Influencing of thin protective coatings on natural frequencies of radial oscillations of claddings of fuel rods of nuclear reactors
The natural frequencies and the modes of the radial oscillations are computed by using the method of grids for the cylindrical claddings made with the thin protective coatings for the fuel rods of the nuclear reactors. It is received the values more than 150 kHz for first natural frequencies of the...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2020 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2020
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Influencing of thin protective coatings on natural frequencies of radial oscillations of claddings of fuel rods of nuclear reactors / Yu.E. Mazurenko, Yu.V. Romashov, A.G. Mamalis // Problems of atomic science and tecnology. — 2020. — № 1. — С. 147-153. — Бібліогр.: 15 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859614187976654848 |
|---|---|
| author | Mazurenko, Yu.E. Romashov, Yu.V. Mamalis, A.G. |
| author_facet | Mazurenko, Yu.E. Romashov, Yu.V. Mamalis, A.G. |
| citation_txt | Influencing of thin protective coatings on natural frequencies of radial oscillations of claddings of fuel rods of nuclear reactors / Yu.E. Mazurenko, Yu.V. Romashov, A.G. Mamalis // Problems of atomic science and tecnology. — 2020. — № 1. — С. 147-153. — Бібліогр.: 15 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | The natural frequencies and the modes of the radial oscillations are computed by using the method of grids for the cylindrical claddings made with the thin protective coatings for the fuel rods of the nuclear reactors. It is received the values more than 150 kHz for first natural frequencies of the radial oscillations of the claddings of fuel rods of the WWER-1000 nuclear reactors. It is shown that the thin protective coatings lead to noticeable increasing of first natural oscillation frequency, but have negligible influencing on the second and higher natural frequencies as well as on the modes of the radial oscillations of the claddings of fuel rods.
Власні частоти і форми радіальних коливань розраховуються з використанням методу сіток для оболонки твелів ядерних реакторів ВВЕР-1000, виконаної з тонкими захисними покриттями. Отримано значення більше 150 кГц для перших власних частот радіальних коливань оболонки твелів ядерних реакторів ВВЕР-1000. Показано, що тонкі захисні покриття призводять до помітного збільшення першої частоти власних коливань, але мають незначний вплив на другу і більш високі частоти, а також на форми власних радіальних коливань оболонок твелів. Збільшення власних частот коливань циліндричних оболонок твелів за рахунок використання тонких захисних покриттів пояснюється істотним збільшенням радіальної жорсткості оболонок завдяки наявності окружних сил у покриттях при незначному збільшенні маси конструкції.
Собственные частоты и формы радиальных колебаний рассчитываются с использованием метода сеток для оболочки твэлов ядерных реакторов ВВЭР-1000, выполненной с тонкими защитными покрытиями. Получены значения более 150 кГц для первых собственных частот радиальных колебаний оболочки твэлов ядерных реакторов ВВЭР-1000. Показано, что тонкие защитные покрытия приводят к заметному увеличению первой частоты собственных колебаний, но оказывают незначительное влияние на вторую и более высокие частоты, а также на формы собственных радиальных колебаний оболочек твэлов. Увеличение собственных частот колебаний цилиндрических оболочек твэлов за счет использования тонких защитных покрытий объясняется значительным повышением радиальной жесткости оболочек благодаря наличию окружных сил в покрытиях при незначительном увеличении массы конструкции.
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| first_indexed | 2025-11-28T16:25:47Z |
| format | Article |
| fulltext |
ISSN 1562-6016. ВАНТ. 2020. №1(125) 147
UDC 621.039:539.3
INFLUENCING OF THIN PROTECTIVE COATINGS ON NATURAL
FREQUENCIES OF RADIAL OSCILLATIONS OF CLADDINGS OF FUEL
RODS OF NUCLEAR REACTORS
Yu.E. Mazurenko
1
, Yu.V. Romashov
1,2
, A.G. Mamalis
3
1
National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine;
2
V.N. Karazin Kharkiv National University, Kharkiv, Ukraine;
3
Project Center for Nanotechnology and Advanced Engineering, NCSR “Demokritos”,
Athens, Greece
E-mail: yu.v.romashov@gmail.com
The natural frequencies and the modes of the radial oscillations are computed by using the method of grids for
the cylindrical claddings made with the thin protective coatings for the fuel rods of the nuclear reactors. It is
received the values more than 150 kHz for first natural frequencies of the radial oscillations of the claddings of fuel
rods of the WWER-1000 nuclear reactors. It is shown that the thin protective coatings lead to noticeable increasing
of first natural oscillation frequency, but have negligible influencing on the second and higher natural frequencies as
well as on the modes of the radial oscillations of the claddings of fuel rods.
