Influence of the End Body Dynamics on Stabilization Processes in the Relative Motion of a Space Tethered System Stabilized by Rotation
Introduction. Space tethered systems (STS) stabilized by rotation is a quite interesting and promising direction in the field of cosmonautics. Such systems are intended for solving a wide range of scientific and research tasks, in particular, those that cannot be solved effectively with the help o...
Збережено в:
| Дата: | 2019 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Видавничий дім "Академперіодика" НАН України
2019
|
| Назва видання: | Наука та інновації |
| Теми: | |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/173961 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Influence of the End Body Dynamics on Stabilization Processes in the Relative Motion of a Space Tethered System Stabilized by Rotation / O.L. Volosheniuk // Наука та інновації. — 2019. — Т. 15, № 2. — С. 17-24. — Бібліогр.: 9 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-173961 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1739612020-12-28T01:26:12Z Influence of the End Body Dynamics on Stabilization Processes in the Relative Motion of a Space Tethered System Stabilized by Rotation Volosheniuk, O.L. Наукові основи інноваційної діяльності Introduction. Space tethered systems (STS) stabilized by rotation is a quite interesting and promising direction in the field of cosmonautics. Such systems are intended for solving a wide range of scientific and research tasks, in particular, those that cannot be solved effectively with the help of the existing space technologies, for example, transport operations, creation of artificial gravity, removal of space debris objects, obtainment of experimental data of functioning tethered systems, etc. Problem Statement. The peculiarities of the STS dynamics models are determined by the specifics of the problems solved by such systems actual among which is the researches the effects of the end body dynamics on the system motion. Purpose. To build a mathematical model of the STS dynamics for considering the general regularities of the system motion and to analyze comprehensively the special features of the end body dynamics. Materials and Methods. The mathematical model of the STS dynamics has been built based on the methods and principles of theoretical mechanics and space flight dynamics. To study the STS dynamics, methods of the theory of oscillations, analytical and numerical integration of differential motion equations have been used. Results. The simplest model of the STS dynamics consisting of the material point and the end body connected by a tether is presented for the motion under consideration. The possibility of the appearance of internal resonances and their influence on the stabilization processes in the relative motion of the system has been considered. Conclusions. The proposed model can apply to analyzing the angular oscillation of the end body relative to the tether attachment point, taking into account the effects of the inertial characteristics of the end body, the tether stiffness and the angular velocity of the proper rotation of the system. Practical problems related to the STS dynamics may include the problems of the stability of the end body orientation, resonance modes in the system motion, as well as the problems in creating the prerequisites for the design of the specific STS. Вступ. Використання космічних тросових систем (КТС), стабілізованих обертанням, є досить новим і перспективним напрямком в галузі сучасної космонавтики. Такі системи призначені для вирішення широкого кола наукових та дослідницьких завдань, зокрема тих, які неможливо або неефективно вирішувати за допомогою наявних засобів космічної техніки, наприклад для транспортних операцій, створення штучної гравітації, відведення об’єктів космічного сміття, отримання експериментальних даних функціонування тросових систем тощо. Проблематика. Особливості моделей динаміки КТС зумовлені специфікою розв’язуваних такими системами завдань, актуальним серед яких є дослідження впливу динаміки кінцевого тіла на рух системи. Мета. Побудова математичної моделі динаміки КТС, яка дозволить розглянути загальні закономірності руху системи та виконати аналіз особливостей динаміки кінцевого тіла. Матеріали й методи. Побудова математичної моделі динаміки КТС базується на методах і принципах теоретичної механіки, методах динаміки космічного польоту. Для дослідження динаміки КТС використано методи теорії коливань, методи аналітичного та чисельного інтегрування диференційних рівнянь руху. Результати. Наведено найпростішу для досліджуваного руху модель динаміки КТС, що складається з матеріальної точки й кінцевого тіла, з’єднаних ниткою. Розглянуто можливість виникнення внутрішніх резонансів та їх вплив на процеси стабілізації у відносному русі системи. Висновки. Запропонована модель динаміки КТС дозволяє виконати аналіз кутових коливань кінцевого тіла відносно точки кріплення до нитки з врахуванням впливу інерційних характеристик кінцевого тіла, жорсткості нитки й кутової швидкості власного обертання системи. До практичних питань, пов’язаних з цією задачею динаміки КТС, можна віднести питання стійкості орієнтації кінцевого тіла, питання про резонансні режими в русі системи, а також питання про створення необхідних передумов для проектування конкретних КТС. Введение. Использование космических тросовых систем (КТС), стабилизированных вращением, является достаточно новым и перспективным направлением в области современной космонавтики. Такие системы предназначены для решения широкого круга научных и исследовательских задач, в частности, тех, которые невозможно или неэффективно решать с помощью имеющихся средств космической техники, например для транспортных операций, создания искусственной гравитации, увода объектов космического мусора, получения экспериментальных данных функционирования тросовых систем и т. д. Проблематика. Особенности моделей динамики КТС обусловлены спецификой решаемых такими системами задач, актуальными среди которых являются исследования влияния динамики концевого тела на