Volume 03 Issue 10-2023
181
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
10
Pages:
181-189
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
A
BSTRACT
This article discusses the analysis of the general equations of the transverse vibration of a piecewise
homogeneous viscoelastic plate obtained in the “Oscillation of bilayer plates of constant thickness” [1].
K
EYWORDS
Analysis, approximate, vibrations, two-layer plate, boundary value problem, stresses, deformation,
oscillation equations.
I
NTRODUCTION
The general equations of oscillation of piecewise
homogeneous viscoelastic plates of constant
thickness, described in [1], are complex in
structure and contain derivatives of any order
concerning x, y coordinates and time t, and,
therefore, are not suitable for solving applied
problems
and
performing
engineering
calculations [2-5].
To solve applied problems, instead of general
equations, it is advisable to use approximate ones
that include one or another finite order in
derivatives [6-8].
The main part
The classical equations of transverse vibration of
a plate contain derivatives of no higher than 4th
order, and for piecewise homogeneous or two-
layer plates, the simplest approximate equation
of vibration is a sixth-order equation [9-12].
Journal
Website:
http://sciencebring.co
m/index.php/ijasr
Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Research Article
EQUATION OF TRANSVERSE VIBRATION OF A PIECEWISE
HOMOGENEOUS VISCOELASTIC PLATE
Submission Date:
October 10, 2023,
Accepted Date:
October 15, 2023,
Published Date:
October 20, 2023
Crossref doi:
https://doi.org/10.37547/ijasr-03-10-30
M.L. Djalilov
Fergana branch of the Tashkent University of Information Technologies named after Muhammad Al-
Khorazmiy, Fergana, Republic of Uzbekistan
Volume 03 Issue 10-2023
182
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
10
Pages:
181-189
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
If in the operators (1.3.8) given in [1] we restrict
ourselves to the first two terms, then from
equation (1.3.11)
1
2
1
(
)
( , , )
L W
F x y t
=
where are the operators
1
L
and
)
t
,
y
,
x
(
F
1
equal to:
1
1( )
2( )
2( )
1( )
3( )
4( )
4( )
3( )
1( )
3( )
3( )
1( )
4( )
2( )
2( )
4( )
1( )
4( )
4( )
1( )
2( )
3( )
3( )
2( )
2( )
3( )
3( )
2( )
4( )
1( )
1( )
4( )
2(
(
)(
)
(
)(
)
(
)(
)
(
)(
)
(
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
L
M
K
M
K
H
E
H
E
M
K
M
K
H
E
H
E
M
K
M
K
H
E
H
E
M
K
M
K
H
E
H
E
M
=
−
−
+
+
−
−
+
+
−
−
−
−
−
−
−
−
)
4( )
4( )
2( )
1( )
3( )
3( )
1( )
3( )
4( )
4( )
3( )
1( )
2( )
2( )
1( )
)(
)
(
)(
);
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
K
M
K
H
E
H
E
M
K
M
K
H
E
H
E
−
−
+
+
−
−
1
1( )
2( )
3( )
3( )
2( )
2( )
3( )
1( )
1( )
3( )
1
(0)
3( )
1( )
2( )
2( )
1( )
0
1( )
2( )
3( )
3( )
2( )
2( )
3( )
1( )
1( )
3( )
3( )
1( )
2( )
2( )
1( )
1
[
(
)
(
)
(
)]{
(
)}
[
(
)
(
)
(
)]{
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
z
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
F
K
H
E
H
E
K
H
E
H
E
K
H
E
H
E
M
f
M
H
E
H
E
M
H
E
H
E
M
H
E
H
E
M
−
= −
−
+
−
+
+
−
+
+
−
+
−
+
+
−
(0)
(0)
1
1( )
2( )
3( )
3( )
2( )
2( )
3( )
1( )
1( )
3( )
1
(1)
3( )
1( )
2( )
2( )
1( )
1
1( )
2( )
3( )
3( )
2( )
2( )
2( )
1( )
1( )
2( )
3( )
1(
(
)}
(
(
)
(
)
(
)]{
(
)}
(
(
)
(
)
(
yz
xz
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
z
n
n
n
n
n
n
n
n
n
n
n
n
f
f
x
y
M
K
E
K
E
M
K
E
K
E
M
K
E
K
E
M
f
M
K
H
K
H
M
K
H
K
H
M
K
−
−
+
−
−
−
+
−
+
+
−
+
+
−
+
−
+
+
(1)
(1)
1
)
2( )
2( )
1( )
1
)]{
(
)};
yz
xz
n
n
n
f
f
H
K
H
M
x
y
−
−
+
we obtain the approximate integral-differential equation
(
)
(
)
4
2
6
4
2
1
2
3
4
5
4
2
6
4
2
2
3
6
7
1
2
( , , ).