INTRODUCTION
The claddings of fuel rods of nuclear reactors have
the most worse operation conditions included the
extreme irradiations of different physical natures, the
chemical corrosive impacts from the surrounding
mediums, the noticeable mechanical loadings as well as
the significant heat flows, such that operability of the
cladding significantly limits the operational time of the
nuclear fuel assemblies in the core of a nuclear reactor,
as well-known [1]. At present, operational conditions of
the claddings of the fuel rods of the nuclear reactors are
corresponded to the limit possibilities of the known
modern wide-used structural materials and the general
problem about developing the structural materials for
claddings of fuel rods is of current interests in the
modern nuclear science and machinery [2] and using the
thin protective coatings is one of effective way to
improve the operability of the claddings [3, 4].
All the influencing factors of the cladding of the fuel
rods are naturally non-stationary and are significantly
depended on the time during operation of the fuel
assemblies in the core of the nuclear reactor. It is well-
known, that the non-stationary factors, influencing on
the cladding of fuel rods, can be represented by the
harmonic time dependencies included a lot of the
summands with the different frequencies. It is well-
known also, that the harmonic time-dependent loadings
can lead in some conditions to the more impacting on
the mechanical systems than the stationary loadings of
same intensities [5], as well as they can induce the
specific damaging impacts like the fretting [6]. Due to
these circumstances, the problems about oscillations of
the fuel rods in assemblies are of current interests, as
well as the theme of this research, which deals with the
natural frequencies and the modes of the radial
oscillations of the claddings of the fuel rods of the
WWER-1000 nuclear reactor. The purpose of this
research is to develop the approach for evaluating the
natural frequencies and the modes of the radial
oscillations for the fuel rod's cladding represented as the
thick-walled cylinder considering with presence the thin
protective coatings, as well as to obtain the quantitative
assessments of these natural frequencies and modes of
the radial oscillations for the fuel rod's cladding of the
WWER-1000 nuclear reactor. These radial oscillations
can have impacting on the width of the gap between the
cladding and the nuclear fuel pellets and can lead to
changes in the temperature state of the nuclear fuel
pellets and of the cladding of the fuel rod [1].
MODELLING THE RADIAL FREE
VIBRATIONS OF CYLINDRICAL
CLADDINGS OF FUEL RODS
CONSIDERING THIN PROTECTIVE
COATINGS
The cladding of fuel rods made as the long thick-
walled cylinder (Fig. 1) represents the typical design [1]
which is widely used in the most of nuclear reactors for
the power industry. The length L , the internal radius a ,
the external radius b of the typical cladding for fuel
rods are satisfied the conditions:
28
1 ab
ab
, (1)
bL . (2)
For example, the cladding of the fuel rods used in the
WWER-1000 nuclear reactors has the well-known sizes:
mm3800L , mm9.3a , mm55.4b [7], and it is
easy to verify that the conditions (1), (2) are satisfied
really.
Fig. 1. Typical design of the cylindrical cladding
of fuel rods for nuclear reactors
148 ISSN 1562-6016. ВАНТ. 2020. №1(125)
The inequality (1) represents the limiting condition
on the sizes for the thick-walled cylindrical structures
and it defines the application area for the equations of
the theory of elasticity [8]. The condition (2) defines the
area of application for the hypotheses of the plane strain
problem well-known in the theory of elasticity [9].
Thus, modeling of the radial free oscillations of the
typical cylindrical claddings of fuel rods for nuclear
reactors can be reduced to the plane strain problem of
the theory of elasticity for the cylinder with the side
surfaces unloaded and free from any fixings.
