CC BY
3Ishmamatov M.R., Kulmuratov N.R.
VIBRATIONS OF A CURVED VISCOELASTIC PIPE WITH
INTERNAL PRESSURE
Ishmamatov M.R. - Navoi State University of Mining and Technologies, Department of Higher Mathematics and Information technologies, Assistant Professor, Kulmuratov N.R. - Navoi State University of Mining and Technologies, Department of Technology Engineering, Assistant Professor
Abstract: The article presents a resolving system of equations of free oscillations of thin-walled multilayer curved composite viscoelastic pipes with internal pressure. For two types of boundary conditions, depending on internal pressure, geometric and structural factors, their proper oscillation is investigated. Taking into account the initial irregularities of the cross section and the rheological properties of the material, the dynamic state of a composite viscoelastic pipe under pressure is considered. The results of numerical analysis of the spectra of lower complex natural frequencies depending on internal pressure, geometric, structural factors, rheological properties and boundary conditions are presented.
Keywords: fluctuations, initial irregularities of the section, boundary conditions, flexibility, viscoelastic pipe, internal pressure.
Аннотация: В статье представлена разрешающая система уравнений свободных колебаний тонкостенных многослойных изогнутых композитных вязкоупругих труб с внутренним давлением. Для двух типов граничных условий, зависящих от внутреннего давления, геометрических и структурных факторов, исследуется их собственное колебание. С учетом начальных неровностей поперечного сечения и реологических свойств материала рассмотрено динамическое состояние композитной вязкоупругой трубы под давлением. Представлены результаты численного анализа спектров нижних комплексных собственных частот в зависимости от внутреннего давления, геометрических, структурных факторов, реологических свойств и граничных условий.
Ключевые слова: колебания, начальные неровности сечения, граничные условия, гибкость, вязкоупругая труба, внутреннее давление.
Introduction
The problem of vibrations is relevant not only for oil and gas pipelines, but also for aircraft. Pulsating flows and intense vibrations associated with them also occur in power plants [1]. Analysis of the behavior of pipelines in nuclear power plants showed [2,3] that vibrations are observed at frequencies from 0.5 to 1000 Hz, vibration amplitudes - 3-5 mm. The phenomena of pulsations in the coolant circulation circuit [4] and vibration of equipment acquire the greatest acuteness. There are cases of violation of the integrity of pipes. The nature of the fractures indicates the fatigue nature of the destruction. The origin and development of cracks is associated with elastic vibrations and hydraulic shocks. Thus, the problem of pipeline vibrations acquires an important national economic significance. When the frequencies of the excitation spectrum coincide with the natural frequencies of the pipeline, resonance develops in the system. Obviously, this approach is justified only for thick-walled structures. The development and creation of composite structures is a more complex problem, which is effectively solved only in an inseparable unity: material -construction -technology. The subject of design becomes the material itself, or rather its structure. The new material is designed taking into account technological capabilities for a given design and a given load. Only in such unity is it possible to realize the potential inherent in the composite. However, the method of calculating the dynamic parameters of multilayer pipes, taking into account the layered-fibrous structure and anisotropy of the material, is practically absent to date. The influence of inhomogeneity's in the structure of the material, as well as initial ovalities and different thicknesses of the cross-section on the stress state and dynamic properties of composite pipes has not been sufficiently studied.
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Problem statement and solution methods
Consider a composite viscoelastic tube whose centerline represents an arc of a circle of radius R with length L. The pipe has a cross-section with a nominal average radius r and a wall thickness h. Let us confine ourselves to thin-walled sufficiently long pipes of small curvature: The pipe is considered as an element of a pipeline conducting a liquid. The internal flow is considered homogeneous, the liquid is single-phase, ideal and incompressible. The end sections of the pipe are closed with absolutely rigid weightless flanges, which are supported by fixed hinge supports. The boundary conditions have the form: when and s=0 and s=L (0<s<L) at points with coordinates and displacements, to describe the viscoelastic properties of the body material, we adopt the linear hereditary Boltzmann-Voltaire theory, while we define the physical relations for the m-th viscoelastic element of the system in the form [5]
O (t) = \ W (t)Smk + 2 Jinm (t) (1)
Here is Voltaire integral operators, which are replaced by one operator below. The Poisson's ratio in the proposed formulation of the problem is assumed to be constant. This means that for a structurally inhomogeneous viscoelastic system, the forms of natural oscillations will be equal to the eigenvectors of the corresponding elastic problem [6,7].