W
W
W
W
Q
Q
Q
W
Q
Q
t
t
t
t
W
Q
Q
W
F x y t
t
+
+
+
+
+
+
+
=
(1)
where are the operators
j
Q
and
(
)
t
,
y
,
x
F
1
equal to:
(
)
2
2
1
1
0
0
1
1
;
Q
M
h
h
−
=
+
Volume 03 Issue 10-2023
183
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
10
Pages:
181-189
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
(
)(
)
(
)
(
)
2
2
1
0
2
0
1
1
0
0
1
1
2
2
2
0
0
0
1
0
0
0
1
1
1
2
(2
1 (
(
)));
Q
M
h P D
h D
h
h
P
h
h
h
h D
h D
−
= −
+
+
+
+
−
+
−
+
2
2
2
3
2
0
2
0
1
1
1
1
0 1 2
0
4(
1)(
2
);
Q
P
h P D
h D
h D
h h P D
=
−
+
+
+
2
2
1
2
2
4
1
0
0
0
1
1
0
0
0
0
1
1
0
2
1
2
2
1
1
1
0
0
1
1
1
1
0
0
1
1
(
(3
(
4
))(2
)
6
(3
(
4
))(2
));
Q
M
h
M
h
h
h
h
D
h
M
h
h
h
h
D
−
−
−
= −
+
+
−
+
+
+
+
−
(2)
2
2
2
2
2
5
1
0
2
0
0
2
0
0
2
0
4
2
2
2
1
1
1
1
1
2
1
1
2
2
1
1
2
0
1
0
1
0
1
2
0
1
2
2
0
2
1
0
1
2
2
1
0
1
0
1
2 0 1
0
1
0
1
(
(2 (4
(1
) (
1)(4
))
6
(2(4
4
1) (
1)
(2
))
6
(
(4(
) (
1)(2 (1
)
(2
)
(1
)))
(
))
2
(2
Q
M
h P
M
P
D
D
P
D
h
M
D
D
P
D
D
h h
M M
P D
D
P
P
D
P D
D
D
D
M
P h h
M
−
−
−
−
−
−
= −
−
+
−
+
−
−
−
− −
−
−
+
+
+
+
−
−
−
−
+
+
+ +
+
+
+
1
1
2
2
2
2
1
0
0
0
1
2
2
1
1
1
2
2
2
2
2
2
0
1
0
2
0
1
0
1
1
1
0
1
)(
(2 4
)
(2
5
))
((
1)(4 3
) 2
(4
)) 2
(4
));
M
h
D
D
h
P
P D
D
D
h h M
P
D
D
D
h
M D
D
−
−
−
−
+
−
+
−
+
−
+
+
−
−
+
−
+
−
2
2
1
2
6
1
0
2
0
0
2
2
0
0
0
2
0
4
1
1
1
1
1
1
1
2
1
1
2
2
0
1
2
0
2
1
1
2
2
1
0
1
2
0
0
1
0
0
1
0
2
0
2
1
(
(2 ((
1)(2 9
3
)) 2
(1 3
4
))
3
(4
(1 2
) 4
(
1)
(3
))
3
((4
(
(1
)
) (
1)(2(
1)
(1
)
(2
2
)))
(4
(1
) (
Q
M
h P
M
P
P
D
D
D
P
D
h
M
D
D
D
P
D
D
h h
P D P
D
D
P
P
D
D
P
D
D D
M
D
D
P D
P
−
−
−
−
=
−
+
−
−
−
+
+
+
−
−
+
−
−
+
+
−
−
−
−
−
−
−
+
−
−
+
+
+
−
−
0
1
2
1
1
2
2
2
0
1
1
1
0 1 2
0
0
0
2
0
0
2
1
0
1
2
1
2
1
2
2
1
1
0
1
2
1
2
1
1
1)(6
(
1)
6
))
) 2
(
(2
((
1)(
2
1)
2
(1
))
(2(
1)
(
3))))
4
(
)(2(
1)(1
)
(1
))))));
D D P
P D
D
M
h h P
M
h
P
D
D
D
D
h
P
D P
M
h
h
P
D
