Volume 02 Issue 04-2022
18
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
02
I
SSUE
04
Pages:
18-28
SJIF
I
MPACT
FACTOR
(2022:
6.
108
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
Oscar Publishing Services
Servi
ABSTRACT
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”.
KEYWORDS
Analysis, approximate, vibrations, two-layer plate, boundary value problem, stresses, deformation, oscillation
equations.
INTRODUCTION
The general equations of vibrations of piecewise
homogeneous viscoelastic plates of constant
thickness, described in [1-7], are complex in structure
and contain derivatives of any order in coordinates x,
y and time t, and therefore are not suitable for solving
applied problems and performing engineering
calculations. To solve applied problems, instead of
general equations, it is advisable to use approximate
ones, which include one or another finite order in
derivatives [8-11]. The classical equations for the
transverse oscillation of a plate contain derivatives of
no higher than the 4th order, and for piecewise
homogeneous or two-layer plates, the simplest
approximate oscillation equation is a sixth-order
equation. If in the operators (3.8) given in [12-17] we
Research Article
ANALYSIS OF THE GENERAL EQUATIONS OF THE TRANSVERSE
VIBRATION OF A PIECEWISE HOMOGENEOUS VISCOELASTIC PLATE
Submission Date:
April 18, 2022,
Accepted Date:
April 25, 2022,
Published Date:
April 29, 2022
Crossref doi:
https://doi.org/10.37547/ajast/Volume02Issue04-03
M.L. Djalilov
Phd, Associate Professor, Fergana Branch Of The Tashkent University Of Information Technologies Named
After Muhammad Al-Khwarizmi, Fergana, Uzbekistan
T. M. Sаbirjаnov
PhD, Associate Professor, Fergana Polytechnic Institute, Fergana, Uzbekistan
Journal
Website:
https://theusajournals.c
om/index.php/ajast
Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Volume 02 Issue 04-2022
19
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
02
I
SSUE
04
Pages:
18-28
SJIF
I
MPACT
FACTOR
(2022:
6.
108
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
Oscar Publishing Services
Servi
confine ourselves to the first two terms, then from
equation (3.11).
)
t
,
y
,
x
(
F
)
W
(
L
1
2
1
where the operators
1
L
and
)
t
,
y
,
x
(
F
1
are equal:
);
)(
(
)
)(
(
)
)(
(
)
)(
(
)
)(
(
)
)(
(
)
(
1
)
(
2
)
(
2
)
(
1
)
(
3
)
(
4
)
(
4
)
(
3
)
(
1
)
(
3
)
(
3
)
(
1
)
(
2
)
(
4
)
(
4
)
(
2
)
(
4
)
(
1
)
(
1
)
(
4
)
(
2
)
(
3
)
(
3
)
(
2
)
(
2
)
(
3
)
(
3
)
(
2
)
(
1
)
(
4
)
(
4
)
(
1
)
(
4
)
(
2
)
(
2
)
(
4
)
(
1
)
(
3
)
(
3
)
(
1
)
(
3
)
(
4
)
(
4
)
(
3
)
(
1
)
(
2
)
(
2
)
(
1
1
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
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
E
H
E
H
K
M
K
M
E
H
E
H
K
M
K
M
E
H
E
H
K
M
K
M
E
H
E
H
K
M
K
M
E
H
E
H
K
M
K
M
E
H
E
H
K
M
K
M
L
)
E
K
E
K
(
M
)
E
K
E
K
(
M
(
)}
y
f
x
f
(
M
)]{
E
H
E
H
(
M
)
E
H
E
H
(
M
)
E
H
E
H
(
M
[
)}
f
(
M
)]{
E
H
E
H
(
K
)
E
H
E
H
(
K
)
E
H
E
H
(
K
[
F
)
n
(
3
)
n
(
1
)
n
(
1
)
n
(
3
)
n
(
2
)
n
(
2
)
n
(
3
)
n
(
3
)
n
(
2
)
n
(
1
)
0
(
yz
)
0
(
xz
1
1
)
n
(
1
)
n
(
2
)
n
(
2
)
n
(
1
)
n
(
3
)
n
(
3
)
n
(
1
)
n
(
1
)
n
(
3
)
n
(
2
)
n
(
2
)
n
(
3
)
n
(
3
)
n
(
2
)
n
(
1
)
0
(
z
1
0
)
n
(
1
)
n
(
2
)
n
(
2
)
n
(
1
)
n
(
3
)
n
(
3
)
n
(
1
)
n
(
1
)
n
(
3
)
n
(
2
)
n
(
2
)
n
(
3
)
n
(
3
)
n
(
2
)
n
(
1
1
)};
y
f
x
f
(
M
)]{
H
K
H
K
(
M
)
H
K
H
K
(
M
)
H
K
H
K
(
M
(
)}
f
(
M
)]{
E
K
E
K
(
M
)
1
(
yz
)
1
(
xz
1
1
)
n
(
1
)
n
(
2
)
n
(
2
)
n
(
1
)
n
(
3
)
n
(
2
)
n
(
1
)
n
(
1
)
n
(
2
)
n
(
2
)
n
(
2
)
n
(
3
)
n
(
3
)
n
(
2
)
n
(
1
)
1
(
z
1
1
)
n
(
1
)
n
(
2
)
n
(
2
)
n
(
1
)
n
(
3
We obtain an approximate integrodifferential equation
).
