ɉɊɈȿɄɌɂɊɈȼȺɇɂȿ
ɂ ɄɈɇɋɌɊɍɂɊɈȼȺɇɂȿ ɋɌɊɈɂɌȿɅɖɇɕɏ ɋɂɋɌȿɆ
.
ɉɊɈȻɅȿɆɕ ɆȿɏȺɇɂɄɂ ȼ ɋɌɊɈɂɌȿɅɖɋɌȼȿ
ɍȾɄ
539.3
Ɉ
.
Ⱥ
.
ȿɝɨɪɵɱɟɜ
,
Ɉ
.
Ɉ
.
ȿɝɨɪɵɱɟɜ
,
ȼ
.
ȼ
.
Ȼɪɟɧɞɷ
ɎȽȻɈɍ ȼɉɈ
«
ɆȽɋɍ
»
ɋȼɈȻɈȾɇɕȿ ɉɈɉȿɊȿɑɇɕȿ ɄɈɅȿȻȺɇɂə ɉɊȿȾȼȺɊɂɌȿɅɖɇɈ
ɇȺɉɊəɀȿɇɇɈɃ ɈɊɌɈɌɊɈɉɇɈɃ ɉɅȺɋɌɂɇɕ
-
ɉɈɅɈɋɕ
Ɋɚɫɫɦɨɬɪɟɧɚ ɧɨɜɚɹ ɩɨɫɬɚɧɨɜɤɚ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ ɨ ɫɨɛɫɬɜɟɧɧɵɯ ɤɨɥɟɛɚɧɢɹɯ ɨɞɧɨɪɨɞɧɨɣ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɧɚɩɪɹɠɟɧɧɨɣ ɨɪɬɨɬɪɨɩɧɨɣ ɩɥɚɫɬɢɧɵ-ɩɨɥɨɫɵ ɫ ɪɚɡɥɢɱɧɵɦɢ ɝɪɚɧɢɱɧɵɦɢ ɭɫ
-ɥɨɜɢɹɦɢ. ȼ ɤɚɱɟɫɬɜɟ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɢɫɩɨɥɶɡɭɟɬɫɹ ɧɨɜɨɟ ɩɪɢɛɥɢɠɟɧɧɨɟ ɝɢɩɟɪɛɨɥɢɱɟ
-ɫɤɨɟ (ɜ ɨɬɥɢɱɢɟ ɨɬ ɛɨɥɶɲɢɧɫɬɜɚ ɪɚɛɨɬ,ɝɞɟ ɢɫɩɨɥɶɡɭɟɬɫɹ ɩɚɪɚɛɨɥɢɱɟɫɤɨɟ)ɭɪɚɜɧɟɧɢɟ ɤɨɥɟ
-ɛɚɧɢɹ ɨɞɧɨɪɨɞɧɨɣ ɨɪɬɨɬɪɨɩɧɨɣ ɩɥɚɫɬɢɧɵ-ɩɨɥɨɫɵ.ɂɫɩɨɥɶɡɭɸɬɫɹ ɜɧɨɜɶ ɜɵɜɟɞɟɧɧɵɟ ɝɪɚɧɢɱ
-ɧɵɟ ɭɫɥɨɜɢɹ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɲɚɪɧɢɪɧɨɝɨ,ɠɟɫɬɤɨɝɨ,ɭɩɪɭɝɨɝɨ (ɜɟɪɬɢɤɚɥɶɧɨɝɨ)ɡɚɤɪɟɩɥɟɧɢɹ,ɚ
ɬɚɤɠɟ ɫɜɨɛɨɞɧɨɝɨ ɨɬ ɡɚɤɪɟɩɥɟɧɢɹ ɤɪɚɹ ɩɥɚɫɬɢɧɵ,ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ Ʌɚɩɥɚɫɚ ɢ ɧɟ ɫɬɚɧɞɚɪɬɧɨɟ
ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɨɛɳɟɝɨ ɪɟɲɟɧɢɹ ɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ. ɉɨɞɪɨɛɧɨ ɢɡɥɨɠɟɧɚ ɡɚɞɚɱɚ ɨ ɫɜɨɛɨɞɧɵɯ ɤɨɥɟɛɚɧɢɹɯ ɨɞɧɨɪɨɞɧɨɣ ɨɪɬɨ
-ɬɪɨɩɧɨɣ ɩɥɚɫɬɢɧɵ-ɩɨɥɨɫɵ,ɠɟɫɬɤɨ ɡɚɤɪɟɩɥɟɧɧɨɣ ɧɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɯ ɫɬɨɪɨɧɚɯ,ɤɪɨɦɟ ɬɨɝɨ,
ɩɪɢɜɟɞɟɧɵ (ɛɟɡ ɜɵɜɨɞɚ) ɱɚɫɬɨɬɧɵɟ ɭɪɚɜɧɟɧɢɹ ɩɥɚɫɬɢɧɵ, ɢɦɟɸɳɟɣ ɫɥɟɞɭɸɳɢɟ ɝɪɚɧɢɱɧɵɟ
ɭɫɥɨɜɢɹ:ɫɦɟɲɚɧɧɨɟ ɠɟɫɬɤɨɟ ɢ ɲɚɪɧɢɪɧɨɟ ɡɚɤɪɟɩɥɟɧɢɟ ɧɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɯ ɫɬɨɪɨɧɚɯ,ɲɚɪ
-ɧɢɪɧɨ ɡɚɤɪɟɩɥɟɧɧɵɣ ɤɪɚɣ ɫ ɨɞɧɨɣ ɫɬɨɪɨɧɵ ɢ ɫɜɨɛɨɞɧɵɣ ɨɬ ɡɚɤɪɟɩɥɟɧɢɹ ɤɪɚɣ ɫ ɞɪɭɝɨɣ,ɫɦɟ
-ɲɚɧɧɨɟ ɠɟɫɬɤɨɟ ɢ ɭɩɪɭɝɨɟ ɡɚɤɪɟɩɥɟɧɢɟ ɧɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɯ ɫɬɨɪɨɧɚɯ.ɉɪɢɜɟɞɟɧɧɵɟ ɪɟɡɭɥɶ
-ɬɚɬɵ ɦɨɝɭɬ ɛɵɬɶ ɩɨɥɟɡɧɵ ɜ ɬɟɯ ɨɛɥɚɫɬɹɯ ɫɬɪɨɢɬɟɥɶɫɬɜɚ ɢ ɬɟɯɧɢɤɢ,ɜ ɤɨɬɨɪɵɯ ɢɫɩɨɥɶɡɭɸɬɫɹ
ɩɥɨɫɤɢɟ ɷɥɟɦɟɧɬɵ ɤɨɧɫɬɪɭɤɰɢɣ.
Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɦɟɯɚɧɢɤɚ ɞɟɮɨɪɦɢɪɭɟɦɨɝɨ ɬɟɥɚ, ɨɪɬɨɬɪɨɩɧɚɹ ɩɥɚɫɬɢɧɚ, ɫɨɛɫɬɜɟɧ
-ɧɵɟ ɤɨɥɟɛɚɧɢɹ,ɚɧɢɡɨɬɪɨɩɢɹ,ɱɚɫɬɨɬɧɨɟ ɭɪɚɜɧɟɧɢɟ,ɤɪɚɟɜɚɹ ɡɚɞɚɱɚ,ɬɟɨɪɢɹ ɭɩɪɭɝɨɫɬɢ,ɜɹɡɤɨ
-ɭɩɪɭɝɨɫɬɶ.
Ɋɚɡɜɢɬɢɟ ɫɨɜɪɟɦɟɧɧɨɣ ɫɬɪɨɢɬɟɥɶɧɨɣ ɬɟɯɧɢɤɢ ɢ ɫɬɪɨɢɬɟɥɶɧɵɯ ɬɟɯɧɨɥɨɝɢɣ ɞɟɥɚɟɬ
ɧɟɨɛɯɨɞɢɦɵɦɢ ɩɨɫɬɨɹɧɧɵɟ ɢɫɫɥɟɞɨɜɚɧɢɹ ɜ ɨɛɥɚɫɬɢ ɦɟɯɚɧɢɤɢ ɞɟɮɨɪɦɢɪɭɟɦɨɝɨ ɬɟɥɚ
.