P D
D
−
−
−
− −
−
+
−
−
−
− −
−
+
−
− +
+
−
−
+
−
−
+
+ +
4
4
7
0
2
0
0
2
1
1
1
2
2
2
0
1
2
0
1
2
2
0
1
2
0
1
1
2
2
0 1 2
0
0
2
1
1
2
2
0
2
(
(4
5(
1)
(4
(
1))
3
3
(8
(
1)((2(
1)
3
(1
)))
4
(
(
1) 2
)
(2(
1) (
1)
)));
Q
h P D
D
P
h D
D
P
h h
P D D
P
P
D D
P D
D
D
h h P D h
P
D
h
P
P
D
=
−
− +
−
−
−
+
−
−
+
−
−
−
−
−
− +
+
− +
+
(3)
Volume 03 Issue 10-2023
184
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
10
Pages:
181-189
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
(
)
2
2
(0)
(1)
1
1
0
0
1
1
2
2
(0)
2
(1)
2
(0)
2
(1)
0
1
1
1
0
0
2
2
2
2
2
(0)
2
(1)
2
(0)
2
(1)
2
2
0
0
0
1
1
1
2
2
2
2
2
1
0
2
( , , )
(
(
))
(
)(
(
)
(
))
(
)((
) (
))
2 (2
(
z
z
yz
yz
xz
xz
yz
yz
xz
xz
F x y t
M
h
h
f
f
t
f
f
f
f
h
h
h
h
x
y
x
y
f
f
f
f
h D
h D
x
y
x
y
M
h P
−
−
=
+
−
+
+
+
+
+
+
+
+
+
+
−
+
−
−
(
)
(0)
(1)
0
1
1
0
1
2
(0)
2
(1)
2
(0)
2
(1)
1
1
2 0 1
0
0
1
1
2
2
2
2
2
(0)
2
(1)
2
(0)
2
(1)
1
2
2
1
0
2
0
1
1
2
2
2
2
(
))
2
)(
(
)
(
))
(
)(
) (
)).
z
z
yz
yz
xz
xz
yz
yz
xz
xz
D
h D
M f
M f
f
f
f
f
P h h
D M
D M
x
y
x
y
f
f
f
f
M
h P D
h D
x
y
x
y
−
−
−
+
−
+
+
+
+
+
+
+
+
+
+
+
(4)
If the plate is homogeneous, and
W
–
is the transverse displacement of the points of the “middle” surface –
the plane of the plate, then, in this case, the dependencies are satisfied
0
1
0
1
2
0
1
0
1
0
1
;
;
1;
;
;
.
N
N
M
M
P
h
h
C
C
D
D
=
=
=
=
=
=
and equation (1) goes into the equation
2
(1)
2
(1)
0
10
0
20
2
(1)
2
(1)
(1)
(1)
0
0
20
0
20
10
20
2
(0)
2
(0)
2
2
0
0
2
2
2
0
2
(1)
2
(1)
1
0
0
0
2
2
((1
)
(1
)
)((
)
((3
(
) ) 4
) 4
(
)))(
)
6
1
(
((
)
(
))
4
((
)
)))
yz
xz
z
yz
xz
z
C
C
h
D
D
W
f
f
M
f
h
h
t
x
y
f
f
D M
f
h
x
y
−
−
−
+ +
+ +
+
+
+
+
+
=
=
+
+
−
−
+
+
(5)
Here on the left is the product of two operators: the first describes the process of longitudinal oscillation,
and the second describes the transverse vibration.
