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Собственные поперечные колебания предварительно напряженной ортотропной пластинки-полосы жестко закрепленной по одному краю и свободной по другому.

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4/2010
ВЕСТНИК
МГСУ
ɋɈȻɋɌȼȿɇɇɕȿ ɉɈɉȿɊȿɑɇɕȿ ɄɈɅȿȻȺɇɂə
ɉɊȿȾȼȺɊɂɌȿɅɖɇɈ ɇȺɉɊəɀȿɇɇɈɃ ɈɊɌɈɌɊɈɉɇɈɃ
ɉɅȺɋɌɂɇɄɂ-ɉɈɅɈɋɕ ɀȿɋɌɄɈ ɁȺɄɊȿɉɅȿɇɇɈɃ ɉɈ
ɈȾɇɈɆɍ ɄɊȺɘ ɂ ɋȼɈȻɈȾɇɈɃ ɉɈ ȾɊɍȽɈɆɍ
OWN TRANSVERSE VIBRATIONS OF A PRESTRESSED
ORTHOTROPIC PLATE-BAND RIGIDLY FIXED ON ONE SIDE
AND FREE ON ANOTHER
Ɉ.Ⱥ. ȿɝɨɪɵɱɟɜ, Ɉ.Ɉ. ȿɝɨɪɵɱɟɜ,
Ɉ.ɂ. ɉɨɞɞɚɟɜɚ, ȼ.ȼ. Ȼɪɟɧɞɷ
O.A. Egorychev, O.Ɉ. Egorychev,
O.I. Poddaeva, V.V. Brende
ȽɈɍ ȼɉɈ ɆȽɋɍ
ɉɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɨɩɪɟɞɟɥɟɧɧɨ ɯɚɪɚɤɬɟɪɧɨɟ ɡɧɚɱɟɧɢɟ ɱɚɫɬɨɬɵ, ɩɨɥɭɱɟɧɵ
ɡɧɚɱɟɧɢɹ ɱɚɫɬɨɬ ɩɪɟɜɵɲɚɸɳɟɟ ɯɚɪɚɤɬɟɪɧɭɸ ɢɥɢ ɦɟɧɶɲɟ ɯɚɪɚɤɬɟɪɧɨɣ ɱɚɫɬɨɬɵ.
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɩɪɟɞɫɬɚɜɥɟɧɨ ɜ ɜɢɞɟ ɬɪɚɧɫɰɟɧɞɟɧɬɧɵɯ ɱɚɫɬɨɬɧɵɯ ɭɪɚɜɧɟɧɢɹɯ.
In solving the problem the characteristic frequency is determined, and then obtained the
frequencies higher or less than the characteristic frequency. The general solution presented
in the form of transcendental frequency equations.
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
A2
2 1 a0 1 c0 2 A13 A331 3 A33 A55 A132 A11 A33 º ;
¼
6 ¬
h2
2 1 c0 A331 A11 A13 A132 ,
6U
ɝɞɟ U - ɩɥɨɬɧɨɫɬɶ, a0 , c0 - ɩɚɪɚɦɟɬɪɵ ɨɞɧɨɪɨɞɧɨɝɨ ɧɚɩɪɹɠɟɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ, Ai , j A3
ɭɩɪɭɝɢɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɨɩɪɟɞɟɥɹɸɳɢɟ ɨɪɬɨɬɪɨɩɢɸ.
267
ВЕСТНИК
МГСУ
4/2010
Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (1) ɛɭɞɟɦ ɢɫɤɚɬɶ ɜ ɜɢɞɟ:
§ b ·
W x, t W0 x exp ¨ i[ t ¸ ,
(2)
© h ¹
ɝɞɟ b - ɫɤɨɪɨɫɬɶ ɩɨɩɟɪɟɱɧɨɣ ɜɨɥɧɵ, [ - ɱɚɫɬɨɬɚ.
Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ (1) ɩɪɟɨɛɪɚɡɭɟɬɫɹ ɜ ɨɛɵɤɧɨɜɟɧɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ:
d 4W0
d 2W0
B
B2W0 0,
(3)
1
dx 4
dx 2
ɝɞɟ
2
2
2
º
A2 § b ·
1 § b· ª § b·
B
A
[
[
(4)
B1
[
;
2
¨
¸ « 1¨
¸ 1» .