движение системы. Цель. Построение математической модели динамики КТС, которая позволит рассмотреть общие закономерности движения системы и выполнить анализ особенностей динамики концевого тела. Материалы и методы. Построение математической модели динамики КТС базируется на методах и принципах теоретической механики, методах динамики космического полета. Для исследования динамики КТС использованы методы теории колебаний, методы аналитического и численного интегрирования дифференциальных уравнений движения. Результаты. Представлена простейшая для исследуемого движения модель динамики КТС, состоящая из материальной точки и концевого тела, соединенных нитью. Рассмотрена возможность возникновения внутренних резонансов и их влияние на процессы стабилизации в относительном движении системы. Выводы. Предложенная модель динамики КТС позволяет выполнить анализ угловых колебаний концевого тела относительно точки крепления к нити с учетом влияния инерциальных характеристик концевого тела, жесткости нити и угловой скорости собственного вращения системы. К практическим вопросам, связанным с данной задачей динамики КТС, можно отнести вопросы устойчивости ориентации концевого тела, вопросы о резонансных режимах в движении системы, а также вопросы о создании необходимых предпосылок для проектирования конкретных КТС. 2019 Article Influence of the End Body Dynamics on Stabilization Processes in the Relative Motion of a Space Tethered System Stabilized by Rotation / O.L. Volosheniuk // Наука та інновації. — 2019. — Т. 15, № 2. — С. 17-24. — Бібліогр.: 9 назв. — англ. 1815-2066 DOI: doi.org/10.15407/scin15.02.017 http://dspace.nbuv.gov.ua/handle/123456789/173961 en Наука та інновації Видавничий дім "Академперіодика" НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Наукові основи інноваційної діяльності Наукові основи інноваційної діяльності |
| spellingShingle |
Наукові основи інноваційної діяльності Наукові основи інноваційної діяльності Volosheniuk, O.L. Influence of the End Body Dynamics on Stabilization Processes in the Relative Motion of a Space Tethered System Stabilized by Rotation Наука та інновації |
| description |
Introduction. Space tethered systems (STS) stabilized by rotation is a quite interesting and promising direction in the
field of cosmonautics. Such systems are intended for solving a wide range of scientific and research tasks, in particular,
those that cannot be solved effectively with the help of the existing space technologies, for example, transport operations,
creation of artificial gravity, removal of space debris objects, obtainment of experimental data of functioning tethered
systems, etc. Problem Statement. The peculiarities of the STS dynamics models are determined by the specifics of the problems
solved by such systems actual among which is the researches the effects of the end body dynamics on the system motion.
Purpose. To build a mathematical model of the STS dynamics for considering the general regularities of the system
motion and to analyze comprehensively the special features of the end body dynamics.
Materials and Methods. The mathematical model of the STS dynamics has been built based on the methods and
principles of theoretical mechanics and space flight dynamics. To study the STS dynamics, methods of the theory of
oscillations, analytical and numerical integration of differential motion equations have been used.
Results. The simplest model of the STS dynamics consisting of the material point and the end body connected by a
tether is presented for the motion under consideration. The possibility of the appearance of internal resonances and their
influence on the stabilization processes in the relative motion of the system has been considered.
Conclusions. The proposed model can apply to analyzing the angular oscillation of the end body relative to the tether
attachment point, taking into account the effects of the inertial characteristics of the end body, the tether stiffness and the
angular velocity of the proper rotation of the system. Practical problems related to the STS dynamics may include the
problems of the stability of the end body orientation, resonance modes in the system motion, as well as the problems in
creating the prerequisites for the design of the specific STS. |
| format |
Article |
| author |
Volosheniuk, O.L. |
| author_facet |
Volosheniuk, O.L. |
| author_sort |
Volosheniuk, O.L. |
| title |
Influence of the End Body Dynamics on Stabilization Processes in the Relative Motion of a Space Tethered System Stabilized by Rotation |
| title_short |
Influence of the End Body Dynamics on Stabilization Processes in the Relative Motion of a Space Tethered System Stabilized by Rotation |
| title_full |
Influence of the End Body Dynamics on Stabilization Processes in the Relative Motion of a Space Tethered System Stabilized by Rotation |
| title_fullStr |
Influence of the End Body Dynamics on Stabilization Processes in the Relative Motion of a Space Tethered System Stabilized by Rotation |
| title_full_unstemmed |
Influence of the End Body Dynamics on Stabilization Processes in the Relative Motion of a Space Tethered System Stabilized by Rotation |
| title_sort |
influence of the end body dynamics on stabilization processes in the relative motion of a space tethered system stabilized by rotation |
| publisher |
Видавничий дім "Академперіодика" НАН України |
| publishDate |
2019 |
| topic_facet |
Наукові основи інноваційної діяльності |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/173961 |
| citation_txt |
Influence of the End Body Dynamics on Stabilization Processes in the Relative Motion of a Space Tethered System Stabilized by Rotation / O.L. Volosheniuk // Наука та інновації. — 2019. — Т. 15, № 2. — С. 17-24. — Бібліогр.: 9 назв. — англ. |
| series |
Наука та інновації |
| work_keys_str_mv |
AT volosheniukol influenceoftheendbodydynamicsonstabilizationprocessesintherelativemotionofaspacetetheredsystemstabilizedbyrotation |
| first_indexed |
2025-07-15T10:49:25Z |
| last_indexed |
2025-07-15T10:49:25Z |
| _version_ |
1837709731746021376 |
| fulltext |
17
Volosheniuk, O.L.