The approximate equation from the general equation (1.3.12) given in [1] is introduced similarly and we
obtain for
−
x
V
y
U
1
1
Volume 03 Issue 10-2023
185
International Journal of Advance Scientific Research
(ISSN
–
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VOLUME
03
ISSUE
10
Pages:
181-189
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
4
2
6
2
1
2
3
4
5
6
2
4
2
6
2
3
1
1
7
8
9
2
(
)(
)
( , , ),
G
G
G
G
G
G
t
t
t
t
U
V
G
G
G
F x y t
y
x
+ +
+
+ +
+
+ + +
−
=
(6)
where are the operators
j
Q
and
(
)
t
,
y
,
x
F
1
equal to:
(
)
1
1
1
0
0
1
1
;
G
M
h
h
−
=
+
(
)
2
0
2
1
;
G
h P
h
= −
+
2
2
1
2
1
3
1
0
0
0
1
1
0
0
1
1
1
0
0
1
1
1
(
(
3
)
(
3
)
);
6
G
M
h h
h
M
h h
h
M
−
−
−
=
+
+
+
2
1
1
1
4
0
2 0
0
0
1
0
0
1
1
2
1
1
1
1
1
1
1
2 0
0
0
1
1
1
(
(2
3 (
))
6
(2
3
(
)));
G
h
P h
M
h
M
M
h
h
M
P h
M
M
−
−
−
−
−
−
= −
+
+
+
+
+
+
2
2
2
5
1
0
2 0
1
1
1
2 0
1
(
(
3 )
(
3
));
6
G
M
h P h
h
h h
P h
−
=
+
+
+
(7)
5
2
2
1
1
5
1
1
1
6
0
2
0
0
1
1
0
0
1
1
1
0
0
1
1
1
1
3
1
2
3
1
2
0 1
0
1
0
1
0
0
0
0
0
1
2
1
1
1
1
1
(
(10
)
(10
)
120
5
(
(3 3
)
(3 3
)));
G
h P
M
M
M
h
M
M
M
h h
M M
h
M
D
D
h P
M
D
D
−
−
−
−
−
−
−
−
−
−
=
+
+
+
+
+
−
−
−
−
−
5
2
2
5
2
2
5
5
1
1
7
0
2
0
0
1
1
1
0
2
1
0
1
0
1
3
1
2
1
1
0 1
0
0
0
0
0
0
0
0
1
1
3
1
2
1
1
1
1
2
1
1
1
1
1
1
0
1
1
0
0
1
( 13(
)
20(
)
120
5
(
((3 3
)
(
4)
)
((3 3
)
(
4)
)));
G
h P
M
h
M
h P
h
M M
h h h
M
D
D
M
D
M
h P
M
D
D
M
D
M
M
−
−
−
−
−
−
−
−
−
−
−
=
−
+
+
+
−
−
−
−
−
−
+
+
−
−
−
−
5
2
1
5
2
2
5
1
5
1
8
0
2
0
0
1
1
1
0
2
1
1
1
0
0
3
1
1
4
1
1
0 1
0
1
1
0
0
0
1
0
0
1
1
1
1
(23(
) 10(
)
120
5
(
(
(
4)
)
(
(
4)
)));
G
h P
M
h
M
h P
M
h
M
h h h
M
D
M
h
M
D
M
−
−
−
−
−
−
−
−
=
+
+
+
+
+
−
−
+
−
−
5
2
2
5
2
2
5
5
1
1
9
0
2
0
0
1
1
1
0
2
1
0
1
0
1
3
1
2
1
1
0 1
0
0
0
0
0
0
0
0
1
1
3
1
2
1
1
1
1
2
1
1
1
1
1
1
0
1
1
0
0
1
( 24(
) 6(
)
120
6
(
((1 3
)
(
2)
)
((3
)
(
2)
)));
G
h P
M
h
M
h P
h
M M
h h h
M
D
D
M
D
M
h P
M
D
D
M
D
M
M
−
−
−
−
−
−
−
−
−
−
−
=
−
+
+
+
−
−
−
−
−
−
+
+
−
−
−
−
and
Volume 03 Issue 10-2023
186
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
10
Pages:
181-189
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
2
(0)
2
(1)
2
(0)
2
(1)
1
1
2
2
0
1
2
2
2
2
2
(0)
2
(0)
2
1
1
2 1
1
1
0
2
2
2
(1)
2
(1)
2
2
1
1
0
0
0
1
2
2
2
2
(0)
2
(0)
2
1
2 1
0
2
( , , )
(
(
)
(
))
1
(
(
)
2
(
))
1
(
(
2
yz
yz
xz
xz
yz
xz
yz
xz
yz
xz
f
f
f
f
F x y t
P N
N
x
y
x
y
f
f
P h
M
N
x
y
f
f
h
M
N
x
y
t
f
f
P h N
x
−
−
−
−
−
−
−
=
−
+
−
+
+
−
−
−
−
−
−
−
2
2
(1)
2
(1)
2
2
1
0
1
2
2
2
)
(
))
.