t
,
y
,
x
(
F
W
Q
t
W
Q
t
W
Q
t
W
Q
W
Q
t
W
Q
t
W
Q
1
3
7
2
2
2
6
4
4
5
6
6
4
2
3
2
2
2
4
4
1
(1)
Volume 02 Issue 04-2022
20
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
02
I
SSUE
04
Pages:
18-28
SJIF
I
MPACT
FACTOR
(2022:
6.
108
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
Oscar Publishing Services
Servi
where the operators
j
Q
and
t
,
y
,
x
F
1
are equal:
;
h
h
M
Q
2
1
1
0
0
2
1
1
)));
D
h
D
h
(
h
h
h
(
1
P
h
h
D
h
D
P
h
2
(
M
2
Q
1
1
2
1
0
0
2
0
1
0
0
0
2
1
1
0
0
1
1
0
2
0
2
1
2
);
)(
(
0
2
1
0
1
2
1
1
2
1
0
2
2
0
2
3
D
P
h
h
2
D
h
D
h
D
P
h
1
P
4
Q
));
D
2
))(
h
4
h
(
h
h
3
(
M
h
)
D
2
))(
h
4
h
(
h
h
3
(
M
h
(
M
6
1
Q
1
0
0
1
1
1
1
2
0
2
0
1
1
1
2
1
0
1
1
0
0
0
0
2
1
2
1
1
0
0
2
0
2
1
4
))
)(
(
)
(
(
(
(
2
0
2
0
0
2
2
0
2
0
2
2
0
2
1
5
D
4
1
P
D
1
D
4
P
2
M
P
h
M
6
1
Q
));
D
4
(
D
M
h
2
))
D
4
(
D
2
)
D
3
4
)(
1
P
((
M
h
h
))
D
D
5
D
P
P
2
(
h
)
D
D
4
2
(
h
)(
M
M
2
(
h
h
P
2
))
(
M
)))
D
1
(
D
)
D
2
(
D
P
)
D
1
(
P
2
)(
1
P
(
)
D
D
P
(
4
(
M
M
(
h
h
6
))
D
2
(
D
)
1
P
(
)
1
D
4
D
4
(
2
(
M
h
1
0
2
1
2
1
2
1
0
1
0
2
2
0
2
1
2
0
2
1
1
1
2
2
2
1
2
0
0
2
0
1
1
1
0
1
0
1
0
2
2
1
2
0
1
1
0
1
0
1
2
0
2
2
1
0
2
2
1
1
1
0
1
0
2
1
2
0
1
1
2
1
2
1
2
1
2
1
4
1
(2)
Volume 02 Issue 04-2022
21
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
02
I
SSUE
04
Pages:
18-28
SJIF
I
MPACT
FACTOR
(2022:
6.
