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The 63rd International Astronautical Congress

The purposes this research are

+ development of the mathematical model for a space elevator taking into account the influence of the atmosphere;

+ study of dynamics of elevator's elements when its ribbon is cut;

+ analysis of the consequences of the rupture of the space elevator ribbon for satellites and objects on the ground.

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- 1. Vladimir S. Aslanov, Alexander S. Ledkov, Arun K. Misra, Anna D. GuermanMotion of the space elevator after the ribbon rupture The 63rd International Astronautical Congress IAC-12.D4.3.9
- 2. 2 Statement of the problemBuilding a space elevator requiresfinding solutions to a significantnumber of complex scientific problems.One of the principal challenges is theguaranteeing the longevity and safetyof the construction. Destruction of thespace elevator due to collision withspace debris is a probable scenario.The elements of the destroyed elevatorcan be a serious threat for bothspacecraft as well as for objectslocated on the Earth.The purposes this research are• development of the mathematical model for a space elevator taking into account the influence of the atmosphere;• study of dynamics of elevators elements when its ribbon is cut;• analysis of the consequences of the rupture of the space elevator ribbon for satellites and objects on the ground.
- 3. 3 Assumptions• The space elevator is located in an equatorial plane of the Earth.• At the moment of tether rupture the elevator cage is on the ground.• Force of the Earths gravity and the aerodynamic force acting on the system. Other disturbing forces are not considered.• Gravitational field of the Earth is Newtonian.• Atmosphere is stationary isometric.• The tether is a homogeneous. It has circular cross-section of variable diameter.• The surface of the Earth is perfectly inelastic.• The space station is considered as a material point.
- 4. 4 Mathematical modelWe divide the tether into N+1 sections of finite length and arrange material pointson the borders of sections. Each tether’s section between material points werepresent as massless viscoelastic bar.The equations of motion of i-thelevators points in rotating Greenwichcoordinate system OXYZ have form (1)where mi – mass of i-th point,Wi - acceleration of i-th point in OXYZ,Gi - gravitational force,Ti - tension in the i-th tether sections,Fi - aerodynamic force operating on i-thpoint,icf - centrifugal force of inertia,iС - Coriolis force of inertia.
- 5. 5 Forces Gravitational force (2)where ri – radius vector of i-th point, - the gravitational constant of the Earth. Tension in the i-th tether sections (3) (4) where Smi – area of cross-section in the middle of i-th tether section, E - modulus of elasticity of the tether, Di - coefficient of internal friction,_________i - elongation of i-th tether section, DLi - length of the i-th section of i i / DL i ri ri 1 the tether in the undeformed state, __________.
- 6. 6 The interaction with the atmosphereWe represent the tether segments whichadjoin to the i-th point in the form of atruncated cone bent in the middle. (5)where N – normal force, - tangential force. (6)where Cj,i, Cnj,i – dimensionlesscoefficients of the tangential and normalaerodynamic forces, Si – area of cross-section of the tether in i-th point,qi – dynamic pressure, j,i - angle betweenthe vectors Vi and i+j. (7)
- 7. 7 Calculation of aerodynamic characteristicsAt a hypersonic use of the method of calculation of aerodynamic characteristics,based on the Newtonian shock theory, gives good results. Taking into account thatthe angle i is small, for a truncated cone we can obtain If di+j<di+j-1 the truncated cone is turned to a flow at 180 , and instead of resulted above expressions for and it is necessary to use
- 8. 8 Selection of initial conditionsFor simplicity, we assume that the tether is deployed along the OY axis. Equatingthe right hand side of (1) to zero, we can obtain stationary solutions (8)Substituting (4) into last equation we have (9)Assuming yN+2=yN+1 and y0 equal to the Earth radius from (9) we have a systemof nonlinear equations, which can be determined from the initial positions ofthe elevators points.
- 9. 9 Simulation: Parameters of the system Parameter of the system ValueElevator length 117000 kmDensity of the tether’s material 1300 kg/m3Tensile strength of the tether’s 130 GPamaterialYoungs modulus 630 GPaMass of the space station 3500 kg Dependence of tether’sThe maximum diameter of the 3,8∙10-4m diameter on distancetether between tether’s point andThe minimum diameter of the 2,6∙10-4m center of the Earthtether