yz
xz
y
f
f
h N
x
y
x
−
−
−
−
(8)
Despite the fact that equation (1) is approximate, it is quite complicated. The operators (2) contain all
parameters and operators characterizing both the mechanical and rheological properties of the piecewise
homogeneous plate material and its geometric dimensions.
Approximate equation (1) is simplified in particular cases when solving specific oscillation problems. For
example, operators (2) are greatly simplified when the Poisson ratios of both components are constant, or
when the thicknesses of both components are equal, and so on.
For example, if
1
0
h
h
=
and
1
0
=
, then the operators
j
Q
in (6) have the form:
(
)
2
2
2
1
1
0
0
1
;
Q
M h
−
=
+
2
2
2
1
0
0
2
0
1
2
0
0
0
1
2
(2
(
1)(
) (
1)(2
(
)));
Q
M h
D P
P
D
−
= −
+
+
+
+
−
−
2
3
2
0
0
2
4(
1)
(3
1);
Q
P
h D
P
=
−
+
2
4
1
2
4
1
0
0
0
0
1
0
0
1
1
2
1
1
0
1
1
0
1
(2
)(
(3
(
4
))
6
(3
(
4
)));
Q
M h
D
M
M
−
−
−
= −
−
+
+
+
+
+
+
(9)
4
2
2
5
0
2
0
0
0
0
2
0
0
2
1
1
2
2
0
0
0
1
0
1
0
2
0
2
2
2
0
0
2
2
0
0
0
0
2
2
2
1
1
0
0
2
0
0
2
0
0
1
(
(4
(4
)
(8
(1
) 5)
6
(
1)(12 6
)) 2
(2(6
(2 5
)
(2 9
)) (
1)
(2 3
)
(1
))
(8(1
) 4
(4
) (
1)
(2
)));
Q
h
P
M
D
D
P
D
D
P
D
D
M M
D
P
D
P
D
D
P
P
D
D
D
D
M
D
D
P D
D
P
D
D
−
−
−
−
= −
−
+
−
+ +
+
−
−
+
+
+
+
+
+
+
−
+
−
−
+
+
+
+
+
+
−
+
−
+
−
−
Volume 03 Issue 10-2023
187
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
10
Pages:
181-189
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
2
1
2
6
0
0
0
2
0
2
0
2
2
2
0
0
1
0
0
1
1
0
0
2
2
0
2
0
2
0
1
(
(4
(2 5
3
(
1)) (
1)(
(20 8
13
)
3
6
(1
)))
(4(4
) 4 (4 2
5
)
17(
1)(
2 (1
))));
Q
h
M
P D
P
D P
P
P
D
D
D
D
M D
D
P
P
D
P
D
P
D
−
−
=
+
−
−
+
−
−
−
+
+
−
+
+
+
+
+
+
+
−
+
−
4
2
7
0
0
0
2
0
2
2
4
(
(4 15
5
) (
1)(1 13 ));
3
Q
h D D
P
P
P
P
=
−
−
+
−
−
(10)
The sixth-order operator in equation (1) can also be represented as the product of second and fourth-order
operators if the plate is elastic and the coefficients
j
Q
connected by addiction
2
4
7
1
5
7
3
4
6
.