ȼ
ɞɚɧɧɨɣ ɪɚɛɨɬɟ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɨɞɧɨɪɨɞɧɚɹ ɨɪɬɨɬɪɨɩɧɚɹ ɩɥɚɫɬɢɧɚ
-
ɩɨɥɨɫɚ ɫ ɪɚɡɥɢɱɧɵ
-ɦɢ ɡɚɤɪɟɩɥɟɧɢɹ-ɦɢ ɧɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɯ ɫɬɨɪɨɧɚɯ
.
ȼ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ
(
x
,
y
,
z
)
ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɨɞɧɨɪɨɞɧɚɹ ɩɪɟɞɜɚɪɢ
-ɬɟɥɶɧɨ ɧɚɩɪɹɠɟɧɧɚɹ ɨɪɬɨɬɪɨɩɧɚɹ ɩɥɚɫɬɢɧɚ
-
ɩɨɥɨɫɚ
,
ɫɪɟɞɢɧɧɚɹ ɩɥɨɫɤɨɫɬɶ ɤɨɬɨɪɨɣ ɫɨɜ
-ɩɚɞɚɟɬ ɫ ɤɨɨɪɞɢɧɚɬɧɨɣ ɩɥɨɫɤɨɫɬɶɸ
XOY
,
ɨɫɶ
Z
ɧɚɩɪɚɜɥɟɧɚ ɜɟɪɬɢɤɚɥɶɧɨ ɜɜɟɪɯ
.
ɉɥɚɫɬɢ
-ɧɚ ɡɚɧɢɦɚɟɬ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɨɛɥɚɫɬɶ
y
f f
(
;
),
x
> @
l l z
; ,
>
h h
;
@
.
Ɉɞɧɨɪɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɨɩɟɪɟɱɧɵɯ ɤɨɥɟɛɚɧɢɣ ɩɥɚɫɬɢɧɵ ɩɪɟɞɫɬɚɜɢɦ ɜ ɜɢɞɟ
[1]
2 2 4 4 4
1 2 3
2 4 2 2 4
0,
6
ɞ
W
h
ɞ
W
ɞ
W
ɞ
W
A
A
A
ɞ
t
ɞ
t
ɞ
t
ɞ
x
ɞ
x
ª
º
«
»
¬
¼
(1)
ɝɞɟ
1 1 1 11
1
0 333 1
0 55,
A
U
ª
¬
c
A
a
A
º
¼
1 1 1 1 22
2 1
01
02
13 333
33 55 11 33 13,
A
¬
ª
c
a
A A
A A
A A
A
º
¼
11
ВЕСТНИК
МГСУ
МГСУ
3 0 33 11 33 13
1
2 1
,
A
ª
¬
c
A
A A
A
º
¼
U
ɝɞɟ
U
—
ɩɥɨɬɧɨɫɬɶ
,
A
ij—
ɤɨɧɫɬɚɧɬɵ ɨɪɬɨɬɪɨɩɧɨɝɨ ɦɚɬɟɪɢɚɥɚ
;
a c
0,
0—
ɧɚɱɚɥɶɧɵɟ ɩɟ
-ɪɟɦɟɳɟɧɢɹ
[2].
1.
ɉɭɫɬɶ ɩɥɚɫɬɢɧɚ ɠɟɫɬɤɨ ɡɚɤɪɟɩɥɟɧɚ
[3]:
0
w
w
x
W
W
ɩɪɢ
ɯ
= 0, .
l
(2)
ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɧɭɥɟɜɵɟ
:
33 2 2
ɞ
t
W
ɞ
ɞ
t
W
ɞ
ɞ
t
ɞ
W
W
,
t
0.
(3)
Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ
(1)
ɛɭɞɟɦ ɢɫɤɚɬɶ ɜ ɜɢɞɟ
0
,
exp
b
,
W x t
W
x
i
t
h
§
[
·
¨
¸
©
¹
(4)
ɝɞɟ
b
—
ɫɤɨɪɨɫɬɶ ɩɨɩɟɪɟɱɧɨɣ ɜɨɥɧɵ
;
[
—
ɛɟɡɪɚɡɦɟɪɧɚɹ ɱɚɫɬɨɬɚ
.