Expressing by known formulas through, and considering that, instead of (1) we get
<k (t) =
1+ K
1 - 2v
(t )^mk + C (t)
(2)
where En is the Volterra operator having the following form:
En(t ) = E0n
t
(t)-{ REn (t-r(r)dr
(3)
here E0n-is the instantaneous modulus of elasticity, and here REn -is the relaxation core. Given (1), the function of time in equality (3) will be with <(t) = exp(-ict) a slowly
to
varying amplitude. Assuming the smallness of JR(r)dr the integral term, using the
0
freezing method [8] we replace the relation (3) with an approximate one:
En<(t) = Eoj [l - Tcn (cR) - iTsn (cR )<(t) = E<(t), (4)
where, rEc (oR ) = J RE (r) cos g)rtcIt,
n
n
0
0
to
r/ (coR) = Jre(r)sina)RTdr- respectively, the cosine and sine of the Fourier images of the
0
relaxation kernel of the material, cR - is the real value. As an example of a viscoelastic
material, we take a three Re (t ) = Ae~Pl t /11- - parameter relaxation core.
For the given boundary conditions, we present the following forms of motion [9]:
w
(5,ç,t) = !! wmn cos nç sin
mns L
S (s ,ç, t )=-±±1
. mns w sinnç sin-
mn
m=1 n=1 n L
u
( Л nr « « m (s t )=—!! — Wmn cos nç cos
m
L
n
mns L
(5)
Here and u, $ and w are the displacements of the points of the median surface of the shell in the axial, circumferential and radial directions. The wave numbers m and n characterize the shape of the oscillations: m is the number of half-waves in the axial direction, n - is the number of waves in the circumferential direction.
The kinetic energy of the pipe motion is determined by the following equation:
1 L 2n 1 « « n 2 + 1
K = -pTrhm J J(u2 + 32 + w2 )h(ç)dsdç = -(mT )!!(—— +
2 0 0 — m-1 n=1 n
n2 +1 m 2n2 r2
n4 L2
)w2
mn
(6)
The viscoelastic potential constructed on the basis of the semi-instant theory of multilayer thin shells and approximations (5) has the form:
1 L 2n
#= 2 Г J J fern*,2 + D 2 mk 2 )dsdç =
2 0 0
—ТУ « «
=nBrrL !!
4 m=1 n=1
! Ê( «2 -1)
mn 2 r n - 2 w m n-1 n + 2 w
(-Г~) -T Wmn +
m n+1
L n2 mn n -1 D n +1 D
+
22 w
mn
m =1 n =1
(7)
Using Lagrange equations:
d( $K ) + $(K-П-Ж) = ^
dt m
and dependences (2) - (10), we obtain a connected system of homogeneous differential equations with complex coefficients of the form
W{w}+Dm (1 -Гп К ))[ф} = 0 ,
(8)
where г«° (œR) = rnC ) + iTns (œR), D1m =
К
1m —--E02, e02 is the instantaneous modulus of
1 -V12V2!
elasticity, is a square diagonal matrix whose elements are determined by recurrent
2
formulas: ann = 2
n
n2 +1 m 2n2 r2
n4 L
, [c ]-is a square matrix whose elements are determined
by the following recurrent formulas :
cnn
+ß+1 ^(n2 _ i)(n2 -1+3pm ),
n
n
6
c
nn+1
-2 ßn
n + n + 1
mn 2( n +1)2
^11 = 2ßm , cnn+2 = -2ßm
(9)
2 ~ ^ n + 2n - 3
2n( n + 2)
Here
ßm = ff Y rR
and
pm =
3 D
are dimensionless parameters, D2m =
h
2 m
1-
"^21
02 ■
Analysis of the structure of the matrix [C] shows that the generalized coordinates are related to each other. The interaction of generalized coordinates is caused by elastic bonds, the intensity of which is characterized by non-diagonal elements of the matrix [C]
and depends on the length of the pipe, where L = * R is the * central angle. The
shorter the pipe, the greater the number of half-waves m on the segment L and the greater the curvature parameter r/R, the stronger their interaction. With an increase in the radius of curvature R, the interaction of generalized wmn coordinates weakens
Evaluation of the flexibility of curved viscoelastic composite pipes
Depending on the boundary conditions and geometric factors, we investigate he flexibility of samples of viscoelastic composite pipes. The geometric characteristics of the samples are given in Table 1.
Table 1. Geometric characteristics of samples
Sample Number R, mm r, mm h, mm % r/R h/r X
1 835 80 4,2 180° 1/10 1/20 0,50
2 1250 80 4,2 180° 1/15 1/20 0,70
The samples are made of Kevlar 49/PR-286 characteristics Ea = 64,10GPa, E
organoplasty
=
physical and
with
5,38GPa, Gap = 2,07 GPa ,va = 0,35
relaxation core parameters A = 0,048; j3x = 0,05; a = 0,10. The number of layers is six. Effective elastic constants with a wall, as a multilayer orthotropic body, depending on the reinforcement, are given in [10].
Conclusion
ID
3
r
m
1. Based on the approximate energy method, a resolving system of equations of free oscillations of thin-walled curved composite viscoelastic pipes with hinged supports is constructed.
2. The layered-fibrous STRUCTURE and anisotropy of the material are taken into account. In a particular case, the resolving equations describe the free oscillations of a cylindrical shell and a hinged straight-rod. The resulting equations correspond to well-known classical solutions.
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