Q Q Q
Q Q Q
Q Q Q
=
+
For a two-layer elastic plate with given parameters of its components, relation (7) gives a 10th-order
algebraic equation concerning the relation
1
2
h
/
h
ℎ
2
/ℎ
1
, the sixth-order operator in (1) can be represented
as the product of two lower-order operators
( )
2
2
2
2
4
4
1
2
3
4
5
1
2
2
2
2
4
4
0,
A
A
A
A
A
A
W
t
x
t
x
t
x
+
+
+
+
=
if the coefficients
j
Q
𝑄
2
and
j
A
𝐴
3
linked by dependencies
1
1
2
2
1
4
2
3
3
2
4
;
;
;
Q
A A
Q
A A
A A
Q
A A
=
=
+
=
4
1
5
5
2
5
6
1
6
7
2
6
;
;
;
;
Q
A A
Q
A A
Q
A A
Q
A A
=
=
=
=
C
ONCLUSIONS
1. The study of vibrations of piecewise-
homogeneous plates in an accurate three-
dimensional formulation allows us to derive the
general and approximate equations of vibration
of such plates based on them without using any
hypotheses.
2. It is shown that the simplest approximate
equation of vibration of a two-layer plate is a
sixth-order equation with respect to derivatives
describing its longitudinal-transverse vibration.
Volume 03 Issue 10-2023
188
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
10
Pages:
181-189
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
3. For an elastic two-layer plate, the sixth-order
operator splits into the product of the second-
longitudinal and fourth-order transverse-wave
operators if the thicknesses of the plate
components satisfy the derived equation
containing the parameters of these components.
4. Formulas are obtained for determining
displacements and stresses through the sought-
for functions at any point of a two-layer plate.
R
EFERENCES
1.
Филиппов, И. Г., & Егорычев, О. А.
(1983). Волновые процессы в линейных
вязкоупругих
средах.
М.:
Машиностроение, 270, 320.
2.
Джалилов
М.Л.
Колебания
прямоугольный
и
безграничной
упругой двухслойной пластинки –
Деп.
В ВНИИНТПИ, 8.02.90
-
№10612. 7 с.
3.
М.Л. Джалилов. С.Ф. Эргашев. (2017).
Общее решение задачи для кусочно
-
однородной
двухслойной
среды
постоянной толщины. НТЖ ФерПИ ( STJ
FerPI), 21(4).
4.
Жалилов, М. Л., & Хаджиева, С. С. (2020,
November). Уравнения поперечного
колебания двухслойной вязкоупругой
пластинки постоянной толщины. In The
3rd International scientific and practical
conference “World science: problems,
prospects and innovations”(November
25-27, 2020) Perfect Publishing, Toronto,
Canada. 2020. 1082 p. (p. 478).
5.
Achenbach, J. D. (1969). An asymptotic
method to analyze the vibrations of an
elastic layer. pp. 37
–
46.
6.
Brunelle, E. J. (1970). The statics and
dynamics of a transversely isotropic
Timoshenko beam. Journal of Composite
Materials, 4(3), 404-416.
7.
Callahan, W. R. (1956). On the flexural
vibrations of circular and elliptical plates.
Quarterly of Applied Mathematics, 13(4),
371-380.
8.
Dong, S. B. (1966). Analysis of laminated
shells of revolution. Journal of the
Engineering Mechanics Division, 92(6),
135-155.
9.
Юлдашев, Б. С., Муминов, Р. А.,
Максудов, А. У., Умаралиев, Н., &
Джалилов, М. Л. (2020). Прогноз
природных катастроф
-
землетрясений,
методом
контроля
вариации
интенсивности потоков нейтронов
и
заряженных частиц. И прикладные
вопросы физики fundamental and
applied problems of physics, 125.
10.
Rakhimov, R. H., & Umaraliev Н, D. M.
(2018). Fluctuations of two-layer plates of
a constant thickness. Computational
Nanotechnology, (2), 2313.
11.
Рахимов, Р. Х., Умаралиев, Н., &
Джалилов, М. Л. (2018). Колебания
двухслойных
пластин
постоянной
толщины.
Computational
nanotechnology, (2), 52-67.
12.
Рахимов, Р. Х., Умаралиев, Н., Джалилов,
М. Л., & Максудов, А. У. (2018).
Регрессионные
модели
для
Volume 03 Issue 10-2023
189
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
10
Pages:
181-189
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
прогнозирования
землетрясений.
Computational nanotechnology, (2), 40-
45.
