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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. 3
- 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. 4
- 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. 5
- 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. 6
- 7. 1.1.5. The research and national economy Interferometer with a long base Installation of radio-repeaters 7
- 8. 1.1.6 Transport operations in space Space Elevator 8
- 9. 1.1.7. Transport operations in space Delivery of payload Space Escalator 9
- 10. 1.1.7. Transport operations in space Braking of the spacecraft by electrodynamics tether 10
- 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 3
- 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 4
- 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. 5
- 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 6
- 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. 7
- 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 8
- 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 ri1 Here ri – density of the atmosphere at an altitude of i-th point, dT – diameter of the tether, k – adjusted coefficient of Newton. 9
- 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 n2 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. 10
- 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 mii Gi FAi Ti Ti 1 Фi Ц Фi К r Inertial forces Фi Ц miω3 (ω3 ri ) Фi К 2miω3 Vi 11
- 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. 12
- 21. 3. Software implementation (TetherCalc) The model is implemented in MatLab package 13
- 22. 4 Analysis of abnormal situations 14
- 23. 4 Analysis of abnormal situations Wrong orientation at the cargo separation 15
- 24. 4 Analysis of abnormal situations Breakage of attitude control system 16
- 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 17
- 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. 18