Mathematical simulation of the stress-strain state of the winding of a closed magnetic system
Mathematical modeling of the stress-strain state of the winding of a closed magnetic system was carried out, which consists in the development of a three-dimensional geometric model of the helical winding and the determination of the values of the characteristics of the magnetic system, which are be...
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irk-123456789-1958742023-12-08T11:52:57Z Mathematical simulation of the stress-strain state of the winding of a closed magnetic system Martynov, S.O. Lukyanova, V.P. Prokhorets, S.I. Yurkin, A.Yu. Khazhmuradov, M.A. Computational and model systems Mathematical modeling of the stress-strain state of the winding of a closed magnetic system was carried out, which consists in the development of a three-dimensional geometric model of the helical winding and the determination of the values of the characteristics of the magnetic system, which are best from the point of view of meeting the requirements of the technical task of the designed object. Проведено математичне моделювання напружено-деформованого стану обмотки замкнутої магнітної системи, котре полягає в розробці тривимірної геометричної моделі гвинтової обмотки та визначенні значень характеристик магнітної системи, найкращих з точки зору задоволення вимог технічного завдання проектованого об’єкта. 2022 Article Mathematical simulation of the stress-strain state of the winding of a closed magnetic system / S.O. Martynov, V.P. Lukyanova, S.I. Prokhorets, A.Yu. Yurkin, M.A. Khazhmuradov // Problems of Atomic Science and Technology. — 2022. — № 5. — С. 119-123. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 52.55.Hc; 52.65.Kj DOI: https://doi.org/10.46813/2022-141-119 http://dspace.nbuv.gov.ua/handle/123456789/195874 en Problems of Atomic Science and Technology Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Computational and model systems Computational and model systems |
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Computational and model systems Computational and model systems Martynov, S.O. Lukyanova, V.P. Prokhorets, S.I. Yurkin, A.Yu. Khazhmuradov, M.A. Mathematical simulation of the stress-strain state of the winding of a closed magnetic system Problems of Atomic Science and Technology |
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Mathematical modeling of the stress-strain state of the winding of a closed magnetic system was carried out, which consists in the development of a three-dimensional geometric model of the helical winding and the determination of the values of the characteristics of the magnetic system, which are best from the point of view of meeting the requirements of the technical task of the designed object. |
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Article |
| author |
Martynov, S.O. Lukyanova, V.P. Prokhorets, S.I. Yurkin, A.Yu. Khazhmuradov, M.A. |
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Martynov, S.O. Lukyanova, V.P. Prokhorets, S.I. Yurkin, A.Yu. Khazhmuradov, M.A. |
| author_sort |
Martynov, S.O. |
| title |
Mathematical simulation of the stress-strain state of the winding of a closed magnetic system |
| title_short |
Mathematical simulation of the stress-strain state of the winding of a closed magnetic system |
| title_full |
Mathematical simulation of the stress-strain state of the winding of a closed magnetic system |
| title_fullStr |
Mathematical simulation of the stress-strain state of the winding of a closed magnetic system |
| title_full_unstemmed |
Mathematical simulation of the stress-strain state of the winding of a closed magnetic system |
| title_sort |
mathematical simulation of the stress-strain state of the winding of a closed magnetic system |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2022 |
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Computational and model systems |
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http://dspace.nbuv.gov.ua/handle/123456789/195874 |
| citation_txt |
Mathematical simulation of the stress-strain state of the winding of a closed magnetic system / S.O. Martynov, V.P. Lukyanova, S.I. Prokhorets, A.Yu. Yurkin, M.A. Khazhmuradov // Problems of Atomic Science and Technology. — 2022. — № 5. — С. 119-123. — Бібліогр.: 4 назв. — англ. |
| series |
Problems of Atomic Science and Technology |
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2025-07-17T00:08:23Z |
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2025-07-17T00:08:23Z |
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ISSN 1562-6016. Problems of Atomic Science and Тechnology. 2022. №5(141) 119
https://doi.org/10.46813/2022-141-119
MATHEMATICAL SIMULATION OF THE STRESS-STRAIN STATE
OF THE WINDING OF A CLOSED MAGNETIC SYSTEM
S.O. Martynov, V.P. Lukyanova, S.I. Prokhorets, A.Yu. Yurkin, M.A. Khazhmuradov
National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine
Mathematical modeling of the stress-strain state of the winding of a closed magnetic system was carried out,
which consists in the development of a three-dimensional geometric model of the helical winding and the determina-
tion of the values of the characteristics of the magnetic system, which are best from the point of view of meeting the
requirements of the technical task of the designed object.