¨
¸
A3 © h ¹ ¬« © h ¹
A3 © h ¹
¼»
Ⱦɥɹ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (3) ɡɚɩɢɲɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ
4
r B1r 2 B2 0
(5)
ȿɝɨ ɪɟɲɟɧɢɟ ɢɦɟɟɬ ɜɢɞ:
r1,2,3,4
2
ª B
º
§ B ·
r « 1 r ¨ 1 ¸ B2 »
« 2
»
© 2¹
¬
¼
1
2
(6)
.
ɂɫɩɨɥɶɡɭɹ ɫɨɨɬɧɨɲɟɧɢɟ (4), ɪɚɜɟɧɫɬɜɨ (6) ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ:
1
2 2
§ b · 1 ª«
§ b· º
A2 r A22 4 A1 A3 4 A3 ¨ [ ¸ » .
(7)
r1,2,3,4 r ¨ [ ¸
© h ¹ 2 A3 «¬
© h ¹ »¼
Ɉɬɦɟɬɢɦ ɯɚɪɚɤɬɟɪɧɨɟ ɡɧɚɱɟɧɢɟ [ 0 , ɩɪɢ ɤɨɬɨɪɨɦ ɜɵɪɚɠɟɧɢɟ (7), ɫɬɨɹɳɟɟ ɩɨɞ
ɤɨɪɧɟɦ, ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɨɥɶ
A3
A3
§h·
[ 0 r2 ¨ ¸
;
! 0.
(8)
2
4 A1 A3 A22
© b ¹ 4 A1 A3 A2
ɂɡ ɚɧɚɥɢɡɚ ɤɨɪɧɟɣ (7), ɜɨɡɦɨɠɧɵ ɞɜɚ ɜɚɪɢɚɧɬɚ.
ȼɚɪɢɚɧɬ I. [ [ 0 , ɬɨɝɞɚ h1 h.
r1,2,3,4
ª
§ b· 1 «
r ¨[ ¸
A2 r
© h1 ¹ 2 A3 ««
¬
1
2
ª § h ·2 º º
2
A2 4 A1 A3 «1 ¨ h ¸ » »» .
«¬ © 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 ,
ɝɞɟ
E1
268
§ b · D2
;
¨[ ¸
© h1 ¹ 2 A3
E2
§ b · D3
¨[ ¸
© h1 ¹ 2 A3
(9)
(10)
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
ВЕСТНИК
МГСУ
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
r1,2,3,4 r D 3 r i E 3 2 r R 2 «cos
r i sin
,
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)
(15)
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 (16)
Ɂɚɞɚɱɚ ɪɟɲɚɟɬɫɹ ɩɪɢ ɫɥɟɞɭɸɳɢɯ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɹɯ [2].
ɩɪɢ x l ;
dW0
W0
0
dx
ɩɪɢ x l ;
d 2W0
K1W0
dx 2
0;
d 3W
dx 3
0
(17)
(18)
269
ВЕСТНИК
МГСУ
4/2010
2
3 A33 2 A13 § b ·
¨[ ¸ .
7 A33 4 A13 © h ¹
Ɋɚɫɫɦɨɬɪɢɦ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ, ɢɫɩɨɥɶɡɭɹ ɜɚɪɢɚɧɬ I, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɪɟɲɟɧɢɟ
ɭɪɚɜɧɟɧɢɹ (3) ɛɭɞɟɦ ɩɪɟɞɫɬɚɜɥɹɬɶ ɜ ɜɢɞɟ (11).
Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɩɪɢɦɭɬ ɜɢɞ:
ɩɪɢ x l
C1 cos E1l1 C2 sin E1l1 C3 cos E 2 l1 C4 sin E 2 l1 0
(19)
C1 E1 sin E1l1 C2 E1 cos E1l1 C3 E 2 sin E 2 l1 C4 E 2 cos E 2 l1 0
(20)
ɩɪɢ x l
C1 K1 E12 cos E1l1 C2 K1 E12 sin E1l1 C3 K1 E 22 cos E 2 l1 C4 K1 E 22 0
(21)
ɝɞɟ K1
U A55
C1 E13 sin E1l1 C2 E13 cos E1l1 C3 E 23 sin E 2 l1 C4 E 23 cos E 2 l1
(22)
0,
l
.