Institute of Technical Mechanics, the NAS of Ukraine and the State Space Agency of Ukraine,
15, Leshko-Popelya St., Dnipro, 49005, Ukraine,
+380 56 372 0640, +380 56 372 0650, office.itm@nas.gov.ua
INFLUENCE OF THE END BODY DYNAMICS
ON STABILIZATION PROCESSES IN THE RELATIVE
MOTION OF A SPACE TETHERED SYSTEM
STABILIZED BY ROTATION
© VOLOSHENIUK, O.L., 2019
Introduction. Space tethered systems (STS) stabilized by rotation is a quite interesting and promising direction in the
field of cosmonautics. Such systems are intended for solving a wide range of scientific and research tasks, in particular,
those that cannot be solved effectively with the help of the existing space technologies, for example, transport operations,
creation of artificial gravity, removal of space debris objects, obtainment of experimental data of functioning tethered
systems, etc.
Problem Statement. The peculiarities of the STS dynamics models are determined by the specifics of the problems
solved by such systems actual among which is the researches the effects of the end body dynamics on the system motion.
Purpose. To build a mathematical model of the STS dynamics for considering the general regularities of the system
motion and to analyze comprehensively the special features of the end body dynamics.
Materials and Methods. The mathematical model of the STS dynamics has been built based on the methods and
principles of theoretical mechanics and space flight dynamics. To study the STS dynamics, methods of the theory of
oscillations, analytical and numerical integration of differential motion equations have been used.
Results. The simplest model of the STS dynamics consisting of the material point and the end body connected by a
tether is presented for the motion under consideration. The possibility of the appearance of internal resonances and their
influence on the stabilization processes in the relative motion of the system has been considered.
Conclusions. The proposed model can apply to analyzing the angular oscillation of the end body relative to the tether
attachment point, taking into account the effects of the inertial characteristics of the end body, the tether stiffness and the
angular velocity of the proper rotation of the system. Practical problems related to the STS dynamics may include the
problems of the stability of the end body orientation, resonance modes in the system motion, as well as the problems in
creating the prerequisites for the design of the specific STS.
K e y w o r d s : space tethered system, mathematical model, stabilization by rotation, end body, and stabilization processes.
ISSN 1815-2066. Nauka innov., 2019, 15(2): 17—24 https://doi.org/10.15407/scin15.02.017
The space tethered systems (STS) stabilized
by rotation is a quite interesting and promising
direction in the field of cosmonautics. Such sys-
tems are intended for solving a wide range of sci-
entific and research tasks (in particular, for sol-
ving tasks that cannot be solved or solved inef-
fectively with the help of the existing space tech-
nologies) [1—4].
However, to study the rotating STS taking in-
to account dynamics of end bodies is a compli-
cated problem of spaceflight dynamics. For the
time being, this problem has not been sufficiently
studied and the current knowledge does not en-
able to set a well-grounded opinion about realiza-
bility of existing projects. Successful solution of
this problem related to creating research met hods
and respective mathematical dynamics models as
18 ISSN 1815-2066. Nauka innov., 2019, 15 (2)
Volosheniuk, O.L.
well as analysis and definition of the main pat-
terns of motion will create necessary precondi-
tions for planning and developing definite STS
projects. So, in the case of some tasks [5, 6] the
creation of STS presupposes compliance with
program requirements concerning accuracy of
mo tion orientation of the system end bodies. Ear-
lier [7], a mathematical model of dynamics rota-
ting STS of two end bodies has been proposed.
The analysis of dynamics of the considered STS
with identical end bodies has shown a possibility
of nonlinear resonances causing a significant in-
fluence on the dynamics of system end bodies in
transient modes of motion. It has been demon-
strated that a long-period energy transfer from
one body to another takes place in the system [8].
And that is why, one of relevant problems of ro-
tating STS dynamics is to study influence caused
by dynamics of end bodies on the process of sys-
tem motion (in particular, this includes study of
interaction between oscillations of STS end bo-
dies and self-rotation of the system).
