Volume 03 Issue 10-2023
175
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
10
Pages:
175-180
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
A
BSTRACT
This article examines the effect of normal load on an infinite piecewise homogeneous two-layer plate when
the materials of the upper and lower layers of the plate are elastic. The transverse displacement of the
points of the contact plane of a two-layer plate is determined, satisfying the approximate equation obtained
in the work, replacing only the viscoelastic operators with elastic Lame coefficients, respectively. For a
rectangular infinite two-layer piecewise homogeneous plate under non-zero initial conditions, the
frequencies of natural oscillations are calculated, and an analytical solution to this problem is constructed.
The theoretical results obtained for solving dynamic problems of transverse vibration of piecewise
homogeneous two-layer plates of constant thickness, taking into account the elastic properties of their
material, make it possible to more accurately calculate the transverse displacement of the points of the
contact plane of the plates under normal external loads.
K
EYWORDS
Vibration equations, two-layer plate, displacement, elastic, viscoelastic, boundary conditions, initial
conditions, operator, Lame coefficients, differential equation.
I
NTRODUCTION
In real structures, the destruction of their
elements is usually accompanied by impact loads.
In this work, a solution is constructed on the
vibrations of an infinite two-layer plate under the
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
INFLUENCE OF SHOCK LOADING ON THE INFINITE
PIECEWISE-HOMOGENEOUS TWO-LAYER PLATE
Submission Date:
October 10, 2023,
Accepted Date:
October 15, 2023,
Published Date:
October 20, 2023
Crossref doi:
https://doi.org/10.37547/ijasr-03-10-29
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
176
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
10
Pages:
175-180
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
action of a normal load applied to the surface of a
two-layer plate [5-12]. The problem is reduced to
solving an approximate equation for the
transverse displacement W of points of the
contact plane of a two-layer plate of constant
thickness, obtained in [1] and [2].
(
)
(
)
(
)
4
2
6
2
1
2
3
4
4
4
6
4
2
2
3
5
6
7
4
2
W
W
W
Q
Q
Δ
Q Δ W
Q
t
t
t
W
W
Q
Δ
Q Δ
Q Δ W
F x, y, t
t
t
+
+
+
+
+
+
+
=
(1)
Here the coefficients
Q
j
are determined by the formula obtained in [2-4].
Assuming the load
F(x, y, t)
to be even in (x, y), the transverse displacement W will be sought in the form
of the Fourier integrals
0
0
0
W
W cos(kx) cos(qy)dkdq
=
(2)
Substituting (2) into equations (1), for
W
0
we obtain the ordinary differential equation
(
)
VI
IV
II
0
1
0
2
0
3
0
0
W
A W
A W
A W
F k, q, t ,
+
+
+
=
(3)
where the coefficients
A
j
and
F
0
(k, q, t)
are equal:
(
)
(
)
(
)
(
)
2
'
2
'
4
'
2
'
'
2
'
2
6
3
7
1
5
1
2
3
'
'
'
4
4
4
0
0
0
γ Q
γ Q
γ Q
γ Q
Q
γ Q
A
; A
; A
Q
Q
Q
F k, q, t
F x, y, t cos(kx) cos(qy)dxdy,
−
−
−
=
