Vladimir S. Aslanov, Alexander S. Ledkov, Arun K. Misra, Anna D. Guerman 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.

1. Vladimir S. Aslanov, Alexander S. Ledkov,
Arun K. Misra, Anna D. Guerman
Motion of the space elevator
after the ribbon rupture
The 63rd International Astronautical Congress
IAC-12.D4.3.9

2. 2
Statement of the problem
Building a space elevator requires
finding solutions to a significant
number of complex scientific problems.
One of the principal challenges is the
guaranteeing the longevity and safety
of the construction. Destruction of the
space elevator due to collision with
space debris is a probable scenario.
The elements of the destroyed elevator
can be a serious threat for both
spacecraft as well as for objects
located 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 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.

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 Earth's 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 model
We divide the tether into N+1 sections of finite length and arrange material points
on the borders of sections. Each tether’s section between material points we
represent as massless viscoelastic bar.
The equations of motion of i-th
elevator's points in rotating Greenwich
coordinate 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-th
point,
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 atmosphere
We represent the tether segments which
adjoin to the i-th point in the form of a
truncated cone bent in the middle.
(5)
where N – normal force,  - tangential force.
(6)
where Cj,i, Cnj,i – dimensionless
coefficients of the tangential and normal
aerodynamic forces, Si – area of cross-
section of the tether in i-th point,
qi – dynamic pressure, j,i - angle between
the vectors Vi and i+j.
(7)

7. 7
Calculation of aerodynamic characteristics
At a hypersonic use of the method of calculation of aerodynamic characteristics,
based on the Newtonian shock theory, gives good results. Taking into account that
the 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 conditions
For simplicity, we assume that the tether is deployed along the OY axis. Equating
the 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 system
of nonlinear equations, which can be determined from the initial positions of
the elevator's points.

9. 9
Simulation: Parameters of the system
Parameter of the system Value
Elevator length 117000 km
Density of the tether’s material 1300 kg/m3
Tensile strength of the tether’s 130 GPa
material
Young's modulus 630 GPa
Mass of the space station 3500 kg
Dependence of tether’s
The maximum diameter of the 3,8∙10-4m diameter on distance
tether between tether’s point and
The minimum diameter of the 2,6∙10-4m center of the Earth
tether

10. 10
Motion of upper part after rupture
We represent a space elevator tether in the form of N=100 material points. spend
a series of calculations, choosing as a rupture point various points of the tether.
Parabolic speed: (10)
The velocity of center of mass of
upper part at the moment of
breakage 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 elevator
where rGEO - radius of the geostationary orbit.
The center of mass of the considered space elevator is at an altitude of
6∙104_km which exceeds the boundary value r*  4,5∙104 km. Therefore the
upper part will pass to a hyperbolic trajectory regardless of the rupture point.

11. 12
Motion of lower part after rupture
Let's consider a boundary situation
when the tether breaks away from
the station for an estimation of
character of motion of the bottom
end of the elevator. In this case the
length of the falling tether is
maximum.
Excluding the influence of the
atmosphere the tether with length
of 117000 km falls to the ground in
about 64500 s. During this time the
tether goes around the Earth two
and a half times.

12. 14
Motion of lower part after rupture
Height of tether rupture
r=3,6∙107m
Initial number of points
N=20
Maximum number of
points
Nmax= 3512
Time of tether falls
t=9453 s
Maximum velocity of
point at landing moment
Vmax= 3355 m/s

13. 15
Evolution of tether’s loops
Since the angles  are small, the main contribution to the aerodynamic force makes the
normal 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 component
of 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 Ya
makes significant impact on the motion of the
section (Fig. c), that can increase its height. It as
though ricochets from the atmosphere,
dragging other parts of the tether that leads to
loop formation.
With increase in height, the atmospheric
density decreases. It leads to reduction of the
aerodynamic force, and the tether part starts to
move again downwards. The loop "falls" to the
Earth (Fig. d).

14. 16
Conclusion
The problem of investigation of dynamics of the space elevator after its destruction
by space debris was considered.
A mathematical model, in which the flexible heavy tether of circular cross-section
is represented as a set of the mass points connected by a set of massless
viscoelastic bars was developed. A distinctive feature of the model is the
consideration 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
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