PACS: 52.55.Hc; 52.65.Kj
INTRODUCTION
The most important property of the magnetic con-
figuration of the system designed to contain high-
temperature plasma is the presence of magnetic surfac-
es, which are formed by the trajectories of the lines of
force of the magnetic field at multiple rotations around
the main axis of the torus. Obtaining three-dimensional
geometric models for mathematical modeling of the
calculation of the stress-strain state of the winding of a
closed magnetic system is the most important and time-
consuming design task. Its solution is associated with
several simplifications and assumptions: the poles of
magnetic windings are modeled by infinitely thin con-
ductors with current; the task of determining the lines of
force is solved under the condition that several hundreds
of revolutions around the main axis of the torus are suf-
ficient; the traces of the lines of force of the magnetic
field in a fixed meridional section are located on a
closed curve; the increment of the length element along
the line of force of the magnetic field in the direction of
the magnetic induction vector, which ensures sufficient
accuracy of the estimation calculations, is taken from a
few millimeters to tens of millimeters. There are also
characteristics that, together with the boundary magnet-
ic surface, decisively affect the retention of the plasma,
these are: the magnitude of the angle of rotation trans-
formation; the rate of fall of the specific magnetic vol-
ume across the magnetic surfaces (magnetic pit); rate of
change of rotational transformation along the radius
(width); modulation of the magnetic field strength along
the force line. Searching for the optimal combination of
these parameters is the main task in mathematical mod-
eling and calculations of the stress-strain state (SSS) of
the winding of a closed magnetic system.
1. THREE-DIMENSIONAL GEOMETRIC
MODEL OF SCREW WINDING
The method of kinematic modeling was used to ob-
tain a three-dimensional geometric model. The essence
of the method is that for the assignment of the surface it
is necessary to describe its frame (creating and guiding
curve), a family of planes that determine the location of
sections and boundary conditions. The description of the
kinematic surface of the helical winding (HW) consists
of the description of the change in the shape of the
drawing curve and the description of the law of move-
ment of this curve in the plane and the law of the change
of the curve in space.
Coordinates of the winding line located on the sur-
face of the torus. Consider the torus (Fig. 1) with pa-
rameters R0 and (a0 + h/2), here h is the height of the
pole [1]. The medial line of a normal section is called
the line of its section with an "overblown" torus. Denote
through ;
і ;
toroidal coordinates of the
ends of the medial line. The dimensions of the pole are
determined by the following parameters – the height h
and the difference in coordinates i . The
following shows how to find the angular width of the
normal pole section, i.e. the difference, from these da-
ta .
Fig. 1. Determination of the length of the medial line:
1 – torus (R0, a); 2 – “overblown” torus (R0, a + h/2);
3 – medial line; 4 – normal section
Suppose that . The x, y, z coordi-
nates of any point of the "overblown" torus are deter-
mined by its angles and by formulas (1)–(3), if in
them the radius а is replaced by the radius (а + h/2):
x = [R0 + (а + h/2)
cos]cos;
y = [R0 + (а + h/2)
cos]sin; (1)
z = (а + h/2)
sin.
If the points with coordinates x, y, z are in the given
normal plane ( ), , which passes through the point M
with coordinates , , then the radius vector of this
point R, offset from the point, lies in the
plane ( ), and, therefore, is orthogonal to the previ-
ously introduced vector . Their scalar product is zero
y( ) 0x x y zR, R R R , (2)
where Rx = x – xм; Ry = y – yм; Rz = z – zм.
The points located on the medial line of the normal
section, by definition, lie on the "overblown" torus and
in the normal plane. Therefore, their coordinates, firstly,
can be written in the form of a system of equations (1),
and, secondly, they satisfy condition (2).
120 ISSN 1562-6016. Problems of Atomic Science and Тechnology. 2022. №5(141)
Then
[R0 + (а + h/2)cos]cosx + [R0 + (а + h/2)cos]siny + (а + h/2) sinz = xMx + yмy + zмz. (3)
The equation connects the angles and points of the medial line lying in the normal section, which is deter-
mined by the angles і . To find the derivative d/d, we differentiate equation (3) by , considering as a func-
tion of :
0 0
( /2)sin cos ( 2)sin sin ( /2)cos
( /2)cos sin ( /2)cos cos 0
x y z
x y
a h a h/ a h
d
R a h R a h .
d
Then
0
sin [ cos sin ] cos
( / 2).