In this research the simplest model of system
dynamics for assessment of influence caused by
end body dynamics on motion of the rotating
STS has been presented. Simplicity of this model
will give an opportunity to carry out a qualitative
analysis of the end body motion relative to the
tether attachment point, to take into account the
influence of end body inertial characteristics and
tether stiffness, and to assess the possibility of
resonances in the system motion.
One end of the tether (material point A) is as-
sumed to move on unperturbed Keplerian orbit,
with the other end of the tether attached to an
absolutely rigid body (Fig. 1).
The tether-body system is boosted to have a
spin motion around the point A with an angular
velocity significantly exceeding that of orbital
motion. The connecting tether is sufficiently
lightweight and in the research mode of motion it
is sufficiently tensioned (i.e. the tether may be
viewed as an elastic weightless connection). Ener-
gy dissipation of the system motion is performed
only by means of internal friction in the elastic
tether. The system is assumed to move in a New-
tonian force field and there are no other external
impacts. It is also presupposed that the system
moves only in the orbital plane.
MOTION EQUATIONS
OF THE SYSTEM
The motion equations of the considered system
are as follows:
(1)
where R
→
A, R
→
1 are radius vectors directed outward
the Newtonian attracting center towards point
A; m1 is mass of end body 1; F
→
tr is tether tension
force; L
→
1 is kinematics momentum of body motion
relative to its own center of mass; ρ→ 1n is radius
vector directed from the body center of mass to
the point of body attachment to the tether; M
→
grav,1
is the Newtonian force field momentum acting on
the system body; μ is the gravitational constant.
The influence caused by tether tension force F
→
tr
and gravitational momentum of forces M
→
grav,1 on
the body 1 is assumed to be defined by formulas
analogous to those presented in [7].
ACTING FORCE AND MOMENTUM
The tether elastic properties are described by
Hook’s law and the energy dissipation in the
tether material is expressed using the formulas of
Fig. 1. Model of the STS motion
O
A
О1
L
→
1 = M
→
grav,1 — ρ→ 1n × F
→
tr,
m1R
→
1 = — — F
→
tr,
R
→
1m1
R1
3
..
,
RА
3
R
→
A= —
R
→
A
..
.
ISSN 1815-2066. Nauka innov., 2019, 15 (2) 19
Influence of the End Body Dynamics on Stabilization Processes in the Relative Motion of a Space Tethered System Stabilized
the equivalent viscous friction:
where r→l is vector directed from the material point
A to the attachment point of the end body 1 (at-
tachment to the tether); rl = |r→l|; d is nominal length
of the tether; c is stiffness coefficient of the tether;
χ is damping coefficient [9].
Momentum of gravitational forces acting on
the end body of the system is described by the
following formula:
where J1 is inertia tensor of the end body 1; e→R1
is
unit vector R
→
1 .
Based on (1), the equation of relative motion
(the center-of-mass motion equation of the end
body relative to point A) has been obtained:
(2)
Presupposing that the end body has a spherical
shape, the equation of end body motion relative
to its center of mass can be written as
(3)
where ω→ 1 is vector of absolutely angular velocity
of the end body motion.
In (2), let us expand into a
power series
neglecting the higher order terms. The symbol
R = RA is introduced for convenience and simpli-
city. As far as r is hundreds meters and R 7021 km,
the value
has an order of 10–8 and therefore,
may be neglected.
Taking into account that ,
the equation for may be presented as follows:
(4)
where e→R is unit vector of the R
→
; e→r is unit vector of
the r→ .
KINEMATICS OF THE SYSTEM
To study the motion of the considerable system
let us introduce right-handed coordinate systems
similar to [7], Fig. 4.
OXuYuZu is inertial coordinate system (ICS)
with the origin in the center of the Earth O. OXu
is directed to the vernal equinox; OZu is directed
along the Earth rotational axis;
OcXoYoZo is orbital coordinate system (OCS)
with the origin in point Oc coinciding with the
center of mass of the system (with point A). OcXo
is directed along the radius vector connecting the
center of mass of the system with the Earth’s cen-
ter OcYo in the plane of instantaneous orbit to-
wards the motion of system center of mass;
OcXcYcZc is moving coordinate system related
with the STS body motion plane (CCS) with the
origin in the Oc coinciding with point А. OcXc is
directed from Oc to the center of mass of the end
body 1, OcZc is axis of instant rotation of the vec-
tor directed from Oc to the center of mass of end
body 1;
O1x1y1z1 is coordinate system related to the end
body (BCS) with the origin in the center of mass
O1 of the end body (Fig. 2). The axis coincides
with the principal central axis of inertia of the body.