- 10. 10 Motion of upper part after ruptureWe represent a space elevator tether in the form of N=100 material points. spenda series of calculations, choosing as a rupture point various points of the tether.Parabolic speed: (10)The velocity of center of mass ofupper part at the moment ofbreakage can be found as (11) Equating the (10) and (11) we obtain the boundary value Trajectories of the center of mass of (12) the upper part of the space elevatorwhere rGEO - radius of the geostationary orbit.The center of mass of the considered space elevator is at an altitude of6∙104_km which exceeds the boundary value r* 4,5∙104 km. Therefore theupper part will pass to a hyperbolic trajectory regardless of the rupture point.
- 11. 12 Motion of lower part after ruptureLets consider a boundary situationwhen the tether breaks away fromthe station for an estimation ofcharacter of motion of the bottomend of the elevator. In this case thelength of the falling tether ismaximum.Excluding the influence of theatmosphere the tether with lengthof 117000 km falls to the ground inabout 64500 s. During this time thetether goes around the Earth twoand a half times.
- 12. 14 Motion of lower part after ruptureHeight of tether rupture r=3,6∙107mInitial number of points N=20Maximum number ofpoints Nmax= 3512Time of tether falls t=9453 sMaximum velocity ofpoint at landing moment Vmax= 3355 m/s
- 13. 15 Evolution of tether’s loopsSince the angles are small, the main contribution to the aerodynamic force makes thenormal component. The points "slide" along the tether (Fig. a ).If the angle is close to 90, strong influence on the motion will have a normal componentof the aerodynamic force. In this case aerodynamic drag Xa will surpass considerably lifting force Ya (Fig. b), and influence of aerodynamic force will be expressed in braking of a falling section.If the angle is not great enough, the force Yamakes significant impact on the motion of thesection (Fig. c), that can increase its height. It asthough ricochets from the atmosphere,dragging other parts of the tether that leads toloop formation.With increase in height, the atmosphericdensity decreases. It leads to reduction of theaerodynamic force, and the tether part starts tomove again downwards. The loop "falls" to theEarth (Fig. d).
- 14. 16 ConclusionThe problem of investigation of dynamics of the space elevator after its destructionby space debris was considered.A mathematical model, in which the flexible heavy tether of circular cross-sectionis represented as a set of the mass points connected by a set of masslessviscoelastic bars was developed. A distinctive feature of the model is theconsideration of aerodynamic forces acting on the tether.The results of the simulation show that• The upper section of the space elevator will pass to a hyperbolic trajectory regardless of the rupture point.• Rupture of the space elevator ribbon can jeopardize the spacecraft in the equatorial plane since the ribbon moves with rather large velocity.• The lower part of the space elevator after the ribbon enters the atmosphere. Most of it slows down and falls smoothly, but some elements reach the Earth surface with rather large velocities. These parts of the tether can put in danger objects on the ground.
- 15. 17 References1. Aslanov, V.S., Ledkov, A.S., Dynamics of tethered satellite systems, Woodhead Publishing, Cambridge, 2012.2. Pearson, J., "The Orbital Tower: A Spacecraft Launcher Using the Earths Rotational Energy", Acta Astronautica, №2, 1975, pp. 785-799.3. Smitherman, D.V. Jr., "Space Elevator: An Advanced Earth-Space Infrastructure for the New Millenium", Marshall Space Flight Center. Huntsville, Alabama. NASA/CP-2000-210429. August 2000.4. Edwards, B.C., "Design and deployment of a space elevator", Acta Astronautica, №2, 2000, pp. 785–799.5. Sadov, Yu.A., Nuralieva, A.B. "On Conception of the Loaded Sectioned Space Elevator", Preprint of Keldysh Institute of Applied Mathematics RAS. Moscow, 2011.6. Edwards, B.C., "The Space Elevator", Final Report on the theme "The Space Elevator" for the NASA Institute for Advanced Concepts (NIAC). 80 p.7. Quine, B.M., Seth, R.K., Zhu, Z.H., "A free-standing space elevator structure: A practical alternative to the space tether", Acta Astronautica, №65, 2009, pp. 365-375.9. Perek, L., " Space elevator: Stability“, Acta Astronautica, Vol.62, №8–9, 2008, p. 514–520.10. Alpatov, A.P., Dranovskii, V.I., Zakrzhevskii, A.E., Pirozhenko, A.V., "Space tether systems. The problem review", Cosmic Science and Technology, Vol.3, №5/6, 1997, pp. 21-29. (in Russian)11. Aslanov, V.S., Ledkov, A.S., Stratilatov, N.R., "Spatial Motion of Space Rope Cago Transport System", Scientific and technical journal "Polyot" ("Flight"), №2, 2007, pp. 28-33. (in Russian)12. Williams, P., "Dynamic multibody modeling for tethered space elevators", Acta Astronautica, №65, 2009, pp. 399–422.13. Beletsky, V.V., Levin, E.M., "Dynamics of space tether systems", Univelt, Inc., San Diego, 1993, p. 499.14. Arjannikov, N.S., Sadekova, G.S., "Aerodynamics of a flying vehicle", Higher School, Moscow, 1983.

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