Due to the cylindrical shape of the cladding it is
suitable to use the cylindrical coordinates, including the
radial coordinate r , the circumferential coordinate
and the axial coordinate z with the corresponding unit
vectors re
, e
, and ze
as shown on the Fig. 2. The idea
of the plane strain is to consider the stress-strain state
far from the edges in the central cross-sections of the
cladding because such consideration allows neglecting
dependence of the stress-strain state on the axial
coordinate z due to the condition (2) and it simplifies
consideration the problem. Besides, the axial symmetry
of the cladding of fuel rods with unloaded side surfaces
leads to independence of the strain-stress state on the
circumferential coordinate and as the result it leads to
zero shear stresses and strains. Due to the hypotheses of
the plane strain, the stress-strain state of the cladding
under the radial axial symmetrical oscillations can be
represented using only the radial displacement u , which
is depending on the radial coordinate r and the time t
only:
truu , . (3)
It is well-known in the theory of elasticity [9, 10] that
the radial displacement (3) of the thick-walled cylinder
considering the plane strain hypotheses must satisfy the
differential equation:
bra
r
u
r
u
rr
u
t
u
E
,
11
22
2
2
22
, (4)
where is the density of the material;
21
E
E and
1
are the effective Young's modulus and the
Poisson's ratio defined corresponding the plane strain
hypothesis thru the values E and of the Young's
modulus and the Poisson's ratio of the material of the
cylinder representing the cladding of fuel rods.
The differential equation (4) must be considered
with the initial conditions, defining the state of the
cylinder at some given moment 0tt of the time t :
brarvtru
t
rutru
,,,, 0000 , (5)
where ru0 is the given radial displacement field and
rv0 is given the radial velocity field in the cylinder
representing the cladding of fuel rods at the initial
moment 0tt of the time.
The boundary conditions required for considering
the differential equation (4) must defining the states of
the cylinder representing the cladding on the internal
and external side boundary surfaces with coordinates
ar and br . As was discussed above, in the case
of the free oscillations of the cylinder representing the
cladding of fuel rods the side surfaces are unloaded and
free from any fixings. These types of the boundary
conditions for the cylinders representing the cladding of
fuel rods without the thin protective coatings is well-
known in the theory of elasticity and they are reduced to
the condition that the radial stress at the side surfaces
are zeroes [9, 10]. The boundary conditions for the
cylinder representing the cladding of fuel rods which
made with the protective thin coatings had been
discussed in the [11] and can be represented in the next
form:
ar
r
u
R
h
E
r
u
r
uE
a
a
a
,0
1 2
, (6)
br
r
u
R
h
E
r
u
r
uE
b
b
b
,0
1 2
, (7)
where aE , ah , and 2aa haR are the Young's
module, the thickness and the middle surface radius of
the internal coating; bE , bh , and 2bb hbR are the
Young's module, the thickness and the middle surface
radius of the external coating.
Fig. 2. The cylindrical cladding of fuel rods
and corresponded cylindrical coordinates
Summarizing, the mathematical model of the free
axial symmetrical radial oscillations of the cladding
with the thin protective coatings for fuel rods of nuclear
reactor is proposed in the form of the partial differential
equation (4) with the initial conditions (5) as well as the
boundary conditions (6) and (7).
FINDING THE NATURAL FREQUENCIES
The solution of the problem (4)–(7) about the free
radial oscillations of the cylinder representing the
cladding of fuel rods can be represented using the
imagine value 12 i in the form [10]:
tierUtru , , (8)
where rU is the mode of the oscillation; is the
cyclic frequency and is the initial phase of the
oscillation.
Substituting the solution of the form (8) into the
equation (4) and into the boundary conditions (6), (7)
allows to obtain the differential equation and the
ISSN 1562-6016. ВАНТ. 2020. №1(125) 149
boundary conditions for the mode of the oscillation
corresponded to the given frequency:
braU
r
U
dr
dU
rdr
Ud
,0
1
22
2
; (9)
ar
r
U
R
h
E
E
dr
dU
a
aa
,0
1 2
; (10)
br
r
U
R
h
E
E
dr
dU
b
bb
,0
1 2
, (11)
where 2
21
E
.
Further, the method of grids [12] will be used for
approximate solving the differential equation (9) with
the boundary conditions (10), (11). Corresponding the
idea of the method of grids, the solution represented by
continuous function rU will be represented by the
discrete nodal values of this function in the given nodes
(points) of the researched domain bra , which are
defined as (Fig. 3):
, , 0,1, 2, , , 1
1
k
b a
r a k r r k n n
n
, (12)
where n is the count of the nodes satisfied the condition
bra (“internal” nodes); r is the step, defined by
the distances between any two the nearest points.