=
=
=
and the coefficients
Q
j
′
are determined by the formulas
(
)
(
)(
) (
) (
) (
)
(
)
(
)
(
)
2
'
2
1
2
'
2
2
2
2
0
1
2
0
1
'
2
3
2
2
0
1
2
0
Q
P 1 hρ ;
Q
2P (2 P D
hD
1 hρ
P 1 1 h
D
hD ρ );
Q
4 P
1 P D
h D
2hP D ;
=
+
= −
+
+
+
−
+
−
+
=
−
+
+
(
)
(
)
(
)
(
)
(
)
(
)
'
2
2
2
2
4
2
0
2
1
1
Q
P ( 3h ρ
1 4hρ
2 D
h P 3 hρ hρ 4
2 D );
6
= −
+ +
−
+ +
+
+
−
(4)
Volume 03 Issue 10-2023
177
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
10
Pages:
175-180
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
(
)
(
)
(
)
(
)
(
)
(
)
(
) (
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)(
)
(
)
(
)
'
2
5
2
2
0
0
2
0
2
2
2
2
1
1
2
1
1
2
2
2
0
1
2
2
0
2
1
0
2
2
1
0
2
2
0
0
2
2
2
2
1
1
1
2
0
2
1
0
2
0
1
1
Q
P ( 2P 4D 1 D
1
P
1 4 D
6
P h ρ
2 4D
4D 1
P 1 D 2 D
6h (ρ(4 P D
D
P 1 (2P 1 D
P D 2 D
D 1 D )) P 1 ρ )
2h(2P ρ 2 4D
D
h
2P
P D
5D
D )
P
1 4 3D
2D 4 D ) 2P hρ D
4 D );
= −
−
+ +
−
−
−
−
−
− −
−
−
+
+
+
+
−
−
−
−
+
+
+
+
+
+ +
+
−
−
−
−
+
−
+
−
−
+
+
−
+
−
(
) (
)
(
)
(
)
(
)
(
) (
)
(
)
(
)
(
)
(
) (
) (
)
(
)
(
)
(
)
(
)
(
)
(
) (
)
(
)
(
)
'
2
6
2
0
2
0
2
0
0
4
1
2
1
1
1
2
1
1
2
2
0
2
1
1
2
2
1
0
2
0
0
1
2
1
0
2
0
2
0
1
2
2
0
1
2
2
1
0
2
0
0
2
1
1
Q
P 2D 3P
4D
1
P
1 2 9D
3D
3
h P ρ 4D 1 2D
4D
P 1 D 3 D
3h (4P D
P 1 D
D
P
1 (2 P
1 D 1 D
P 2 D
4D D ) P ρ(4D 1 D
P D
P
1 6D D P
1
6P D
D ))
2hP (2 2D 1 2D
P
1 1 2D
D
h
2
=
−
− +
−
+
−
+
+
−
−
+
−
−
+
+
−
−
−
−
−
−
−
−
−
−
+
+
+
−
−
−
− −
+
+
+
+
+
−
+
−
+
+
(
)
(
)
(
)
(
)(
)
(
)
(
)
(
)
2
2
1
2
2
0
0
2
1
2
1
1
P
1
D P
3
4P ρD 1 h
2 P 1 1 D
P D
1 D
));
− +
+
−
+
−
−
+
+ +
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
) (
)
(
)
'
4
7
2
0
0
2
1
1
1
2
2
2
0
1
2
2
0
2
0
1
1
1
2
2
0
1
2
1
2
2
1
2
Q
(P D (4D
5 P
1
h D 4D
P
1
3
3h
8P D D
P
1 3P D
2P
1 D D
D 1 D
4hP D
2D
P
1
h
2 P
1
P
1 D ));
=
−
− +
−
−
−
−
+
−
−
+
−
−
−
−
+
−
+
− +
+
and
γ is determined by the formula
(
)
2
2
2
0
γ
h k
q ,
=
+
and immeasurable parameters were introduced:
(
)
(
)
0
0
1
1
2
0
1
0
0
1
1
0
1
b
μ
h
ρ
1
1
h
; ρ
; b
; P
; D
; D
.
h
ρ
b
μ
2 1 v
2 1 v
=
=
=
=
=
=
−
−
For
ξ
from equation (3) we obtain the frequency equation
6
4
2
1
2
3
ξ
A ξ
A ξ
A
0
+
+
+
=
(5)
frequency equation (5) has purely imaginary roots, i.e., frequencies of own fluctuations.
Then, the common decision of the homogeneous differential equation (4) is equal
Volume 03 Issue 10-2023
178
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
10
Pages:
175-180
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
og
1
1
2
1
3
2
4
2
5
3
6
3
W
C cos(ξ t) C sin(ξ t) C cos(ξ t) C sin(ξ t)
C cos(ξ t) C sin(ξ t).
=
+
+
+
+
+
+
(6)
Applying a method of a variation of any constants, for
C
j
′
we will receive:
(
)(
)
(
)(
)
(
)(
)
(
)(
)
(
)(
)
(
)(
)
'
1
0
1
2
2
2
2
1
1
2
1
3
'
2
0
1
2
2
2
2
1
1
2
1
3
'
3
0
2
2
2
2
2
2
1
2
1
3
'
4
0
2
2
2
2
2
2
1
2
1
3
'
5
0
3
2
2
2
2
3
2
3
1
3
'
6
0
3
2
2
2
2
3
2
3
1
3
1
C
F sin(ξ t);
ξ ξ
ξ
ξ
ξ
1
C
F cos(ξ t);
ξ ξ
ξ
ξ
ξ
1
C
F sin(ξ t);
ξ ξ
ξ
ξ
ξ
1
C
F cos(ξ t);
ξ ξ
ξ
ξ
ξ
1
C
F sin(ξ t);
ξ ξ
ξ
ξ
ξ
1
C
F cos(ξ t)
ξ ξ
ξ
ξ
ξ
=
−
−
= −
−
−
= −
−
−
=
−
−
=
−
−
= −
−
−
.