108
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
Oscar Publishing Services
Servi
))))));
D
1
(
D
P
)
D
1
)(
1
P
(
2
)(
h
h
(
M
4
))))
3
P
(
D
)
1
P
(
2
(
h
))
D
1
(
D
2
)
1
D
2
D
)(
1
P
((
h
2
(
M
(
P
h
h
2
)
M
))
D
D
P
6
)
1
P
(
D
D
6
)(
1
P
(
)
D
P
D
1
(
D
4
(
M
)))
D
D
2
D
2
(
P
)
D
1
(
D
)
1
P
(
2
)(
1
P
(
)
D
)
D
1
(
P
(
D
P
4
((
h
h
3
))
D
3
(
D
)
1
P
(
D
4
)
D
2
1
(
D
4
(
M
h
))
D
4
P
3
1
(
D
2
))
D
3
D
9
2
)(
1
P
((
P
2
(
M
P
h
(
M
3
1
Q
1
1
2
1
2
2
1
2
0
1
1
1
2
1
2
2
1
0
1
0
2
0
2
2
0
1
0
0
2
1
0
1
1
1
1
0
2
2
1
0
2
0
2
0
1
1
0
0
1
0
0
2
0
1
2
2
1
1
2
0
2
2
1
2
0
1
1
2
1
1
1
1
1
1
4
1
0
2
0
2
0
0
2
2
1
0
0
2
2
0
2
1
6
)));
D
)
1
P
(
)
1
P
(
2
(
h
)
D
2
)
1
P
(
h
(
D
P
h
h
4
)))
D
1
(
D
D
P
3
D
D
)
1
P
(
2
)((
1
P
(
D
D
P
8
(
h
h
3
))
1
P
(
D
4
(
D
h
)
1
P
(
5
D
4
(
D
P
h
(
3
2
Q
0
2
2
2
1
1
2
2
0
0
2
1
0
1
1
0
2
1
0
2
2
1
0
2
2
1
2
0
2
1
1
4
1
2
0
0
2
4
0
7
and
))
f
M
f
M
(
D
h
D
P
h
(
M
2
(
2
))
y
f
x
f
(
)
y
f
x
f
)((
D
h
D
h
(
))
y
f
x
f
(
h
)
y
f
x
f
(
h
)(
h
h
(
))
f
f
(
h
h
(
t
M
)
t
,
y
,
x
(
F
)
1
(
z
1
)
0
(
z
0
1
1
0
2
0
2
1
2
)
1
(
yz
2
2
)
1
(
xz
2
2
)
0
(
yz
2
2
)
0
(
xz
2
1
1
2
1
0
0
2
0
2
)
1
(
yz
2
2
)
1
(
xz
2
0
0
2
)
0
(
yz
2
2
)
0
(
xz
2
1
1
1
0
)
1
(
z
)
0
(
z
1
1
0
0
2
2
2
1
1
(3)
)).
y
f
x
f
(
)
y
f
x
f
)(
D
h
D
P
h
(
M
))
y
f
x
f
(
M
D
)
y
f
x
f
(
M
D
)(
h
h
P
2
2
)
1
(
yz
2
2
)
1
(
xz
2
2
)
0
(
yz
2
2
)
0
(
xz
2
1
2
1
0
2
2
0
1
1
2
)
1
(
yz
2
2
)
1
(
xz
2
1
1
1
2
)
0
(
yz
2
2
)
0
(
xz
2
1
0
0
1
0
2
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 dependences
.
D
D
;
C
C
;
h
h
;
1
P
;
M
M
;
N
N
1
0
1
0
1
0
2
1
0
1
0
Volume 02 Issue 04-2022
22
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
02
I
SSUE
04
Pages:
18-28
SJIF
I
MPACT
FACTOR
(2022:
6.
108
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
Oscar Publishing Services
Servi
and equation (1) becomes the equation
)))
y
f
x
f
h
)
f
((
M
D
4
))
y
f
x
f
(
h
)
f
((
t
M
(
h
1
)
W
)))(
(
4
)
D
4
)
)
(
D
3
((
6
h
)
)((
)
C
1
(
)
C
1
((
2
)
1
(
yz
2
2
)
1
(
xz
2
0
z
1
0
0
2
)
0
(
yz
2
2
)
0
(
xz
2
0
z
2
2
2
0
0
)
1
(
20
)
1
(
10
)
1
(
20
0
2
)
1
(
20
0
2
0
)
1
(
20
2
0
)
1
(
10
2
0
(4)
Where on the left is the product of two operators: the first describes the process of longitudinal oscillation, and the
second describes the transverse oscillation [18-27].
Similarly, an approximate equation is introduced from the general equation (1.3.12) given in [1] and we obtain for
x
V
y
U
1
1
),
t
,
y
,
x
(
F
)
x
V
y
U
)(
G
G
G
t
G
G
t
G
t
G
G
t
G
(
2
1
1
3
9
2
8
7
6
6
6
2
5
2
2
4
4
4
3
2
2
1
(5)
Where the operators
j
G
and
t
,
y
,
x
F
2
are equal:
;
h
h
M
G
1
1
0
0
1
1
1
;
h
P
h
G
1
2
0
2
);
M
)
h
3
h
(
h
M
)
h
3
h
(
h
(
M
6
1
G
1
1
1
0
0
1
1
2
1
1
0
0
1
1
0
0
2
0
2
1
3
Volume 02 Issue 04-2022
23
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
02
I
SSUE
04
Pages:
18-28
SJIF
I
MPACT
FACTOR
(2022:
6.