In accordance with the task set, the system
center of mass is assumed to move on the Keple-
rian orbit, only in the plane of the orbit. In this
case, the mutual orientation of coordinate sys-
tems can be described as (Fig. 2): OcXoYoZo and
OXuYuZu are Euler angles (true anomaly angle ν),
(ν = ωout, ωou is angular velocity of the center of
mass motion on the orbit or ν = t); Oc XcYcZc
and Oc XoYoZo are Euler angles (pure rotation ang-
le ' ); O1x1y1z1 and Oc XcYcZc are Krylow ang les
(yaw angle ѱ1).
MOTION EQUATIONS OF IN THE SCALAR FORM
For the purpose of numerical investigations
concerning the end body dynamics in the rota-
tional motion of the system let us develop for-
mulas for and in the projections on the CCS
axis.
Vector r→ may be presented as r→ = re→r .
Having differentiated r→ on time, we receive the
formulas for definition of , :
r→ = [3re→R (e→R, e→r) — re→r] — F
→
tr /m1,
R3
..
J1 ω
→
1 = –ρ→ 1n × F
→
tr,
.=
M
→
grav,1 = 3 e→R1
× J1 e
→
R1
,R1
3
F
→
tr = — c — r
.
l , =
r→l
rl
r→l
rl d
(rl — d)
0,rl ≤ d,
1,rl > d,[ ] {
r→
..
r→
..
r→
..
r→
..
r→
.
r
R
( )
r
R
( ) 2
= 1 — 3(e→R, e→r)
1
R1
3
1
R3
r
R
( )
R
→
1
R1
3
R
→
A
RA
3
r→ = R
→
1 — R
→
A = — + — F
→
tr /m1.
.. .. .. [ ]
R2
R
r→ = r·e→r + r ·e→Yc
,
.
.
ω→ 1
20 ISSN 1815-2066. Nauka innov., 2019, 15 (2)
Volosheniuk, O.L.
where e→Yc
is unit vector of the axis Yc.
(5)
where · is angular velocity of CCS relative to ICS
(ωcu = · , = ν + ' ).
Let’s write expressions for F
→
tr in the projections
on the CCS axis as
(6)
where r· = .
Proceeding from the geometry of the system
(Fig. 2)
r→l = r→ + ρ→ 1n.
Having differentiated r→I on time we receive:
where
The orientation of body 1 in CCS can be con-
veniently provided by means of radius vector ρ→ 1n
and angle (Fig. 2).
In accordance with [8], the orientation of ra-
dius vector ρ→ 1
*
n = –ρ→ 1n (directed from the point of
end body attachment (to the tether) to the cen-
ter of mass of the end body 1) in the CCS is deter-
mined by two angles 1, 1 (Fig. 3):
1 is angle between ρ→ 1
*
n and the axis Oс Xс ;
1 is angle between the projection of the radius
vector ρ→ 1* n — P ρ1
*
n on the plane OсXсYс and the axis
OсXс, respectively.
In this case orientation of ρ→ 1n in CCS is defined
by means of a column of direction cosines of the
unit vector
where
.
In projections on CCS axis the expression for
shall be presented as
where is unit vector of the axis O1Y1 in pro-
jections on CCS axis,
In a similar way, r→ , in the projections on CCS
axis
The expressions for rl and r. l included into Ftr in
(6) can be easily received by means of permuting
(adding) and to them
Momentum of the tether tension force, M
→
tr =
= –ρ→ 1n × F
→
tr, in the projections on CCS axes is
where is unit vector of the axis Oc Zc .
ρ→ 1n = ω→ 1 × ρ→ 1n.
.
Fig. 2. Mutual orientation of coordinate systems Fig. 3. Orientation of radius vector ρ→ 1
*
n in CCS
.. ..r→ = (r— r
·)e→r + (r+ 2r
.
·)e→Yc
,
..
. . .
r→l = r→ + ρ→ 1n,
Ftr er
l
= c + r
.
l , =(с)
d
(rl — d)
1,rl > d,
0,rl < d,[ ] { rl
(c)
r→l
ey1
(c)
en
(c)
ρ1n
= ρ1n
,
(c)
en
(c)
en
= – er cos — eYc
sin(c)
rl
= (r — ρ1n cos ) er — ρ1n sineYc
,
ey1
= – er sin + eYc
cos.
(c)
r→l , r
→
l
r→l
.
( )
→
r→
.
ρ1n
= ω1ρ1n
,
(c)
ey1
(c)
.
ρ→ 1n
.
rl
= (r — ω1ρ1n sin) er + (r· + ω1ρ1n cos) eYc
.
.
rl
.
rl
rl = r 2 — 2rρ1n cos+ ρ1n ,2
r· r — ρ1n cos— rρ1n sin (ω1 — ·)
r·l = .
r 2 — 2rρ1n cos+ ρ1n
2
eZc
Mtr = Ftr eZc
,
ρ1n r sin
rl
(c)
Хc
Zc
Oc
O1
x1
1
y1
'
Хu
Хo
Хc
1
ρ→ 1
*
n
ρ→ 1n
1
Pρ1
*
n
→
Yc
Yc
Zc
Oc
O
R
Yо
Yu
r
→
l
r
→
ISSN 1815-2066. Nauka innov., 2019, 15 (2) 21
Influence of the End Body Dynamics on Stabilization Processes in the Relative Motion of a Space Tethered System Stabilized
—
Let us write down the right side of the equa-
tion (4) in the projections on CCS axes.