ɝɞɟ l1
h1
Ɋɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɨɞɧɨɪɨɞɧɵɯ ɭɪɚɜɧɟɧɢɣ (19), (20), (21) ɢ (22) ɩɪɢɜɨɞɢɬ ɤ
ɬɪɚɧɫɰɟɧɞɟɧɬɧɨɦɭ ɭɪɚɜɧɟɧɢɸ, ɨɩɪɟɞɟɥɹɸɳɟɦɭ ɢɫɤɨɦɨɟ ɱɚɫɬɨɬɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɢɞɚ:
ª K1 E12 E 24 K1 E 22 E14 º sin 2E1l1 sin 2E 2 l1 ª K1 E12 E13 E 2 ¬
¼
¬
(23)
2
3
K1 E 2 E1 E 2 º¼ cos 2E1l1 sin 2E 2 l1 1 0.
ɂɫɩɨɥɶɡɭɹ ɪɚɜɟɧɫɬɜɨ (16) ɞɥɹ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (3), ɬ.ɟ. ɪɟɲɚɹ ɜɚɪɢɚɧɬ II ɢ ɢɫɩɨɥɶɡɭɹ
ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ (17) ɢ (18), ɩɨɥɭɱɢɦ ɫɢɫɬɟɦɭ ɨɞɧɨɪɨɞɧɵɯ ɭɪɚɜɧɟɧɢɣ.
ɋ1e 2D 4l1 cos E 4 l1 C2 e 2D 4l1 sin E 4 l1 C3 cos E 4 l1 C4 sin E 4 l1 0
ɋ1 D 42 cos E 4 l1 2D 4 E 4 sin E 4 l1 E 42 cos E 4 l1 K1 cos E 4 l1 ɋ2 D 42 sin E 4 l1 2D 4 E 4 cos E 4 l1 E 42 sin E 4 l1 K1 sin E 4l1 C3 e 2D 4l1 D 42 cos E 4l1 2D 4 E 4 sin E 4 l1 E 42 cos E 4 l1 K1 cos E 4 l1 C4 e2D 4 l1 D 42 sin E 4 l1 2D 4 E 4 cos E 4 l1 E 42 sin E 4 l1 K1 sin E 4 l1 2D 4 l1
0
2D 4 l1
D 4 cos E 4l1 E 4 sin E 4 l ɋ2 e D 4 sin E 4l1 E 4 cos E 4l C3 D 4 cos E 4 l1 E 4 sin E 4 l1 C4 D 4 sin E 4 l1 E 4 cos E 4 l1 0
ɋ1e
(24)
ɋ1 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 ɋ2 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 C3 e 2D 4 l1 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 C4 e 2D 4 l1 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 0.
Ɋɟɡɭɥɶɬɚɬɨɦ ɪɟɲɟɧɢɹ ɨɞɧɨɪɨɞɧɨɣ ɫɢɫɬɟɦɵ (24), ɩɨɥɭɱɢɦ ɬɪɚɧɫɰɟɧɞɟɧɬɧɨɟ
ɭɪɚɜɧɟɧɢɟ ɜɢɞɚ:
P
e8D 4 l1 1 e 4D 4 l1 1 0,
(25)
P2
ɝɞɟ
270
4/2010
P1
2sin 2 E 4 l1 D 45 2D 42 E 43 D 4 E 44 D 4 sin 2E 4 l1 E 4 ,
P1
E 42 D 42 E 42 K1 E 4 3D 42 E 42 .
ВЕСТНИК
МГСУ
Ɋɚɡɥɨɠɢɜ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟ ɮɭɧɤɰɢɢ ɢ eD x ɜ ɫɬɟɩɟɧɧɵɟ ɪɹɞɵ ɜ ɭɪɚɜɧɟɧɢɹɯ
(23) ɢ (25), ɩɨɥɭɱɢɦ ɱɚɫɬɨɬɧɵɟ ɚɥɝɟɛɪɚɢɱɟɫɤɢɟ 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: ɩɨɩɟɪɟɱɧɵɟ ɤɨɥɟɛɚɧɢɹ, ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɧɚɩɪɹɠɟɧɧɨɟ ɫɨɫɬɨɹɧɢɟ,
ɨɪɬɨɬɪɨɩɧɚɹ ɩɥɚɫɬɢɧɤɚ-ɩɨɥɨɫɚ
Key words: natural transverse vibrations, prestressing, orthotropic plate – band
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