Development of a mathematical model describing the motion of the space tether system. Creation of the program complex designed to analyze the dynamics of the space tether system.

1. Vladimir S. Aslanov, Aleksandr S. Ledkov
aslanov_vs@mail.ru, ledkov@inbox.ru
Mathematical models and analysis
of the space tether systems motion
Theoretical mechanics department
www.termech.ru
Samara State Aerospace Univercity
www.ssau.ru
2012

2. 1.1. Area of application of space tether systems
Dynamics of space tether systems has been studied by: Beletsky V. V., Levin E. M., Cartmell M.P., Cosmo M.L. , Lorenzini
E.C., Misra A.K., Modi. V.J., Williams P., Fujii H. A., Edwards B. C., Kumar K. D., Kumar R., McCoy J. E., Sorensen K.,
Zimmermann F. 2

3. 1.1.1. Creation of certain conditions onboard the
spacecraft
Artificial gravity Generation of electricity
Creating artificial gravity through Generation of electricity by the interaction of
centrifugal force of inertia tether with an electromagnetic field of the Earth.
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4. 1.1.2. Creation of certain conditions onboard the
spacecraft
Gravitational stabilisation of the spacecraft
Between the spacecraft and the spherical hinge dissipation force acts.
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5. 1.1.3. The research and national economy
Studying of an upper atmosphere
Due to the influence of the atmosphere the probe can not survive for long at a high
of 80-100 km. These heights are not available for aircraft.
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6. 1.1.4. The research and national economy
Sounding of a surface of the Earth
By reducing the height of the probe can be got a higher resolution scan.
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7. 1.1.5. The research and national economy
Interferometer with a long base Installation of radio-repeaters
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8. 1.1.6 Transport operations in space
Space Elevator
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9. 1.1.7. Transport operations in space
Delivery of payload Space Escalator
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10. 1.1.7. Transport operations in space
Braking of the spacecraft by electrodynamics tether
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11. 1.2. Experiment with space tethers
There are currently executed more than 20 experiments using the STS
Mission Orbit Year of Full tether Deployed
implementa length tether
tion length
Gemini-11 LEO 1966 36 m 36 m
Gemini-12 LEO 1966 36 m 36 m
TPE-1 Suborbital 1980 400 m 38 m
TPE-2 Suborbital 1981 400 m 103 m
Charge-1 (TPE-3) Suborbital 1983 418 m 418 m
Charge-2 (TPE-4) Suborbital 1984 426 m 426 m
T-REX
Oedipus-A Suborbital 1989 958 m 958 m
Charge-2B Suborbital 1992 426 m 426 m
TSS-1 LEO 1992 20 km 268 m TiPS PICOSAT
SEDS-1 LEO 1993 20 km 20 km
PMG LEO 500 m 500 m
SEDS-2 LEO 1994 20 km 20 km YES-2 SEDS
Oedipus-C Suborbital 1995 1174 m 1174 m
TSS-1R LEO 1996 19.7 km 19.7 km
TiPS LEO 1996 4 km 4 km
YES GTO 1997 35 km -
ATEx LEO 1998 6.05 km 22 m
PICOSAT 1.0 LEO 2000 30 m 30 m
PICOSAT 1.1 LEO 2000 30 m 30 m
YES2 LEO 2007 31.7 km 29 km
T-REX Suborbital 2010 300 m 300 m
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12. 1.3. Modern materials and tethers
Material Density , Tensile Elastic Carbon nanotubes
g/cm3 strength, GPa modulus, GPa
Aluminum 2.7 0.6 70
Diamond 3.5 54 1050
Dyneema 0.99 3 172
Graphite 2.2 20 690 1977 – M.U. Kornilov
Kevlar 29 1.44 3.6 83
1952 – L.V. Radushkevich and V.M. Lukyanivich
1991 - S.Iijima
Kevlar 49 1.44 3.6 124 1993 – N. Pungo
Silica 2.19 6 74 Existing technologies
Spectra-2000 0.97 3.34 124
Stainless steel 7.9 2 200
Tungsten 19.3 4 410
Zylon AS 1.54 5.8 180
Zylon HM 1.56 5.8 270
Carbone Nanotube 1.3 150 630
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13. 1.4. Problem formulation
• Development of a mathematical model describing the motion of the space
tether system.
• Creation of the program complex designed to analyze the dynamics of the
space tether system.
• Analysis abnormal situations in the problem of cargo delivery from the
orbit.
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14. 2. Mathematical models
Types of mathematical models
1. Tether is considered as the rod.
2. Tether is considered as the set of point masses connected by weightless
viscoelastic rod segments.
3. Tether is considered as the heavy thread.
The choice of the model is due to the specifics of the problem
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15. 2.1 Model with the massless rod
3 ( A  B) c
  2
 sin 2  (l  l0 )sin(   ),
2 C C
   c(l  l0 )   2  2 C   2 cos(2  2 )  
l  
2C  m2 
3 2 cos  (l cos    cos  )   2 l 

3 2 sin 2 sin(   )
2l   (  2 )cos(   ) 
   ,
2C
2c  (  2 )sin(   )
 

 (l  l0 )sin(2  2 )  
2lC l
3 2 sin  (l cos    cos  ) 2l(   ) 3 2 ( A  B)sin 2

   .
l l 2lC
Here  – angular velocity of the carrying spacecraft in
circular orbit, A, B, C – principal moments of inertia of
the spacecraft, l0 – length of the unstrained tether.
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16. 2.2 Multipoint model of the tether
Discrete representation of the tether
 