[ ( / 2)cos ][ cos sin ]
x y x
y x
d
a h
d R a h
(4)
The length of the arc of the medial line is determined by the formula
2 2 2 2
2
0
2 2 2 2
0
2 2 2 2
0
[ ( / 2)sin cos
[ ( / 2)cos ]sin ] [ ( / 2)sin sin
[ ( / 2)cos ]cos ] ( / 2) cos
( / 2) [ ( / 2)cos ] ,
dS dx dy dz a h
R a h d a h d
R a h d a h d
a h d R a h d
from which
2
2 2
0( / 2) [ ( / 2)cos ] .
dS d
a h R a h
d d
(5)
Substituting the value of the derivative into expression (5)
d
d
we get the differential dependence between the
angle and the arc of the medial line S
2
1 1
.
/ 2
sin [ cos sin ] cos
1
cos sin
x y z
y x
d
dS a h
From here
2
0
sin ( cos sin ) cos 1
.
[ ( / 2)cos ]( cos sin ) sin ( cos sin ) cos
1
cos sin
x y z
y y
x y z
y z
d d d
dS d dS R a h
After performing the appropriate mathematical transformations, we obtain:
22
cos sin1
;
2
( cos sin ) sin [ cos sin ] cos
y x
y x x y z
d
dS a h/
(6)
220
sin ( cos sin ) cos1
.
( 2)cos
(cos sin ) sin [ cos sin ] cos
x y z
y x x y z
d
dS R a h/
(7)
Relations (6), (7) can be considered as a system of
two nonlinear differential equations of the first order.
Solving this system under initial conditions
, = for S = 0, we find dependencies = (S)
and = (S) and with their help we will find the desired
value of S from the condition + = (S).
You can greatly simplify the task if you use a raster
representation of the surface. In this case, the model of
the HW surface is presented in the form of a grid con-
sisting of characteristic intersecting lines belonging to
the surface. These lines are meridional cross-sections
and segments connecting characteristic points when
dividing the closed lines limiting the cross-sections.
Intersection points of closed curves, which limit the
cross-section and segments, form raster nodes, and a set
of such points on the modeled surface is a raster. If the
distance between the raster nodes is small, then the ras-
ter points describe the surface of the HW quite accurate-
ly. The three-dimensional geometric model of the HW
is shown in Figs. 2a, 2b.
ISSN 1562-6016. Problems of Atomic Science and Тechnology. 2022. №5(141) 121
Fig. 2a. Three-dimensional geometric model of the HW
(front view)
Fig. 2b. Three-dimensional geometric model of the HW
(arbitrary direction of gaze)
The formation of the complex surface of the HW is
carried out by a set of meridional sections (Fig. 3) [2, 3].
Infinitely thin conductors, considered as calculated, are
divided into 720 discrete currents of the element. The
length of each element ranges from 10 to 16 mm. The
division is carried out by a cutting plane = const.
Fig. 3. Formation of the surface of the HW by a set of
meridional sections
2. DETERMINATION OF THE CHARAC-
TERISTIC VALUES OF THE MAGNETIC
CONFIGURATION
Formulation of the problem. The optimization
problem of determining the values of the characteristics
of the magnetic configuration is a multi-criteria problem
of nonlinear programming due to the presence of five
initial parameters and is formulated as the problem of
determining the values of the characteristics of the field,
which are the best from the point of view of meeting the
requirements of the technical task with an unchanged
structure of the designed object. Among the initial pa-
rameters, we will single out the most critical (separate
criterion) – the topology of the magnetic field lines [4]
and, thereby, reduce the solution of the multi-criteria
problem to a single-criterion one, and the conditions for
the performance of the other initial parameters (rota-
tional transformation angle, width, magnetic pit, voltage
modulation along of the line of force of the magnetic
field [2]) can be attributed to the limitations of the prob-
lem, which are determined analytically.
To solve the problem of mathematical modeling of
SSS, we will introduce partial criteria for the optimality
of characteristic3
where k = 1, 2, 3, 4; i = 1, 2, 3 – main stresses
acting on the faces of the selected pole element of the
magnetic windings; – tangential stresses on a
selected element; – deformations in the direc-
tions of the axes, which are applied to the characteristic
points of the end elements; – angles of rota-
tion of the calculated points of the final elements in the
direction of the coordinate axes.
The solution to the task of determining the partial
optimality criteria of SSS characteristics is reduced to
the determination of parameters , which
are in the range of available values and at the same
time provide a minimum of all optimality criteria .
For optimality based on a set of criteria, we will intro-
duce a vector optimality criterion . Then the objec-
tive function can be written as
min
X D
Q X ,
where – vector of controlled param-
eters; – objective function or system performance
criterion.