Using the grid (12), it is possible to define formally
the unknown nodal values of the mode (see Fig. 3):
, 0,1, 2, , , 1k kU U r k n n . (13)
To finding the nodal values (13) it is used the finite
differences technique [12]; the follows finite differences
are used for the internal surface boundary node ( 0k ),
for the “internal” nodes ( 1, 2, ,k n ) and for the
external surface boundary node ( 1 nk ) [12]:
22100
2
43
ro
r
UUU
dr
dU
; 14)
2
2
11
2
2 2
ro
r
UUU
dr
Ud kkkk
;
nkro
r
UU
dr
dU kkk ,,2,1,
2
211
; (15)
2111
2
43
ro
r
UUU
dr
dU nnnn
. (16)
Leading to the method of grids [12], the derivative (14)
is substituted into the boundary condition (10), and the
derivatives (15) are substituted into the differential
equation (9), as well as the derivative (16) is substituted
into the boundary condition (11). As the results of these
substitutions, the next relations between the nodal
values (13) are obtained:
0201000 UUU ; (17)
1 1 0, 1, 2,...k k k k k k kU U U U k n ; (18)
011111 nnnnnn UUU , (19)
where k , k , and k are the values defines for all
numbers of 0,1, 2, , , 1k n n as follows:
rraRE
hE
ra a
aa
2
1
,
2
,
1
2
3
00
2
0 ;
2 2 2
1 1 2 1
;
2
k k
kr r r r r
;
2
1 1
; 1, 2, ,
2
k
k
k n
r r r
;
1 1
1 2
;
2
n n
r r
;
b
bb
n
bRE
hE
br
2
1
1
2
3
. (20)
Fig. 3. The cross section of the cladding, as well as the
grid nodes and the nodal values of the vibration mode
Using the relations (14) and (16) it is possible to
represent the nodal values 0U and 1nU thru some of
the “internal” nodal values:
2
0
0
1
0
0
0 UUU
,
n
n
n
n
n
n
n UUU
1
1
1
1
1
1
. (21)
Relations (21) allow excluding the nodal values 0U and
1nU from the relations (15) and allow representing
these relations (15) in the matrix-vector form as follows:
nnnn 0uIA , (22)
where nA is the some given matrix and nI is the unit
diagonal square matrix are with the size nn ; nu is
the nodal values vector and n0 is the zero vector are
with the size n .
The matrix nA and the vector nu from the relation
(22) are defined as:
nn
nnn
n
00000
0000
0000
0000
00000
111
333
222
11
A ,
Tnnn UUU 1u , (23)
where 1 , 1 , n , and n are the values defined
taking into account the relations (20) as follows:
1 1
1 1 0 1 1 0
0 0
;
;
1 1 1 0
1 1
;n n
n n n
n n
. (24)
150 ISSN 1562-6016. ВАНТ. 2020. №1(125)
Relation (21) represents the homogeneous linear
equations for defining the “internal” nodal values,
which allow defining the “boundary” nodal values by
the relations (21). The condition of existing of the non-
zero solution of the homogeneous linear equations (22)
defining the “internal” nodal values of the radial
oscillation mode has the follows form:
0det nn IA . (25)
The condition (25) represents the non-linear algebraic
equation for defining the parameters introduced in
the differential equation (9); the count of these
parameters is equal to the number n of the “internal”
nodes of the grid (12). By using these parameters
nk ,,,,, 21 , it is possible to define the natural
oscillation frequencies of the radial oscillations:
21 , 1, 2, ,k k
E
k n
. (26)
Solving the algebraic equation represented in the form
(25) is well-known as the eigenvalues problem [13]. It is
interesting that the matrix nA defined in the relations
(23) has the Hessenberg's form and the eigenvalue
problem for such matrices can be approximately solved
using the numerical QR-method directly without the
required transformations for the common form matrices
[13]. The well-known procedure HQR2 from the
handbook [13] is used to solve numerically the problem
(25) and to find approximately the eigenvalues and the
eigenvectors required for computing the values (25) of
the natural frequencies and the natural vibrations modes
of the radial vibrations of the cladding of fuel rods. All
necessary programs are developed using the FORTRAN
programming language which is very suitable for
scientific and engineering computing [14].
RESULTS FOR NATURAL OSCILATIONS
FREQUENCIES AND MODES
OF THE CLADDING OF FUEL RODS
The mathematical formulation (9)–(11) allow us to
consider the natural oscillations and modes of the radial
vibrations of the cylindrical claddings of fuel rods made
with and without the thin protective coatings. Really,
influencing the thin protective coatings on the radial
oscillations of the cladding is defined by the items with
the multipliers
a
aa
R
hE
and
b
bb
R
hE
presented in the
boundary conditions (10) and (11). The particular cases
for the zeroes values 0aahE and 0bbhE are
corresponded to the cladding without the internal and
external coatings. These circumstances allow us to use
the same computing software for evaluating the natural
oscillations frequencies and modes both for the
claddings with and without the thin protective coatings
by the necessary choices of the computing input data.
