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5TWO-DIMENSIONAL PROBLEM ON THE MOTION OF A STRUCTURE IN AN ELASTIC MEDIUM AT A WAVE PASSES THROUGH
IMAMALIYEV ASIM ELMAN Ph.D. student, Yildiz Teknik Universitesi (YTU), Yildiz Technical University (YTU), Turkey, Istanbul
Abstract. Modern constructions are becoming more and more complex, the intensity of machines and mechanisms on or near them is increasing. The economy of construction materials often leads to the reduction of the strength and rigidity properties of structures. With the introduction of new, not fully studied building materials into the construction practice, the probability of destruction of structures from vibrations and loss of stability increases. Since the requirements are often made to the structures to take into account the influence of vibrations on the exact technological processes, to ensure the strength and stiffness, the calculation of structures on the vibrations and stability is an important and urgent task. Vibrations of buildings, bridges, aircraft, etc., which cause additional stresses and deformations of structures, have a harmful effect on people. Continuous growth of traffic speeds, power of machines and equipment lead to an increase in the level of vibration of structures, increase the risk of their destruction. Intensive enough vibrations can lead to serious consequences. The study of vibrations is of great practical importance because it allows to use positive properties of vibrations in technics and technology, to avoid undesirable consequences by limiting their level.
Key words: strength, rigidity, vibration, structure, load, wave.
Dynamic loads cause in the elements of constructions considerable forces of inertia, arising as a result of rapid changes in time of their magnitude, direction or point of application. Consequently, there are vibrations of structures, which should be taken into account when calculating the structural elements of structures. Periodical repetition of dynamic influences leads to the accumulation of energy by a mechanical system, a gradual increase in the amplitude of the vibrations, leading to the phenomenon of resonance, at which a failure may occur from low-intensity influences.
The most important characteristics determining the behavior of buildings and structures under the action of external dynamic loads are the frequencies and forms of vibration. All methods of calculation for wind and seismic loads are based on the determination of these parameters.
A large number of problems on the diffraction of an elastic wave on a rigid inclusion has been studied, and the problems on the dynamics of a structure with an oscillator after the passage of the wave are less affected. In order to study the dynamics of complicated engineering systems during wave passage, we have to neglect the diffraction phenomenon and consider the latter as an initial transient process. In this paper, we consider a two-dimensional problem of circular structure motion with and without the presence of oscillating elements in elastic medium after the wave passing through.
In the two-dimensional case, the radial u and circular v displacements
1 5 >
A v =
at2 (1) 1 av
ь2 at2
as follows:
u=
v=
дф l дщ
Dr дщ Dr
+
r дв l дф
(2)
where: r - the distance of the particle from the pole; 9 - polar angle; t - time. E (1 -v)
a =
p(l - 2v)(l + v)
- velocity of propagation of longitudinal waves;
b2 =
E
- shear wave propagation velocity;
p(1 + v)
E - Young's module; v -Poisson's ratio; p- density.
In the case of planar deformation, the latter are related to the stress as follows:
= Q [ (1 ~v)£r + v^] °e= Q[(1 ~v)^e+v^r ]
= Gsr0
Q =
E
where:
or considering
G =
(l - 2v)(l + v)
E
l + v
Du Dr
l д v u
+ —
r 89 r
the expressions for the plane stresses will take the form:
„ „ ,8u v 8u u
°r = Q
(l - v)~~ +
в Q
G
+ v—
Dr r дв r _
l -vDu Du l -v + v--1--u
тв
r Dr Dr
l Du дв u +
r
(3)
r дв Dr r
To describe the boundary conditions of this problem, it is assumed that the particles adjacent to the inclusion move without separation. The pressure of the medium on a circular inclusion of radius ro of unit thickness is
2П
p = r0 J qd0
о
(4)
where: q - is the specific pressure, due to symmetry, considered in the direction of the initial displacement along the x-axis.