Let us present e→R in the projections on CCS
axes. The transition from OCS to CCS is presen-
ted in Fig. 2.
Having been transformed, the expression for
in (6) is presented as
(7)
As a result of substitution of (7) in (5) and pro-
jecting the obtained expression on CCS axis we
shall receive
(8)
Taking into account that and con si-
dering the fact that unit vectors and
CCS
and BCS coincide, (3) can be presented as
(9)
where .
Fig. 4. Change of r ( = 0.01)
Fig. 6. Change of r ( = 0.1)
Fig. 5. Change of angle ( = 0.01)
Fig. 7. Change of angle ( = 0.1)
eZc
eZ1
eR = ercos ' — eYc
sin'. (c)
r
..
52.85
52.9
V, tеrn/orb
52.95
53
53.05
53.1
53.15
53.2
r, m
0 1 2 3 4 5 6 7 8
V, tеrn/orb
, grad
–
–4
–6
–8
0 1 2 3 4 5 6 7 8
52.85
52.9
V, tеrn/orb
52.95
53
53.05
53.1
53.15
53.2
r, m
0 1 2 3 4 5 6 7 8
V, tеrn/orb
, grad
–
–4
–6
–8
0 1 2 3 4 5 6 7 8
. 1 = + .. ..
— r ·2= r (3cos2 ' — 1) — (r — ρ1n cos), r
..
R3
m1r1
Ftr
(r — ρ1n cos) er — ρ1n sin eYcm1r1
Ftr [ ]
r = r (3cos2 ' — 1) er — sin2' eYc
—
R3
3
2
.. [ ]
r + 2r·· = — r sin2' + ρ1n sin.
R3
3
2 m1r1
Ftr..
ρ1n r sin
rl
. 1 = — Ftr,
1
Jz
ω→ 1 = . 1e
→
Z1
.
—
— .
22 ISSN 1815-2066. Nauka innov., 2019, 15 (2)
Volosheniuk, O.L.
So, taking into account (8), (9) the system of
equations (1) has the following form:
(10)
The presented STS motion equations (10) give
an opportunity to obtain a closed system of first
order equations with 6 unknown variables.
The possibility of internal resonances in the
system motion and their influence on the pro-
cesses of stabilization of the end body oscillations
of rotating STS with energy dissipation of longi-
tudinal oscillations has been considered. In the
motion of such system, internal resonances may
occur when the following frequencies are com-
mensurate: the orbital motion of the system cen-
ter of mass, the spin motion of STS around its
own center of mass (point A), changing distance
between the ends of the tether (longitudinal os-
cillations), and free (angular) oscillations of the
end body.
It is obvious that the possible frequency reso-
nance between the longitudinal oscillations of
the tether and the angular oscillations of the end
body is of the greatest interest. Conditions of res-
onant motions between longitudinal oscillations
of the tether and angular oscillations of the end
body are determined analytically on the basis of
expressions presented in [10]. In this case, when
the STS moves in the orbital plane, the internal
resonance of the cosidered oscillations will be ob-
served near the following values of the system
characteristics (see Table).
The stabilization process of angular oscilla-
tions of the end body in the resonant commensu-
rability with longitudinal oscillations of the tet-
her has been numerically estimated taking into
account the influence of longitudinal oscillations
damping. In Figs. 4—7, there are presented sche-
dules of longitudinal oscillations of the tether
(change of length — r, m) and schedules of angu-
lar oscillations of the end body (change of angle
, grad) in time at V = 8 periods (turns) of STS
motion in orbit
Based on the obtained results it is possible to
make comparative estimations of the influence of
damping coefficient of longitudinal oscillations ()
on the stabilization of angular oscillations of the
end body. The estimates have been done for val-
ues equal to 0.01 and 0.1 kg/s. The data va lues
are chosen because of the following reasons [9]:
for various structural materials of the tether,
the characteristic value is ~ 0.01 kg/s;
more energy losses are caused by friction bet-
ween the tangential details of special damping
devices — structural damping, for which the
cha racteristic value is ~ 0.1 kg/s.
Figs. 4—7 show that for fast damping of angu-
lar oscillations of the end body, it is necessary to
ensure their strong connection with longitudi-
nal oscillations of the tether and, accordingly,
the intensive transfer of energy to longitudinal
oscillations. As can be seen from Figs. 6, 7, the de-
pendence of logarithmic decrement of longitu di -
nal oscillations = Tk on the coefficient of dam-
ping = 0.1 kg/s provides the optimum in the
rate of damping of the end body oscillations. Un-
der such , the amplitude of longitudinal oscilla-
tions and angular oscillations of the end body
"radically" decreases in one period of the STS
motion in orbit.