Thus, all possibilities are available for us to research
influencing the thin protective coatings on the natural
oscillations frequencies of the claddings of fuel rods of
nuclear reactors. Next, the quantitative estimations
about influencing the thin protective coatings on the
natural frequencies and the natural modes of the radial
oscillations of the claddings of fuel rods are presented
for the WWER-1000 nuclear reactors made without the
protective thin coatings and made with these coatings as
possible. It is considered the typical cladding of the fuel
rods of the WWER-1000 nuclear reactor with the next
parameters:
3.855 mm; 4.55 mma b ;
396 GPa; 0.33; 6500kg mE . (27)
Influencing on natural oscillations frequencies and
modes of the possible thin protective coatings made
from the stainless steel like discussed in [4] with the
next value of the Young's modulus:
GPa210 ba EE . (28)
Comparison between the natural oscillation frequencies
and the modes for the cladding of fuel rods without the
thin protective coatings and with these coatings of
different thicknesses ah and bh is the methodology
basis for estimating the influence of the protective thin
protective coatings on the oscillation characteristics of
the claddings.
Using the approximate numerical solutions of the
eigenvalues and eigenvectors problem (25) to evaluate
the natural oscillations frequencies (26) and the modes
of radial oscillations of the cladding of fuel rods
requires substantiating the accuracy of obtained results.
The accuracy of the obtained results for the natural
oscillations frequencies and the modes depends on the
count n of the “internal” nodes of the grid (see Fig. 3).
Increasing the count n of the grid nodes leads to
increasing the accuracy of the numerical solutions due
to decreasing the approximations errors in the used
finite differences (14)–(16) taking into account
decreasing the grid step (12) with increasing the nodes
number. Due to this depending, substantiating the
accuracy of the numerical solutions of the eigenvalues
and eigenvectors problem (25) is reduced to
substantiating the number n of the grid nodes providing
the required accuracy of the results for the natural
oscillations frequencies and modes of the cladding of
fuel rods. Thus, the accuracy of the approximate
numerical solutions of the problem (25) can be
estimated by comparing the results obtained by using
the different number n of the grid nodes. This
comparing (Table) shows that the results with 500n
have the error about %100.2 5 , and it is possible to
use these results in the further analyses.
To represent the results for the natural frequencies
(26) it is used the next values of frequencies:
, 1, 2, , .
2
k
k k n
(29)
The results were obtained for fist natural frequencies of
the radial oscillations of the claddings made with the
different thin protective coatings are presented on the
Fig. 4. It is obtained the large value about 150 kHz of
first natural frequency of radial oscillations for the
cladding without protective coatings. The protective
thin coatings lead to noticeable increasing the value of
first natural frequency of the radial oscillations of the
cladding of fuel rods. It is shown that the internal
coating has the more effect on first natural frequency
than the external coating, but effect of presence the both
internal and external coatings is approximately equals to
ISSN 1562-6016. ВАНТ. 2020. №1(125) 151
superposition of the separate effects from the internal
and external coatings.
Convergence of the results for the natural oscillation
frequencies with increasing the grid nodes count
Count of the
nodes, n
Natural oscillation frequencies, Hz
0 ba hh ha = hb = 100 m
3 154187.1211 193572.9636
10 154608.6745 193873.0273
500 154703.0281 193939.6472
1000 154703.0681 193939.6752
Continuation of the Table
Count of the
nodes, n
Natural oscillation frequencies, Hz
ha = 100 m hb = 100 m
3 178571.7676 171302.5818
10 178904.0075 171671.3978
500 178978.2469 171753.5551
1000 178978.2763 171753.5910
0**;0* ab hh .