Controlled parameters include:
– geometric characteristics of the torus;
– the law of winding poles on a toroidal surface
where – the angle characterizing
the position of the winding line point in the meridional
section; і – modulation coefficients; – the num-
ber of steps of the helical conductor along the length of
the torus; – the number of occurrences of helical con-
ductors;
– quantities included in the Biot-Savart law
where S – radius vector drawn from the element to
the point at which the magnetic field vector is calculat-
ed;
122 ISSN 1562-6016. Problems of Atomic Science and Тechnology. 2022. №5(141)
– a number of constants characterizing the properties
of the materials that make up the winding pole;
– the square of thecross section of winding poles.
The main requirement for the mathematical model of
the SSS is that the mathematical description should ac-
curately reflect the magnitudes of stresses and strains
occurring in the elements of the magnetic system. The
accuracy of the model is determined by the reliability of
the results obtained during the tests.
3. CALCULATION MODEL OF SSS
1. It is assumed that the currents flowing through the
conductors of the magnetic windings are concentrated in
an infinitely thin conductor located in the center of the
magnetic windings.
2. Infinitely thin conductors are divided into 720
discrete current cells. The length of each element varies
from 10 mm to 16 mm. The division is carried out by a
cutting plane (Fig. 4).
3. In the selected coordinate system, the coordinates
of the ends of the elements and the coordinates of the
midpoints of the segments of each discrete element are
calculated.
4. Using the analytical expression of the Biot-
Savart-Laplace law
3S
Sld
C
J
B
,
where
0S r r , radius vector drawn from the current
element Jdl to the point of observation, the vector of the
magnetic field B and forces acting on the middle of the
discrete element of each conductor of the helical wind-
ing is calculated.
Fig. 4. Vector representation of the calculation of forces
acting on discrete current elements
The results of the calculations of the distribution of
the magnetic field and the forces acting on the discrete
current elements (n is the number of the current ele-
ment) are shown in Figs. 5–9.
Fig. 5. Distribution of the modulus of the induction vec-
tor of the magnetic field Вn(1) along the length of the
1st half-pole
Fig. 6. Distribution of the modulus of forces dFn(1)
along the length of the 1st conductor of the 1st half-pole
Fig. 7. Distribution of the radial force component
dFn(1) along the length of the 1st conductor of the 1st
half-pole
Fig. 8. Distribution of the azimuthal component of the
force dFn(1) along the length of the 1st conductor of the
1st half-pole
Fig. 9. Distribution of the poloidal force component
dFn(1) along the length of the 1st conductor of the 1st
half-pole
CONCLUSIONS
Mathematical models and methods of determining
the stress-strain state of the winding of a closed magnet-
ic system have been developed. The practical value of
the obtained results is that the developed models and
methods can be used to calculate electrodynamic forces
in the conductors of the helical winding of a closed
magnetic system, which allows solving a number of
design, technological and operational tasks, namely: – at
which currents the forces reach extreme values; in
which deformations of the poles lead to distortions of
the geometry of the winding poles and how these per-
turbations in the geometry are reflected in the properties
of the magnetic field that hold the plasma.
REFERENCES
1. E.D. Volkov, V.A. Suprunenko, A.A. Shishkin.
Stellarator. Kyiv: “Nauk. Dumka”, 1983, 310 p. (in
Russian).
2. V. Kelton, A. Lowe. Simulation modeling. St.
Petersburg: BHV Petersburg; Kyiv: ed. BHV group.
2004, 847 p.
ISSN 1562-6016. Problems of Atomic Science and Тechnology. 2022. №5(141) 123
3. M.B. Shubin. A complex of programs for the
formation of surfaces. M.: CS AS USSR. 1979, 102 p.
4. D.I. Batishchev. Search methods for optimal de-
sign. M.: “Soviet radio”, 1975, 215 p.
Article received 19.08.2022
МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ НАПРУЖЕНО-ДЕФОРМОВАНОГО СТАНУ ОБМОТКИ
ЗАМКНУТОЇ МАГНІТНОЇ СИСТЕМИ
С.О. Мартинов, В.П. Лук’янова, С.І. Прохорець, А.Ю. Юркін, М.А. Хажмурадов
Проведено математичне моделювання напружено-деформованого стану обмотки замкнутої магнітної си-
стеми, котре полягає в розробці тривимірної геометричної моделі гвинтової обмотки та визначенні значень
характеристик магнітної системи, найкращих з точки зору задоволення вимог технічного завдання проекто-
ваного об'єкта.
|