɥɟɛɚɧɢɣ ɩɥɚɫɬɢɧ. ɉȽɋ ʋ9. 2004.
2. ȿɝɨɪɵɱɟɜ Ɉ.Ɉ. Ʉɨɥɟɛɚɧɢɹ ɩɥɨɫɤɢɯ ɷɥɟɦɟɧɬɨɜ ɤɨɧɫɬɪɭɤɰɢɣ. ɂɡɞ-ɜɨ Ⱥɋȼ. Ɇɨɫɤɜɚ. 2005.
239 ɫ.
3. Ɏɢɥɢɩɩɨɜ ɂ.Ƚ., ɑɟɛɚɧ ȼ.Ƚ. Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɬɟɨɪɢɹ ɤɨɥɟɛɚɧɢɣ ɭɩɪɭɝɢɯ ɢ ɜɹɡɤɨɭɩɪɭɝɢɯ ɩɥɚɫɬɢɧ ɢ ɫɬɟɪɠɧɟɣ. Ʉɢɲɢɧɟɜ. ɒɌɂɂɇɐȺ. 1988. 190 ɫ.
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.
References
1. Egorychev O.A., Egorychev O.O. Influence of formulation of boundary conditions in
determining the natural frequencies of plates. PGS ʋ 9. 2004.
2. Egorychev O.O. Fluctuations of plane structural elements. Publishing house ASV. Moscow.
2005. 239 pp.
3. Filippov I.G., Cheban V.G. The mathematical theory of vibrations of elastic and viscoelastic
plates and rods. Chisinau. Shtiintsa. 1988. 190 pp.
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. 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: ɫɨɛɫɬɜɟɧɧɵɟ ɩɨɩɟɪɟɱɧɵɟ ɤɨɥɟɛɚɧɢɹ, ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɧɚɩɪɹɠɟɧɧɨɟ
ɫɨɫɬɨɹɧɢɟ, ɭɩɪɭɝɨɟ ɡɚɤɪɟɩɥɟɧɢɟ
Key words: natural transverse vibrations, prestressing, elastic fixing
ɉɨɱɬɨɜɵɣ ɚɞɪɟɫ: 129337 ɝ. Ɇɨɫɤɜɚ əɪɨɫɥɚɜɫɤɨɟ ɲɨɫɫɟ ɞɨɦ 26
Ʉɨɧɬɚɤɬɧɵɟ ɞɚɧɧɵɟ: (495) 739-33-63, e-mail: misi@mgsu.ru
Ɋɟɰɟɧɡɟɧɬ: ɩɪɨɮɟɫɫɨɪ ɤɚɮɟɞɪɵ ɜɨɥɧɨɜɨɣ ɢ ɝɚɡɨɜɨɣ ɞɢɧɚɦɢɤɢ ɦɟɯ.-ɦɚɬ ɮ-ɬɚ ɆȽɍ, ɞ.ɮ.-ɦ.ɧ.
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