Fig. 4. Influencing the thicknesses h of the coatings on
the first natural frequency 1μ of the radial oscillations
of the cylindrical cladding of fuel rods
Let denote 0 , 1, 2, ,k k n the natural
frequencies (29) of the radial oscillations of the cladding
of fuel rods with design parameters (27) made without
the thin protective coatings. To estimate and to
represent the results for influencing the thin protective
coatings on the higher natural frequencies of the radial
oscillations of the cladding of fuel rods there are used
the values of percentile increasing of the frequencies of
the claddings with thin coatings comparing with the
cladding without the coatings, which are defined as:
0
0
100%, 1,2, , .k k
k
k
k n
(30)
The results of comparing for influencing the thin
coatings on first and some higher natural frequencies of
the radial oscillations of the cladding of fuel rods are
presented on the Fig. 5 using the logarithm coordinates.
Due to these results, it is seen (see Fig. 5) that the thin
protective coatings are having significant influencing on
first natural oscillation frequency only, but influencing
on second frequency is ten times smaller than for first
frequency and influencing on third frequency is about
hundred times smaller than for first frequency.
Let denote as
, 1, 2, ,
k
n k nu the vectors
representing the solutions of the next homogeneous
linear equations:
, 1, 2, ,
k
n k n n n k n A I u 0 . (31)
Equations (31) are the equations (22) with substituted
values nk ,,,,, 21 which are the solutions of
the equation (25). Due to this circumstance, the linear
systems (31) are having the nonzero solutions such that
the components of any vector
, 1, 2, ,
k
n k nu are
defined thru any one of their component. It is suitable to
normalize the vectors
, 1, 2, ,
k
n k nu such as the
absolute maximum component will be equaled to unit.
These vectors
, 1, 2, ,
k
n k nu are representing the
nodal values of the modes , 1, 2, ,
k
U r k n of
the natural radial oscillations at the grid (12) (see Fig.
3). Each of these natural oscillation modes corresponds
to one of the natural oscillation frequencies (26) or (29).
a
b
c
Fig. 5. Influencing the thickness of the coatings on the
natural oscillation frequencies of the cladding of fuel
rods with the outer (a) and inner coatings (b) only as
well as both the inner and outer coatings with the equal
thicknesses (c)
h, m
h, m
ha, m
hb, m
152 ISSN 1562-6016. ВАНТ. 2020. №1(125)
Obtained results allow us to conclude that the thin
protective coatings are having no noticeable influencing
on the natural radial oscillation modes of the cylindrical
cladding of fuel rods with design parameters (27). The
results for some of the modes of the natural radial
oscillations of the cladding of fuel rods are shown on
the Fig. 6. The presented results are approved with the
well-known fundamental properties [15] of the natural
oscillations modes. Really, the each of modes has some
number of the crossings with horizontal zero axis: the
mode corresponded to first natural frequency has no
crossing with zero axis, the mode corresponded to
second natural frequency has one crossing with zero
axis, the mode corresponded to third natural frequency
has two crossing with zero axis and so on (see Fig. 6).
Fig. 6. The modes of the natural radial oscillations of
the cladding of fuel rods corresponded to the
frequencies with the numbers k :
1 – k = 1; 2 – k = 2; 3 – k = 3; 4 – k = 4
DISCUSSION THE RESULTS
It is obtained the large values about 150 kHz for first
natural frequency of the radial oscillations of the typical
design of cylindrical cladding of fuel rods. It is well-
known in the theory of vibrations [13, 14] that the
natural frequencies of oscillations of the structure are
defined by relation between the rigidness and the mass
of this structure. It is well-known that the elastic
cylinder representing the cladding of fuel rods has the
high rigidness on the radial direction and it is this high
rigidness is the reason for using the claddings with the
cylindrical shape, because due to this shape the cladding
with small wall thickness has no noticeable strains
under the operational pressures from the gaseous fission
products and the moving heat carrier. At the same time,
the small thickness of the wall leads to the smaller mass
of the cladding. Thus, the large values of the natural
frequencies are due to the well-known high rigidness on
the radial direction of the elastic cylinder representing
the cladding of fuel rods with the small thickness of the
wall. It is necessary to notice that the model of the thin
protective coatings used to formulate the boundary
conditions (22), (23) has no considering the inertia of
the coating, but considering only the rigidness on the
coatings. It seems that the inertia of the claddings is
negligible due to the significantly smaller masses, but
such neglecting the inertia of the claddings must be
substantiated by quantitative results for the natural
frequencies considering the inertia of the coatings in
further researches.