Ɍɨɝɞɚ ɩɨɥɭɱɢɦ ɨɛɵɤɧɨɜɟɧɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ
4 2
0 0
1 2 0
4 2
0,
ɞ
W
ɞ
W
B
B W
ɞ
x
ɞ
x
(5)
ɝɞɟ
2 2 1 3;
A
b
B
A
h
§
[
·
¨
¸
©
¹
2 2 1 2 3 31
.
A
b
b
B
h
A
h
A
ª
º
§
·
§
·
[
¨
¸
«
¨
[
¸
»
©
¹
«
¬
©
¹
»
¼
(6)
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ
(5)
ɡɚɩɢɲɟɦ ɜ ɜɢɞɟ
0 1 0 1
1 2
0 1 0 1
0 1 0 1
3 4
0 1 0 1
cos
cos
cos
cos
,
sin
sin
sin
sin
,
n n n n
n n n n
l
l
l
l
W x t
ɋ
ɋ
l
l
l
l
ɋ
ɋ
D
D
D
D
ª
º
ª
º
«
D
D
»
«
D
D
»
¬
¼
¬
¼
D
D
D
D
ª
º
ª
º
«
»
«
»
D
D
D
D
¬
¼
¬
¼
(7)
ɝɞɟ
ɋ
i—
ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ
;
i
D
0,1—
ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ
4 2
1 2
0,
B
B
D D
(8)
ɝɞɟ
1,02 1 12 2.
2
4
B
B
B
D
r
ɐɟɥɵɟ ɱɢɫɥɚ
(
m
,
n
)
ɜɵɛɢɪɚɸɬɫɹ ɩɪɢ ɭɞɨɜɥɟɬɜɨɪɟɧɢɢ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɩɪɢ
x
= 0,
ɚ
ɞɪɭɝɢɟ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɩɪɢ
x
= l,
ɩɪɢɜɨɞɹɬ ɤ ɬɪɚɧɫɰɟɧɞɟɧɬɧɨɦɭ ɭɪɚɜɧɟɧɢɸ
.
ɉɪɢ ɩɨɦɨɳɢ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ
(2)
ɢ ɢɫɩɨɥɶɡɭɹ ɪɟɲɟɧɢɟ
(7),
ɩɨɥɭɱɢɦ
n
= 0,
m
=1,
ɋ ɋ
1 30,
ɢɡ ɭɫɥɨɜɢɣ ɩɪɢ
x
= l
ɩɨɥɭɱɢɦ
0 1
2 0 1 4
0 1
sin
sin
cos
cos
l
l
0.
ɋ
¬
ª
D
l
D
l
º
¼
ɋ
ª
«
D
D
D
D
º
»
¬
¼
(9)
2 0
sin
0 1sin
1 4 0cos
0cos
10.
ɋ
ª
¬
D
D
l
D
D
l
¼
º
ɋ
¬
ª
D
D
l
D
l
º
¼
ɂɡ ɭɫɥɨɜɢɣ ɧɟɬɪɢɜɢɚɥɶɧɨɫɬɢ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ
(9)
ɩɨɥɭɱɚɟɦ ɬɪɚɧɫɰɟɧɞɟɧɬɧɨɟ
ɭɪɚɜɧɟɧɢɟ
2 2 0 1
0 1 0 1 0 1
2
¨
§
D D
·
¸
sin
D
l
sin
D
l
2cos
D
l
cos
D
l
0.
D D
©
¹
(10)
ɉɪɟɞɫɬɚɜɢɦ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟ ɮɭɧɤɰɢɢ ɜ ɜɢɞɟ ɫɯɨɞɹɳɢɯɫɹ ɪɹɞɨɜ ɢ
,
ɢɫɩɨɥɶɡɭɹ
ɬɨɥɶɤɨ ɩɟɪɜɵɟ ɞɜɚ ɱɥɟɧɚ ɪɚɡɥɨɠɟɧɢɹ ɢ ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɩɪɟɞɫɬɚɜɥɟɧɢɟɦ
(6),
ɩɨɥɭ
-ɱɢɦ ɱɚɫɬɨɬɧɨɟ ɭɪɚɜɧɟɧɢɟ
12 ISSN 1997-0935. Vestnik MGSU. 2012. № 2
ВЕСТНИК
2/2012
МГСУ
2 2 4
4 3 2 2 2 3
1
2 2
1 2 3 3 1 2
6
2
3
0.