Thus, controlling the tensile strength of the
tether or using a longitudinal damper in the reso-
nant setting is an effective means of extinguish-
ing angular oscillations of the end body.
Characteristics of the STS
Characteristics of the tether
Nominal length d = 50 m
Stiffness c = 732 Н
Equilibrium length rl = 50.83 m
Inertial characteristics of the end body
Mass mi = 24 kg
Coordinates →ρ 1n in projec tions
on the axis BCS
ρ1n = [2.165 m; 0; 0]
— r ·2= r (3cos2 ' — 1) — (r — ρ1n cos), r
..
R3
m1rl
Ftr
r + 2r·· = — r sin2' + ρ1n sin,
R3
3
2 m1rl
Ftr..
ρ1n r sin
rl
. 1 = – Ftr.
1
Jz
V = 2 T, T is time interval, s .
ou( )
2
ISSN 1815-2066. Nauka innov., 2019, 15 (2) 23
Influence of the End Body Dynamics on Stabilization Processes in the Relative Motion of a Space Tethered System Stabilized
CONCLUSION
Hence, the simplest mathematical model of dy-
namics of the rotating STS that consists a mate-
rial point and an end body connected with an
elastic weightless tether has been presented. This
model gives an opportunity to carry out analysis
of the end body motion relative to the tether at-
tachment point. The possibility of internal reso-
nances in the rotating STS motion and their in-
f luence on the processes of stabilization of the end
body oscillations with the energy dissipation of
longitudinal oscillations has been considered. In
the resonant modes, damping of longitudinal os-
cillations by several orders of magnitude has been
shown to reduce the amplitude of longitudinal
os cillations and angular oscillations of the end
body and, respectively, the duration of the stabi-
lization processes in relative motion of the STS.
REFERENCES
1. Linskens, H. T. K., Mooij, E. (2016). Tether Dynamics Analysis and Guidance and Control Design for Active Space-
Debris Removal. Journal of Guidance, Control, and Dynamics, 39(6), 1232—1243. URL: https://doi.org/10.2514/1.G001651
(Last accessed: 12.03.2018).
2. Cartmell, M. P., McKenzie, D. J. (2008). A review of Space Tether research. Progress in Aerospace Sciences, 44(1),
1—21. URL: https://doi.org/10.1016/j.paerosci.2007.08.002 (Last accessed: 12.03.2018).
3. Volosheniuk, O. L., Pirozhenko, A. V., Khramov, D. A. (2011). Space Tethered Systems — a promising trend in the
sphere of space engineering and technologies. Space science and Technology, 17(2), 32—44. URL: https://doi.org/10.15407/
knit2011.02.032. (Last accessed: 10.04.2018).
4. Troger, H., Alpatov, А. P., Beletsky, V. V., Dranovskii, V. I., Khoroshilov, V. S., Pirozhenko, A. V., Zakrzhevskii, A. E.
(2010). Dynamics of tethered space systems. Boca Raton London New York: CRC Pres. Taylor & Francis Group.
5. Levin, E. M. (2007). Dynamic analysis of space tether missions. San Diego, Calif.: Published for the American Astro-
nautical Society by Univelt, Inc.
6. James, H. G., Yau, A. W., Tyc, G. (1995, April). Space research in the BICEPS experiment. Fourth International Con-
ference on tether in Space, Washington. P. 1585—1598.
7. Volosheniuk, О. L., Pirozhenko, A. V. (2004). Mathematical model of dynamics of a space tethered system subjected
to spin stabilization. Technical Mechanics, 2, 17—27 [in Russian].
8. Volosheniuk, О. L., Pirozhenko, A. V. (2005). Analysis of Frequencies and Characteristics of Transient Responses of
a Space Tethered System Stabilized by Rotation. Technical Mechanics, 1, 13—21 [in Russian].
9. Beletskiy, V. V. (2009). Profiles on Movement of Space Bodies. Мoscow: Publishing house LKI [in Russian].
Received 25.07.18
O.Л. Волошенюк
Інститут технічної механіки Національної академії наук України
і Державного космічного агентства України (ІТМ НАНУ та ДКАУ),
вул. Лешко-Попеля, 15, Дніпро, 49005, Україна,
+380 56 372 0640, +380 56 372 0650, office.itm@nas.gov.ua
ВПЛИВ ДИНАМІКИ КІНЦЕВОГО ТІЛА НА ПРОЦЕСИ СТАБІЛІЗАЦІЇ
У ВІДНОСНОМУ РУСІ КОСМІЧНОЇ ТРОСОВОЇ СИСТЕМИ,
СТАБІЛІЗОВАНОЇ ОБЕРТАННЯМ
Вступ. Використання космічних тросових систем (КТС), стабілізованих обертанням, є досить новим і перс-
пективним напрямком в галузі сучасної космонавтики. Такі системи призначені для вирішення широкого кола
наукових та дослідницьких завдань, зокрема тих, які неможливо або неефективно вирішувати за допомогою наяв-
них засобів космічної техніки, наприклад для транспортних операцій, створення штучної гравітації, відведення
об’єктів космічного сміття, отримання експериментальних даних функціонування тросових систем тощо.