 ESi   i  1  Di i ,  i  1,
Ti   t (1)
0,  i  1,

where Ti - tension of the i-th section of the
tether, E - modulus of elasticity, i -
elongation, ri - length of the i-th part of
strained tether, li - length of the i-th part of
unstrained tether, Si - area of i-th tether part
tether cross-secrion, h - loss factor , mi -
mass of the i-th point, Di - coefficient of
internal friction for the case of longitudinal
vibrations of the tether section. Point i=0 is
correspond to spacecraft, and i=N+1 - to
cargo.
Di  ESi mili 1h
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17. 2.2 Multipoint model of the tether
The interaction with the atmosphere
2 Approximate formulas for calculating the aerodynamic
c  0, cn ( )  k sin 2  ,
3 coefficients
raiVi kdT  sin 1,i sin 2,i  ni , j  (Vi  ρ j )  ρ j
FAi   ni ,i  ni ,i 1 
6  ri ri1 
Here ri – density of the atmosphere at an altitude of i-th point, dT – diameter of the
tether, k – adjusted coefficient of Newton.
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18. 2.2 Multipoint model of the tether
Accounting singularities Earth's gravitational field
G i  gradU gravitational force acting on the i-th point of tether
The gravitational potential of the Earth, in the form of an expansion in
spherical harmonics:
 
n n

 rE   n
 rE  ( k )
U 1   J n   Pn sin      Pn sin  (Cnk cos k l  Snk sin k l ) 
ri  n2  ri  n  2 k 1  r  
 i 
Here  , l – geocentric latitude and longitude,
Pn - Legendre polynomial of the n-th order,
Pn(k) - associated Legendre function,
Jk - zonal harmonic coefficient,
Cnk, Snk - dimensionless coefficients, called for
n≠k tesseral harmonic coefficients, and when
n=k - coefficients of sectoral harmonics.
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19. 2.2 Multipoint model of the tether
The equations of motion of the tether's points
The general equation of dynamics for
noninertial Greenwich geocentric coordinate
system
mii  Gi  FAi  Ti  Ti 1  Фi Ц  Фi К
r
Inertial forces
Фi Ц  miω3  (ω3  ri )
Фi К  2miω3  Vi
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20. 2.2 The equations of motion of the end-bodies
Dynamic equations
dK i
 ωi  K i  M Ai  MGi  Δi  Ti , i  A, B
dt
Ki – angular momentum vector,
i – angular velocity of i-th body, MAi –
moment of aerodynamic forces, MGi –
moment of gravitational forces, Ti –
vector of the tension force of tether's part
adjacent to the body.
Kinematic equation
  l ω , m  m ω , n  l m .
li i i
i i i i i i
Here li, mi, ni – unit vectors of the
coordinate system OXYZ, specified as
projections on the axis associated with
the i-th body coordinate system.
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21. 3. Software implementation (TetherCalc)
The model is implemented in MatLab package
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22. 4 Analysis of abnormal situations
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23. 4 Analysis of abnormal situations
Wrong orientation at the cargo separation
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24. 4 Analysis of abnormal situations
Breakage of attitude control system
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25. 3 Analysis of abnormal situations
The consequences of jamming Diadram of the consequences of jamming
a - tether break
b - winding the tether on the spacecraft
c - impact tethered cargo and spacecraft
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26. The main results were published in the following
articles
• Aslanov V.S. and Ledkov A.S. Dynamics of the Tethered Satellite Systems, Woodhead Publishing
Limited, Cambridge, UK, (2012) 275 pages. ISBN-10: 0857091565 | ISBN-13: 978-0857091567)
• Aslanov V.S. and Ledkov A.S. Chaotic oscillations of spacecraft by elastic radial oriented tether -
Cosmic Research ISSN 0010-9525, No. 2, 2012, , Vol. 50, No. 2, 2012, 188-198.
• Aslanov V.S. Orbital oscillations of an elastic vertically-tethered satellite, Mechanics of Solids, Vol.
46, Number5, 2011, pp. 657-668, DOI: 10.3103/S0025654411050013.
• Aslanov V.S. Oscillations of a Spacecraft with a Vertical Elastic Tether - AIP Conference Proceedings
1220, CURRENT THEMES IN ENGINEERING SCIENCE 2009: Selected Presentations at the World
Congress on Engineering-2009, Published February 2010; ISBN 978-0-7354-0766-4, Vol. 1, 1-16.
• Aslanov V.S. The effect of the elasticity of an orbital tether system on the oscillations of a satellite -
Journal of Applied Mathematics and Mechanics 74 (2010) 416–424.
• Aslanov V.S. The Oscillations of a Spacecraft under the Action of the Tether Tension Moment and
the Gravitational Moment - American Institute of Physics (AIP) conference proceedings 1048,
ICNAAM, Melville, New York, pp.56-59, 2008
• Aslanov V. S. The oscillations of a body with an orbital tethered system - Journal of Applied
Mathematics and Mechanics 71 (2007) 926–932.
• V.S. Aslanov, A.V. Pirozhenko, B.V. Ivanov, A.S. Ledkov. Chaotic Motion of the Elastic Tether System
- Vestnik SSAU, ISSN 1998-6629, 2009, No. 4(20), 9-15.
• V.S. Aslanov, A.S. Ledkov, N.R. Stratilatov. The Influence of the Cable System Dedicated to Deliver
Freights to the Earth of the Rotary Motion of Spacecraft -Scientific and technical journal "Polyot"
("Flight"), ISSN 1684-1301, 2009, No. 1, 54-60.
• V.S. Aslanov, A.S. Ledkov, N.R. Stratilatov. Spatial Motion of Space Rope Cago Transport System -
Scientific and technical journal "Polyot" ("Flight"), ISSN 1684-1301, 2007, No. 2, 28-33.
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