A
A
A
A
h
h
A
b
A A
l
A
A
l
b
A A
§
ª
º
·
§ ·
§ ·
[
¨ ¸
© ¹
¨
¨
«
»
¸
¸
[
¨ ¸
© ¹
¬
¼
©
¹
(11)
2.
ɉɭɫɬɶ ɩɥɚɫɬɢɧɚ ɠɟɫɬɤɨ ɡɚɤɪɟɩɥɟɧɚ ɩɪɢ
x
= l
ɢ ɲɚɪɧɢɪɧɨ ɩɪɢ
x
= 0.
ɉɪɨɞɟɥɚɜ ɚɧɚɥɨɝɢɱɧɨɟ ɪɚɧɟɟ ɢɡɥɨɠɟɧɧɨɦɭ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
,
ɬɚɤɠɟ ɢɫɩɨɥɶɡɭɹ
ɬɨɥɶɤɨ ɞɜɚ ɩɟɪɜɵɯ ɱɥɟɧɚ ɪɚɡɥɨɠɟɧɢɹ
,
ɩɨɥɭɱɢɦ ɱɚɫɬɨɬɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɢɞɚ
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2
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3.
ɉɭɫɬɶ ɩɥɚɫɬɢɧɚ ɲɚɪɧɢɪɧɨ ɡɚɤɪɟɩɥɟɧɚ ɩɪɢ
x
= 0
ɢ ɫɜɨɛɨɞɧɚ ɨɬ ɧɚɩɪɹɠɟɧɢɣ ɩɪɢ
x
= l,
ɬɨ ɱɚɫɬɨɬɧɨɟ ɭɪɚɜɧɟɧɢɟ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ
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2 2
33 13 55 1
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2
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4
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7
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1
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4
12
1
0.
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(13)
4.
ɉɭɫɬɶ ɩɥɚɫɬɢɧɚ ɠɟɫɬɤɨ ɡɚɤɪɟɩɥɟɧɚ ɩɪɢ
x
= 0
ɢ ɭɩɪɭɝɨ ɡɚɞɟɥɚɧɚ ɩɪɢ
x
= l,
ɬɨ ɱɚɫ
-ɬɨɬɧɨɟ ɭɪɚɜɧɟɧɢɟ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ
10 1 8 2 6 3 4 4 2 5
0 0 0 0 0
0,
D
D
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D
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D
D
D
D
[
[
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[
[
(14)
ɝɞɟ
D
i—
ɟɫɬɶ ɮɭɧɤɰɢɹ ɨɬ ɩɚɪɚɦɟɬɪɨɜ
:
A
11,
A
31,
A
13,
A
33,
A
55, , , , ,
U
h b l a c
0,
0.
Ⱥɧɚɥɢɡɢɪɭɹ ɩɨɥɭɱɟɧɧɵɟ ɪɟɲɟɧɢɹ
,
ɫɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ
,
ɱɬɨ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɜ ɩɨ
-ɥɭɱɟɧɧɵɯ ɪɟɲɟɧɢɹɯ ɜɥɢɹɸɬ ɧɚ ɩɨɪɹɞɨɤ ɱɚɫɬɨɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
.
Ȼɢɛɥɢɨɝɪɚɮɢɱɟɫɤɢɣ ɫɩɢɫɨɤ
1.
ȿɝɨɪɵɱɟɜ
O.
Ɉ
.
Ʉɨɥɟɛɚɧɢɹ ɩɥɨɫɤɢɯ ɷɥɟɦɟɧɬɨɜ ɤɨɧɫɬɪɭɤɰɢɣ
.
Ɇ
. :
ɂɡɞ
-
ɜɨ Ⱥɋȼ
, 2005.
ɋ
. 45—49.
2.
Arun K Gupta, Neeri Agarwal, Sanjay Kumar.