Obtained numerical results allow us to approve that
the thin protective coatings lead to increasing the natural
frequencies of the radial oscillations of the typical
cylindrical claddings of fuel rods. Increasing first
natural frequency of radial oscillations for the typical
cylindrical cladding of fuel rods due to using the thin
coatings is really noticeable, but increasing the higher
natural frequencies is practically negligible comparing
with increasing of first frequency. Increasing the natural
frequencies due to using the thin protective coating can
be explained by increasing the radial rigidness of the
cladding through presence the circumferential forces in
the coatings as it seen from the boundary conditions (6),
(7) or (10), (11) considering the boundary surfaces of
the cladding of fuel rods with the thin protective
coatings. At the same time, the inertia of the coating is
neglected in the boundary conditions (6), (7) modelling
of the thin coatings and it is required the additional
researches.
CONCLUSIONS
The natural oscillation frequencies of the radial
vibrations of the cladding made with the thin protective
coatings for the fuel rods of the WWER-1000 nuclear
reactors are computed by using the method of grids. It is
received the values about 150 kHz for first natural
oscillation frequencies of the radial vibrations of the
cladding for the fuel rods of the WWER-1000 nuclear
reactors. It is shown that the thin protective coatings
lead to noticeable increasing of first natural oscillation
frequency, but have negligible influencing on the
second and higher natural oscillation frequencies as well
on the natural modes of the radial oscillations of the
typical cylindrical cladding of fuel rods.
The mathematical model proposed for the thin
protective coatings and used for formulating the
boundary conditions considering with influence of the
thin coatings is not took into account the inertia of the
coating, but it is took into account only the rigidness on
the coatings. It is necessary to substantiate neglecting
the inertia of the coatings in further researches, although
it is seem that the coatings has really the significantly
smaller masses comparing with the mass of the
cladding. Besides, it is interesting to estimate the
mechanical stresses in the cladding of fuel rods
occurring due to the time harmonics of the outer
pressure of the moving heat carrier with the frequencies
proportional the rotation velocity of the main circulation
pump.
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Статья поступила в редакцию 04.11.2019 г.
ВЛИЯНИЕ ТОНКИХ ЗАЩИТНЫХ ПОКРЫТИЙ НА СОБСТВЕННЫЕ ЧАСТОТЫ
РАДИАЛЬНЫХ КОЛЕБАНИЙ ОБОЛОЧЕК ТВЭЛОВ ЯДЕРНЫХ РЕАКТОРОВ
Ю.Е. Мазуренко, Ю.В. Ромашов, А.Г. Мамалис
Собственные частоты и формы радиальных колебаний рассчитываются с использованием метода сеток
для оболочки твэлов ядерных реакторов ВВЭР-1000, выполненной с тонкими защитными покрытиями.
Получены значения более 150 кГц для первых собственных частот радиальных колебаний оболочки твэлов
ядерных реакторов ВВЭР-1000. Показано, что тонкие защитные покрытия приводят к заметному
увеличению первой частоты собственных колебаний, но оказывают незначительное влияние на вторую и
более высокие частоты, а также на формы собственных радиальных колебаний оболочек твэлов. Увеличение
собственных частот колебаний цилиндрических оболочек твэлов за счет использования тонких защитных
покрытий объясняется значительным повышением радиальной жесткости оболочек благодаря наличию
окружных сил в покрытиях при незначительном увеличении массы конструкции.
ВПЛИВ ТОНКИХ ЗАХИСНИХ ПОКРИТТІВ НА ВЛАСНІ ЧАСТОТИ РАДІАЛЬНИХ
КОЛИВАНЬ ОБОЛОНОК ТВЕЛІВ ЯДЕРНИХ РЕАКТОРІВ
Ю.Є. Мазуренко, Ю.В. Ромашов, А.Г. Мамаліс
Власні частоти і форми радіальних коливань розраховуються з використанням методу сіток для
оболонки твелів ядерних реакторів ВВЕР-1000, виконаної з тонкими захисними покриттями. Отримано
значення більше 150 кГц для перших власних частот радіальних коливань оболонки твелів ядерних
реакторів ВВЕР-1000. Показано, що тонкі захисні покриття призводять до помітного збільшення першої
частоти власних коливань, але мають незначний вплив на другу і більш високі частоти, а також на форми
власних радіальних коливань оболонок твелів. Збільшення власних частот коливань циліндричних оболонок
твелів за рахунок використання тонких захисних покриттів пояснюється істотним збільшенням радіальної
жорсткості оболонок завдяки наявності окружних сил у покриттях при незначному збільшенні маси
конструкції.