q = <Jr cos 6 + <Jre sin 6 (5)
The equations of motion of the circular inclusion of mass M1 and the spring stiffness L with it of mass M2 are
m
m
d x^ 1 ~d?~ d
2 IF
= p + L ( x2 - x J
— L(x2 x
The problem is solved taking into account oscillatory elements, at L^0. Solving equations
2 A 5 V A a Av--— = 0
V 5t2
U2K 5W A b Aw--— = 0
W 5t2
is written as
(6)
p — pl (r)cos^ ¥ = ¥1 (r)sin 0
(7)
Formulas (2) for displacements u and vtake the form
u = (p — — ¥\ ) cos 0 r
v = (щ[ — — p ) sin 0 r
(8)
After the Laplace-Carson transform of equations (1), the solutions for the external problem in images are
Pi = CK1
v\= DK1
where: k1 - is a first-order MacDonald function. Substituting (8) in relation (3) we obtain
v а у
fpr\
v b У
(9)
err
Q cos0
(1 -v)
d2p v dp 1 - 2v dyx 1 - 2v
dr2
+
r dr
r
dr
+
-¥1
r
v
- P1
(10)
ar0 d ¥ 2 dp 1 dp 2 ¥
-= —2---------2 p1 +—2
crsm0 dr r dr r dr r r
2
r
From (4), (5) and (10) by integrating it is possible to obtain
p = ^(а0 + ав >
(11)
where:
a0 =
r
a r cose a
_0 ^ гв
агв =
sin 9
Turning to the images and from (9) defining
r
dr d
dr
— c
ko ~ — K
\
P a
V Pr J
—c - 4)k -
P
r
a
ar
dr
— D
V
k 0 k1 pr
P Ь
(12)
d 2yx dr2
— D
^ - ^ - fko r b br
Given (12) in images (10) and (11), after some transformations (1 + v)P
Tir^E
6v — 2 1 1 — v P2
--7 +--r
V1 - 2v r2 1 - 2v a2 J
ki - 3 p ko
a2
Ci +
Vr2 " b2 J
ki - 3ko b
D (13)
Movement of medium particles adjacent to the circular inclusion
X = u cos9 + v sin 9
or taking into account (8)
X^ —
^ ^ cos2 в + (д^1 ^
dr
r
V
dr
r
sin 9
J
Take into account the independence of the displacement xi from the polar angle 9
dpi y/1 _ d^i
dr r
x,
dr r
whence, taking into account (12), it follows that
(14)
cbh,
V a J
= Dakn
v b j
(15)
x j —
Pko
v а у
К
v a у
Pro
k
v a у
Vo^ v b у
a
rakn
v b у
At L=0, the transformed equation (6) has the form
(16)
2 p p xi - — — Pxo
m
(17)
where: x 0 - velocity of particles of the passed wave.
Substituting in (17) the images of the corresponding values p from (13) and Xj from (16) it is obtain:
PKK
c — ■
v b у
A iko
'Pro
k0
v a у
Pr
b
A 2k 1
Vo
k0
v b у
v a у
PJ0
ь
A 3k0
v b у
v a у
PJ0
ь
(18)
v b у
where:
A i
A
A
P
A
p 2
P 2+6A
v
+ ■
r
A
o у
f
r 1 - 2v
6v - 2 1 - v ^ 2
-:---P
v ro
a
у
P A — + A
r
v ro
1_ En.
2 i2
A = nroE
M (1 + v)
In the case of
Pr
a
))1, then the MacDonald functions of zero and first genera will take the form
ж
ko(z) ~ k1(z) e-
and consequently, from (18) and (15) it follows that
Pxo
A1 A2 A3
2 Pr, ^
Ш
D
Px o
b 2 Pr, b
A1 —A2 —A3 a V Ж
(19)
(20)
c
Substituting (19) and (20) in (9) at r - r0 it is obtain
PI =
Pxo
Ai - Л2 - A3
(21)
¥ =
px o
Л - Л - Л а
(22)
Formula (21) taking into account the expressions^;; A2; As
ax0p
Pi =
P3
a + b l-v A
V r0
l - 2v a
p2 + 6Ap _ (6v- 2)a - 5 (l - 2v)b A
r
0
l - 2v
r0
(23)
Further it is necessary to investigate in formula (23) the denominator as a cubic polynomial. For a qualitative estimation and construction of the original function 91, however, we will limit ourselves to a practically possible special case.
To find the roots of the denominator in (23), consider the equation by first substituting expression A from (18).
P
a + b(l - u) 2 6ßb2 (6v- 2)a + 5 (l - v)b 6u—
r
-P + 2 roP
l - 2v
= 0
where / = — ; — = - the density of inclusion.
l
At V = — equation (24) will take the form:
+1 -u 2 6u 2 30u 3 ^
-— bp +-Çb p--f-b = 0
(24)
(25)
Assuming j = 0,568, equation (25) can be represented as
b
2,458-
p -1,546—
'0 j
r0 j
0
where the solutions and, hence, the original function are easily found 91
(26)
where
Pi =
ax„
3ß2
a = l,546— r0
ß = 2,458 —
ea+ß - 2e
(2a-ß) 2
sin
ß
6 2
j
(27)
The function y is similarly found. To find the displacements it is necessary to use the formulas
b
0
3
r
0
t
r,
0
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