Free transverse vibrations of orthotropic
visco-elastic rectangular plate with continuously varying thickness and density // Institute of
Thermomechan-ics AS CR, Prague, Czech Rep 2010 # 2.
3.
Ɏɢɥɢɩɩɨɜ ɂ.Ƚ.,
ɑɟɛɚɧ ȼ.Ƚ.
Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɬɟɨɪɢɹ ɤɨɥɟɛɚɧɢɣ ɭɩɪɭɝɢɯ ɢ ɜɹɡɤɨɭɩɪɭɝɢɯ
ɩɥɚɫɬɢɧ ɢ ɫɬɟɪɠɧɟɣ
.
Ʉɢɲɢɧɟɜ
:
ɒɬɢɧɢɰɚ
, 1988.
ɋ
. 27—30.
ɉɨɫɬɭɩɢɥɚ ɜ ɪɟɞɚɤɰɢɸ ɜ ɮɟɜɪɚɥɟ 2012 ɝ.
Ɉ ɛ ɚ ɜ ɬ ɨ ɪ ɚ ɯ
:
ȿɝɨɪɵɱɟɜ Ɉɥɟɝ Ⱥɥɟɤɫɚɧɞɪɨɜɢɱ
—
ɞɨɤɬɨɪ ɬɟɯɧɢɱɟɫɤɢɯ ɧɚɭɤ
,
ɩɪɨɮɟɫɫɨɪ
,
ɎȽȻɈɍ ȼɉɈ
«
Ɇɨɫɤɨɜɫɤɢɣ ɝɨɫɭɞɚɪɫɬɜɟɧɧɵɣ ɫɬɪɨɢɬɟɥɶɧɵɣ ɭɧɢɜɟɪɫɢɬɟɬ
»
, 129337
əɪɨ
-ɫɥɚɜɫɤɨɟ ɲɨɫɫɟ
, 26,
ɬɟɥ
. 8(495)320-4302;
ȿɝɨɪɵɱɟɜ Ɉɥɟɝ Ɉɥɟɝɨɜɢɱ
—
ɞɨɤɬɨɪ ɬɟɯɧɢɱɟɫɤɢɯ ɧɚɭɤ
,
ɩɪɨɮɟɫɫɨɪ
,
ɎȽȻɈɍ ȼɉɈ
«
Ɇɨɫ
-ɤɨɜɫɤɢɣ ɝɨɫɭɞɚɪɫɬɜɟɧɧɵɣ ɫɬɪɨɢɬɟɥɶɧɵɣ ɭɧɢɜɟɪɫɢɬɟɬ
»
, 129337
əɪɨɫɥɚɜɫɤɨɟ ɲɨɫɫɟ
, 26,
ɬɟɥ
.
8(495) 287-49-14, misi@mgsu.ru;
Ȼɪɟɧɞɷ ȼɥɚɞɢɦɢɪ ȼɥɚɞɢɫɥɚɜɨɜɢɱ
—
ɫɬɚɪɲɢɣ ɩɪɟɩɨɞɚɜɚɬɟɥɶ
,
ɎȽȻɈɍ ȼɉɈ
«
Ɇɨɫɤɨɜ
-ɫɤɢɣ ɝɨɫɭɞɚɪɫɬɜɟɧɧɵɣ ɫɬɪɨɢɬɟɥɶɧɵɣ ɭɧɢɜɟɪɫɢɬɟɬ
»
, 129337
əɪɨɫɥɚɜɫɤɨɟ ɲɨɫɫɟ
, 26,
ɬɟɥ
.
8(499)161-2157, brende@mail.ru.
Ⱦ ɥ ɹ ɰ ɢ ɬ ɢ ɪ ɨ ɜ ɚ ɧ ɢ ɹ
:
ȿɝɨɪɵɱɟɜ Ɉ
.
Ⱥ
.,
ȿɝɨɪɵɱɟɜ Ɉ
.
Ɉ
.,
Ȼɪɟɧɞɷ ȼ
.
ȼ
.