|
| id | nasplib_isofts_kiev_ua-123456789-194749 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-28T16:25:47Z |
| publishDate | 2020 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Mazurenko, Yu.E. Romashov, Yu.V. Mamalis, A.G. 2023-11-29T14:33:13Z 2023-11-29T14:33:13Z 2020 Influencing of thin protective coatings on natural frequencies of radial oscillations of claddings of fuel rods of nuclear reactors / Yu.E. Mazurenko, Yu.V. Romashov, A.G. Mamalis // Problems of atomic science and tecnology. — 2020. — № 1. — С. 147-153. — Бібліогр.: 15 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/194749 621.039:539.3 The natural frequencies and the modes of the radial oscillations are computed by using the method of grids for the cylindrical claddings made with the thin protective coatings for the fuel rods of the nuclear reactors. It is received the values more than 150 kHz for first natural frequencies of the radial oscillations of the claddings of fuel rods of the WWER-1000 nuclear reactors. It is shown that the thin protective coatings lead to noticeable increasing of first natural oscillation frequency, but have negligible influencing on the second and higher natural frequencies as well as on the modes of the radial oscillations of the claddings of fuel rods. Власні частоти і форми радіальних коливань розраховуються з використанням методу сіток для оболонки твелів ядерних реакторів ВВЕР-1000, виконаної з тонкими захисними покриттями. Отримано значення більше 150 кГц для перших власних частот радіальних коливань оболонки твелів ядерних реакторів ВВЕР-1000. Показано, що тонкі захисні покриття призводять до помітного збільшення першої частоти власних коливань, але мають незначний вплив на другу і більш високі частоти, а також на форми власних радіальних коливань оболонок твелів. Збільшення власних частот коливань циліндричних оболонок твелів за рахунок використання тонких захисних покриттів пояснюється істотним збільшенням радіальної жорсткості оболонок завдяки наявності окружних сил у покриттях при незначному збільшенні маси конструкції. Собственные частоты и формы радиальных колебаний рассчитываются с использованием метода сеток для оболочки твэлов ядерных реакторов ВВЭР-1000, выполненной с тонкими защитными покрытиями. Получены значения более 150 кГц для первых собственных частот радиальных колебаний оболочки твэлов ядерных реакторов ВВЭР-1000. Показано, что тонкие защитные покрытия приводят к заметному увеличению первой частоты собственных колебаний, но оказывают незначительное влияние на вторую и более высокие частоты, а также на формы собственных радиальных колебаний оболочек твэлов. Увеличение собственных частот колебаний цилиндрических оболочек твэлов за счет использования тонких защитных покрытий объясняется значительным повышением радиальной жесткости оболочек благодаря наличию окружных сил в покрытиях при незначительном увеличении массы конструкции. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Physics and the technology of construction materials Influencing of thin protective coatings on natural frequencies of radial oscillations of claddings of fuel rods of nuclear reactors Вплив тонких захисних покриттів на власні частоти радіальних коливань оболонок твелів ядерних реакторів Влияние тонких защитных покрытий на собственные частоты радиальных колебаний оболочек твэлов ядерных реакторов Article published earlier |
| spellingShingle | Influencing of thin protective coatings on natural frequencies of radial oscillations of claddings of fuel rods of nuclear reactors Mazurenko, Yu.E. Romashov, Yu.V. Mamalis, A.G. Physics and the technology of construction materials |
| title | Influencing of thin protective coatings on natural frequencies of radial oscillations of claddings of fuel rods of nuclear reactors |
| title_alt | Вплив тонких захисних покриттів на власні частоти радіальних коливань оболонок твелів ядерних реакторів Влияние тонких защитных покрытий на собственные частоты радиальных колебаний оболочек твэлов ядерных реакторов |
| title_full | Influencing of thin protective coatings on natural frequencies of radial oscillations of claddings of fuel rods of nuclear reactors |
| title_fullStr | Influencing of thin protective coatings on natural frequencies of radial oscillations of claddings of fuel rods of nuclear reactors |
| title_full_unstemmed | Influencing of thin protective coatings on natural frequencies of radial oscillations of claddings of fuel rods of nuclear reactors |
| title_short | Influencing of thin protective coatings on natural frequencies of radial oscillations of claddings of fuel rods of nuclear reactors |
| title_sort | influencing of thin protective coatings on natural frequencies of radial oscillations of claddings of fuel rods of nuclear reactors |
| topic | Physics and the technology of construction materials |
| topic_facet | Physics and the technology of construction materials |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/194749 |
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