(7)
private decision of the differential equation (3) we will write down in a kind
(
)(
)(
)
(
)
(
)
(
)
(
)
(
)
(
)
2
2
t
2
3
0
1
2
2
2
2
2
2
0
1
1
2
2
3
3
1
2
2
t
3
1
0
2
0
2
2
2
t
1
2
0
3
0
3
ξ
ξ
1
W
{
F k, q, ζ sin[ξ t
ζ ]dζ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
F k, q, ζ sin[ξ
t ζ ]dζ
ξ
ξ
ξ
F k, q, ζ sin[ξ
t ζ ]dζ}.
ξ
−
=
−
+
−
−
−
−
+
−
+
−
+
−
(8)
Satisfying with a zero initial condition, i.e.,
2
5
0
0
0
0
2
5
W
W
W
W
...
0,
t
dt
t
=
=
= =
=
(9)
We find that
C
1
′
= C
2
′
=. . . = C
6
′
= 0.
then, the decision of a problem for displacement W looks like
Volume 03 Issue 10-2023
179
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
10
Pages:
175-180
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
(
)(
)(
)
(
)
(
)
(
)
(
)
(
)
(
)
2
2
2
3
2
2
2
2
2
2
0
0
1
1
2
2
3
3
1
t
0
1
0
2
2
t
3
1
0
2
0
2
2
2
t
1
2
0
3
0
3
ξ
ξ
cos(kx) cos(qy)
W
{
ξ
ξ
ξ
ξ
ξ
ξ
ξ
F k, q, ζ sin[ξ t
ζ ]dζ
ξ
ξ
F k, q, ζ sin[ξ
t ζ ]dζ
ξ
ξ
ξ
F k, q, ζ sin[ξ
t ζ ]dζ}dkdq
ξ
−
=
−
−
−
−
+
−
+
−
+
−
+
−
(10)
Let, if
(
)
( ) ( ) ( )
0
F x, y, t
σ δ x δ y δ z ,
=
Here
–
a
σ
0
constant of dimension of pressure;
δ(ζ)
- delta - function of Diraka.
Then, problem decisions will register in a kind
(
)(
)(
)
2
2
2
3
0
1
2
2
2
2
2
2
0
0
1
1
2
2
3
3
1
2
2
2
2
3
1
1
2
2
3
2
3
ξ
ξ
cos(kx) cos(qy)
W
σ
sin(ξ t
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
sin(ξ t)
sin(ξ t)]dkdq
ξ
ξ
−
=
+
−
−
−
−
−
+
+
+
(11)
C
ONCLUSIONS
From the analytical decision of a problem on influence of normal loading on a surface of a two-layer plate
follows that the deflection depends on geometrical and mechanical characteristics of a material of a plate,
and also allows to describe precisely enough tensely - the deformed status of a plate in its any point
eventually.
R
EFERENCES
1.
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).
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).
Volume 03 Issue 10-2023
180
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
10
Pages:
175-180
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
4.
Филиппов, И. Г., & Егорычев, О. А. (1983). Волновые процессы в линейных вязкоупругих
средах.
М.: Машиностроение
,
270
, 320.
5.
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
.
6.
Achenbach, J. D. (1969). An asymptotic method to analyze the vibrations of an elastic layer. pp. 37
–
46.
7.
Rakhimov, R. H., & Umaraliev Н, D. M. (2018). Fluctuations of two
-layer plates of a constant
thickness.
Computational Nanotechnology
, (2), 2313.
8.
Brunelle, E. J. (1970). The statics and dynamics of a transversely isotropic Timoshenko beam.
Journal
of Composite Materials
,
4
(3), 404-416.
9.
Callahan, W. R. (1956). On the flexural vibrations of circular and elliptical plates.
Quarterly of Applied
Mathematics
,
13
(4), 371-380.
10.
Dong, S. B. (1966). Analysis of laminated shells of revolution.
Journal of the Engineering Mechanics
Division
,
92
(6), 135-155.
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 прогнозирования землетрясений.
Computational nanotechnology
, (2), 40-45.