ɋɜɨɛɨɞɧɵɟ ɩɨɩɟɪɟɱɧɵɟ
ɤɨɥɟɛɚɧɢɹ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɧɚɩɪɹɠɟɧɧɨɣ ɨɪɬɨɬɪɨɩɧɨɣ ɩɥɚɫɬɢɧɵ
-
ɩɨɥɨɫɵ
//
ȼɟɫɬɧɢɤ ɆȽɋɍ
. 2012.
ʋ
2.
ɋ
. 11—14.
O.A. Egorychev, O.O. Egorychev, V.V. Brendje
NATURAL TRANSVERSE VIBRATIONS OF A PRESTRESSED ORTHOTROPIC PLATE-STRIPE
The article represents a new outlook at the boundary-value problem of natural vibrations of a homogeneous pre-stressed orthotropic plate-stripe. In the paper, the motion equation represents a
13
ВЕСТНИК
МГСУ
МГСУ
Проектирование и конструирование строительных систем. Проблемы механики в строительствеnew approximate hyperbolic equation (rather than a parabolic equation used in the majority of pa-pers covering the same problem) describing the vibration of a homogeneous orthotropic plate-stripe. The proposed research is based on newly derived boundary conditions describing the pin-edge, ri-gid, and elastic (vertical) types of fixing, as well as the boundary conditions applicable to the unfixed edge of the plate. The paper contemplates the application of the Laplace transformation and a non-standard representation of a homogeneous differential equation with fixed factors. The article pro-poses a detailed representation of the problem of natural vibrations of a homogeneous orthotropic plate-stripe if rigidly fixed at opposite sides; besides, the article also provides frequency equations (no conclusions) describing the plate characterized by the following boundary conditions: rigid fixing at one side and pin-edge fixing at the opposite side; pin-edge fixing at one side and free (unfixed) other side; rigid fixing at one side and elastic fixing at the other side. The results described in the ar-ticle may be helpful if applied in the construction sector whenever flat structural elements are consi-dered. Moreover, specialists in solid mechanics and theory of elasticity may benefit from the ideas proposed in the article.
Key words: solid mechanics, orthotropic plate, natural vibrations, anisotropy, frequency equa-tion, boundary value problem, theory of elasticity, viscoelasticity.
References
1. Egorychev O.O. Kolebanija ploskih elementov konstrukcij [Vibrations of Two-Dimensional Struc-tural Elements]. Moscow, ASV, 2005, pp. 45—49.
2. Arun K Gupta, Neeri Agarwal, Sanjay Kumar. Free transverse vibrations of orthotropic visco-elastic rectangular plate with continuously varying thickness and density// Institute of Thermomechanics AS CR, Prague, Czech Rep, 2010, Issue # 2.
3. Filippov I.G., Cheban V.G. Matematicheskaja teorija kolebanij uprugih i vjazkouprugih plastin i sterzhnej [Mathematical Theory of Vibrations of Elastic and Viscoelastic Plates and Rods]. Kishinev, Shti-nica, 1988, pp. 27—30.
A b o u t t h e a u t h o r s : Oleg Aleksandrovich Egorychev — Doctor of Technical Sciences, Professor, Moscow State University of Civil Engineering (MSUCE), 26 Jaroslavskoe shosse, Moscow, 129337, Rus-sia, 8 (495) 320-4302;
Oleg Olegovich Egorychev — Doctor of Technical Sciences, Professor, Moscow State University of Civil Engineering (MSUCE), 26 Jaroslavskoe shosse, Moscow, 129337, Russia, misi@mgsu.ru, 8 (495) 287-4914;
Vladimir Vladislavovich Brendje — Senior Lecturer, Moscow State University of Civil Engineering (MSUCE), 26 Jaroslavskoe shosse, Moscow, 129337, Russia, brende@mail.ru, 8 (499) 161-2157. F o r c i t a t i o n : Egorychev O.A., Egorychev O.O., Brendje V.V. Svobodnye poperechnye kolebanija predvaritel''no naprjazhennoj ortotropnoj plastiny-polosy [Natural Transverse Vibrations of a Prestressed Orthotropic Plate-Stripe], Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering], 2012, Issue # 2, pp. 11—14.
14 ISSN 1997-0935. Vestnik MGSU. 2012. № 2