Проблематика. Особливості моделей динаміки КТС зумовлені специфікою розв’язуваних такими системами
завдань, актуальним серед яких є дослідження впливу динаміки кінцевого тіла на рух системи.
Мета. Побудова математичної моделі динаміки КТС, яка дозволить розглянути загальні закономірності руху
системи та виконати аналіз особливостей динаміки кінцевого тіла.
24 ISSN 1815-2066. Nauka innov., 2019, 15 (2)
Volosheniuk, O.L.
Матеріали й методи. Побудова математичної моделі динаміки КТС базується на методах і принципах теоретич-
ної механіки, методах динаміки космічного польоту. Для дослідження динаміки КТС використано методи теорії
коливань, методи аналітичного та чисельного інтегрування диференційних рівнянь руху.
Результати. Наведено найпростішу для досліджуваного руху модель динаміки КТС, що складається з ма те-
ріальної точки й кінцевого тіла, з’єднаних ниткою. Розглянуто можливість виникнення внутрішніх резонансів та їх
вплив на процеси стабілізації у відносному русі системи.
Висновки. Запропонована модель динаміки КТС дозволяє виконати аналіз кутових коливань кінцевого тіла
відносно точки кріплення до нитки з врахуванням впливу інерційних характеристик кінцевого тіла, жорсткості нитки
й кутової швидкості власного обертання системи. До практичних питань, пов’язаних з цією задачею динаміки КТС,
можна віднести питання стійкості орієнтації кінцевого тіла, питання про резонансні режими в русі системи, а також
питання про створення необхідних передумов для проектування конкретних КТС.
Ключові слова : космічна тросова система, математична модель, стабілізація обертанням, кінцеве тіло, процеси
стабілізації.
O.Л. Волошенюк
Институт технической механики Национальной академии наук Украины
и Государственного космического агентства Украины (ИТМ НАНУ и ГКАУ),
ул. Лешко-Попеля, 15, Днепр, 49005, Украина,
+380 56 372 0640, +380 56 372 0650, office.itm@nas.gov.ua
ВЛИЯНИЕ ДИНАМИКИ КОНЦЕВОГО ТЕЛА НА ПРОЦЕССЫ СТАБИЛИЗАЦИИ
В ОТНОСИТЕЛЬНОМ ДВИЖЕНИИ КОСМИЧЕСКОЙ ТРОСОВОЙ СИСТЕМЫ,
СТАБИЛИЗИРОВАННОЙ ВРАЩЕНИЕМ
Введение. Использование космических тросовых систем (КТС), стабилизированных вращением, является
достаточно новым и перспективным направлением в области современной космонавтики. Такие системы предназна-
чены для решения широкого круга научных и исследовательских задач, в частности, тех, которые невозможно или
неэффективно решать с помощью имеющихся средств космической техники, например для транспортных операций,
создания искусственной гравитации, увода объектов космического мусора, получения экспериментальных данных
функционирования тросовых систем и т. д.
Проблематика. Особенности моделей динамики КТС обусловлены спецификой решаемых такими системами
задач, актуальными среди которых являются исследования влияния динамики концевого тела на движение системы.
Цель. Построение математической модели динамики КТС, которая позволит рассмотреть общие закономернос-
ти движения системы и выполнить анализ особенностей динамики концевого тела.
Материалы и методы. Построение математической модели динамики КТС базируется на методах и принципах
теоретической механики, методах динамики космического полета. Для исследования динамики КТС использованы
методы теории колебаний, методы аналитического и численного интегрирования дифференциальных уравнений
движения.
Результаты. Представлена простейшая для исследуемого движения модель динамики КТС, состоящая из мате-
риальной точки и концевого тела, соединенных нитью. Рассмотрена возможность возникновения внутренних резо-
нансов и их влияние на процессы стабилизации в относительном движении системы.
Выводы. Предложенная модель динамики КТС позволяет выполнить анализ угловых колебаний концевого
тела относительно точки крепления к нити с учетом влияния инерциальных характеристик концевого тела, жест-
кости нити и угловой скорости собственного вращения системы. К практическим вопросам, связанным с данной
задачей динамики КТС, можно отнести вопросы устойчивости ориентации концевого тела, вопросы о резонанс-
ных режимах в движении системы, а также вопросы о создании необходимых предпосылок для проектирования
конкретных КТС.
Ключевые слова: космическая тросовая система, математическая модель, стабилизация вращением, концевое
тело, процессы стабилизации.
|