108
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
Oscar Publishing Services
Servi
)));
M
M
(
h
P
3
M
h
2
(
h
))
M
M
(
h
3
M
h
P
2
(
h
(
6
1
G
1
1
1
1
0
0
0
2
1
1
1
1
2
1
1
1
1
1
0
0
1
1
0
0
0
2
2
0
4
));
h
P
3
h
(
h
)
h
3
h
P
(
h
(
M
6
1
G
0
2
1
2
1
1
0
2
2
0
2
1
5
(6)
)));
D
D
3
3
(
M
P
h
)
D
D
3
3
(
M
h
(
M
M
h
h
5
)
M
M
10
(
M
h
)
M
M
10
(
M
P
h
(
120
1
G
2
1
1
1
1
1
2
3
1
2
0
0
1
0
0
3
0
1
1
1
0
1
0
1
0
1
1
1
1
0
0
1
1
1
5
1
1
0
0
1
1
1
2
0
2
0
2
5
0
6
)));
M
M
)
4
D
(
M
)
D
D
3
3
((
M
P
h
)
M
)
4
D
(
M
)
D
D
3
3
((
M
h
(
h
h
5
M
M
)
h
P
h
(
20
)
M
h
M
P
h
(
13
(
120
1
G
1
0
0
1
1
1
0
1
1
1
2
1
1
1
1
1
2
3
1
1
1
1
0
1
0
0
2
0
0
1
0
0
3
0
1
0
1
1
1
0
1
0
5
1
2
5
0
2
1
2
1
5
1
2
0
2
0
2
5
0
7
)));
M
)
4
D
(
M
(
h
)
M
)
4
D
(
M
(
h
(
h
h
5
)
M
h
M
P
h
(
10
)
M
h
M
P
h
(
23
(
120
1
G
1
1
1
1
1
0
0
4
1
1
0
0
0
1
1
1
3
0
1
0
1
0
0
5
1
1
1
1
2
5
0
2
1
2
1
5
1
1
0
2
0
2
5
0
8
)));
M
M
)
2
D
(
M
)
D
D
3
((
M
P
h
)
M
)
2
D
(
M
)
D
D
3
1
((
M
h
(
h
h
6
M
M
)
h
P
h
(
6
)
M
h
M
P
h
(
24
(
120
1
G
1
0
0
1
1
1
0
1
1
1
2
1
1
1
1
1
2
3
1
1
1
1
0
1
0
0
2
0
0
1
0
0
3
0
1
0
1
1
1
0
1
0
5
1
2
5
0
2
1
2
1
5
1
2
0
2
0
2
5
0
9
and
Volume 02 Issue 04-2022
24
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
02
I
SSUE
04
Pages:
18-28
SJIF
I
MPACT
FACTOR
(2022:
6.
108
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
Oscar Publishing Services
Servi
.
x
))
y
f
x
f
N
(
h
)
y
f
x
f
N
(
h
P
(
2
1
t
))
y
f
x
f
N
(
M
h
)
y
f
x
f
N
(
M
h
P
(
2
1
))
y
f
x
f
(
N
)
y
f
x
f
(
N
(
P
)
t
,
y
,
x
(
F
2
2
2
)
1
(
yz
2
2
)
1
(
xz
2
1
1
2
0
2
)
0
(
yz
2
2
)
0
(
xz
2
1
0
2
1
2
2
2
2
)
1
(
yz
2
2
)
1
(
xz
2
1
1
1
0
0
2
0
2
)
0
(
yz
2
2
)
0
(
xz
2
1
0
1
1
1
2
1
2
2
)
1
(
yz
2
2
)
1
(
xz
2
1
1
2
)
0
(
yz
2
2
)
0
(
xz
2
1
0
2
2
Approximate equation (1) is simplified in special cases when solving specific vibration problems [28-31]. For example,
operators (2) are greatly simplified when the Poisson's ratios of both components are constant, or when the thicknesses
of both components are equal, and so on [32-35].
For example, if
1
0
h
h
and
1
0
, operators in
j
Q
в (6) have the form:
;
h
M
Q
2
1
0
2
0
2
1
1
)));
(
D
2
)(
1
P
(
)
)(
1
P
(
D
2
(
h
M
2
Q
1
0
0
0
2
1
0
2
0
2
0
2
1
2
);
1
P
3
(
D
h
)
1
P
(
4
Q
2
0
2
0
2
3
)));
4
(
3
(
M
))
4
(
3
(
M
)(
D
2
(
h
M
6
1
Q
0
1
1
2
0
1
1
1
1
0
0
2
1
1
0
0
0
4
0
2
1
4
(7)
)
5
)
D
1
(
D
8
(
P
)
D
4
(
D
4
(
M
P
(
h
6
1
Q
0
0
2
0
0
2
0
2
0
2
4
0
5
Volume 02 Issue 04-2022
25
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
02
I
SSUE
04
Pages:
18-28
SJIF
I
MPACT
FACTOR
(2022:
6.
108
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
Oscar Publishing Services
Servi
)));
D
2
(
D
)
1
P
(
)
D
4
(
D
P
4
)
D
D
1
(
8
(
M
))
D
1
(
D
)
D
D
3
2
(
P
)
1
P
(
))
D
D
9
2
(
P
)
D
5
2
(
P
D
6
(
2
(
M
M
2
))
D
D
6
12
)(
1
P
(
0
0
2
0
0
2
2
0
0
2
1
2
1
0
0
2
0
0
2
2
2
0
0
2
0
2
2
0
1
1
1
0
1
0
2
0
0
2
)
D
13
D
8
20
(
P
)(
1
P
(
))
1
P
(
D
3
P
5
2
(
D
P
4
(
M
(
h
3
1
Q
2
0
0
2
2
2
0
2
0
2
1
0
0
2
0
6
))));
D
1
(
P
2
D
)(
1
P
(
17
)
D
5
P
2
4
(
P
4
)
D
4
(
4
(
D
M
)))
D
1
(
D
6
0
2
0
2
0
2
2
0
0
1
1
1
0
0
));
P
13
1
)(
1
P
(
)
P
5
P
15
4
(
D
(
D
h
3
4
Q
2
2
2
0
2
0
0
4
0
7
(7)
The sixth order operator in equation (1) can also be represented as a product of second and fourth-order operators if
the plate is elastic and the coefficients
j
Q
are related by
.
Q
Q
Q
Q
Q
Q
Q
Q
Q
6
4
3
7
5
1
7
4
2
For a two-layer elastic plate with given parameters of its components, relation (7) gives a 10th order algebraic equation
with respect to the ratio, while the sixth order operator in (1) can be represented as a product of two lower-order
operators
,
0
W
x
A
t
A
x
A
t
A
x
A
t
A
4
4
1
4
4
5
2
2
4
2
2
3
2
2
2
2
2
1
if the coefficients
j
Q
and
3
A
are related by dependencies
;
A
A
Q
;
A
A
A
A
Q
;
A
A
Q
4
2
3
3
2
4
1
2
2
1
1
;
A
A
Q
;
A
A
Q
;
A
A
Q
;
A
A
Q
6
2
7
6
1
6
5
2
5
5
1
4
Volume 02 Issue 04-2022
26
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
02
I
SSUE
04
Pages:
18-28
SJIF
I
MPACT
FACTOR
(2022:
6.
108
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
Oscar Publishing Services
Servi
Despite the fact that equation (1) is approximate, it is rather complicated. Operators (2) contain all the parameters and
operators that characterize both the mechanical and rheological properties of the material of a piecewise homogeneous
plate and its geometric dimensions.
CONCLUSION
The study of vibrations of piecewise-homogeneous
plates in an exact three-dimensional formulation
allows, without involving any hypotheses, to derive
the general and based on them approximate
equations for the vibrations of such plates.
It is shown that the simplest approximate equation
for the oscillation of a two-layer plate is the
equation of the sixth order in terms of derivatives,
which
describes
its
longitudinal-transverse
oscillation.
For an elastic two-layer plate, the sixth-order
operator decomposes into the product of the
operators of the second - longitudinal and fourth -
transverse oscillations, if the thicknesses of the
plate components satisfy the derived equation
containing the parameters of these components.
Formulas
were
obtained
for
determining
displacements and stresses through the desired
functions at any point of a two-layer plate.
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