MECHANICS
ÓÄÊ 539.3
Academician S. A. Ambartsumian, M. V. Belubekyan, K. B.
Ghazaryan,
V. Ts. Gnuni
On Design Problem of a Space Elevator Cable
(Submitted 13/V 2004)
The concept of the space elevator comprises a
very long cable/ribbon (twenty times the Earth radius) with one end fixed to the
Earth and the other end free out in the space [1]. Based on this concept we
present here the mathematical study and formulation of mechanics problems of
strength and elastic stability arising in a simple elastic rod, with a view of a
better understanding of this physical phenomenon that could lead to a new
knowledge in the field of space elevator studies.
Let us consider the equilibrium state of a
very long elastic cable anchored on the Earth equatorial point. The cable
subject to both the action of Earth gravity inward force F_{1}, defined
by Newton gravity law
F_{1}(g) = 
rg_{0}R_{0}^{2}
(R_{0 }+ g)^{2}

 
(1) 
and
centrifugal outward force F_{2}, due to Earth daily spinning
F_{2}(g) = rw^{2}(R_{0 }+ g).  
(2) 
In
(1), (2) g is a coordinate along cable length counted
from Earth surface, r is the bulk density of the cable
material, R_{0 }= 6378 km is the Earth equatorial radius,
g_{0 }= 980 sm·sec^{2} is the gravity
force acceleration on the Earth surface, w = 2p/T is the circular spinning frequency of the Earth, T = 86146
sec is the period of the Earth spinning.
For cables anchored at the points of the
Earth interstitial latitudes centrifugal outward force is equal to
F_{2}(g) = rw^{2}(R + g),  
where
R is the radius of the corresponding latitude circumference, which changes from
zero at the pole to R_{0} (at the equator).
When the crosssection area S(x) of the cable
is the function of cable length (tapered cable), using the dimensionless
notations the equation determining the cable elastic stress s(x) can be written as,

d[S(x)s(x)]
dx

+ S(x)F(x) = 0,  
(3) 
F(x) = rg_{0}R_{0}g(x),
g(x) = 
é ê ë 
a(1 + x)  
1
(1 + x)^{2}


ù ú û 
.  
(4) 
Here the following dimensionless
notations are used:
x = g/R_{0},
L = l/R_{0}, a = w^{2}R_{0}/g_{0} » 1/288, l is the cable length.
Coefficient a characterizes the ratio of the gravity and centrifugal
accelerations on the Earth surface. For the Mars and the Moon this coefficient
equals to
a_{1} » 1/218, a_{2} »
1/176.  
Equation (3) is to be considered with the following boundary condition
at free end
When
S(x)=const, the elastic stress is defined by the following function, satisfying
to (35)
s(x) = rg_{0}R_{0}(x), 

(6) 
(x) = 
(L  x)[(1 + L)(1 + x)(2 + L + x)  576]
576(1 + L)(1 + x)

.  
Since function
(x) may
have only one zero in the interval x Î [0,L], then from
condition
(0) = 0, (1 + L)(2 + L) = 576
, follows that for all L ³ L_{0 }= 22.5
(x) ³ 0,
x Î
[0,L_{0}].  
(7) 
Therefore, when the length of the cable is more than
l_{0 }= 143.325 km the cable is in a pure tension state. For cables with
L = L_{0}, s(0) = 0 and mechanical tension stress
reaches its maximum value s_{0} » 0.78rg_{0}R_{0},
at point x_{0} » 5.602 (where
g(x_{0}) = 0), which corresponds to the Earth geosynchronous orbit g_{0 }= 35.785 km. For cables with L ³ L_{0}, we have the following condition
sup_{x}[s(x)] ³ s_{0}.
Let us note
that the "limit" lengths for space elevator cable located on the Mars or the
Moon equators are the following:
L_{01 }= 19.38
(l_{01 }= 65698)
km  
L_{02 }= 17.21
(l_{02 }= 29893)
km  
To minimize the cable maximum tension stress and its critical length we
need to consider a tapered cable, crosssection area of which is the function of
cable length S = S(x).
Based on solution of equation (3) the stress function can be written as
s(x) =


(8) 
The solution (8) satisfies the
boundary condition (5) at a free end.
Let us consider the cable with the following
cross section area function
S_{*}(x)S_{0}exp 

(9) 
where
is a constant one.
Substituting (8) into (9)
leads to
s(x) = 

(10) 
From (10) follows that analogous to the case of the non tapered cable,
s(x ³ 0), when L ³ L_{0} » 22.4; for
L = L_{0}, s(0) = 0 and mechanical tension stress
reaches to its maximum value at x_{0} » 5.60
max s(x) = 

(11) 
On the other hand we have

max S_{* }= S_{0}exp 

(12) 
From (12), (13) one can note that
where h is
a tapering ratio of the cable, s_{0 }= 0.76rg_{0}R_{0} is the maximum stress of the of
non tapered cable.
On the cable ends we have
S_{*}(0) = S_{*}(L) = S_{0}  
Based on formula (13) the dependence of the maximum stress value from
tapering ratio is given in Table 1. As it follows from Table 1. data, the
thickening of the cable at the point of the Earth geosynchronous orbit leads to
the decreasing of the maximum tension stress. E.g. for a continuous cable of a
circular cross section the two times increase of the cable radius at the
geosynchronous orbit point leads to the two times decrease of the maximum
stress.
Table 1
Dependence of Maximum Stress upon Tapered
Ratio
h 
1.1 
1.6 
2.5 
4 
8 
10 
20 
x 
0.95 
0.79 
0.65 
0.54 
0.42 
0.39 
0.31 
The abovementioned results determine the projects of the cable design
when the "limit" length does not change, while the cable form permits decrease
of maximum stress.
We can reach the
decreasing both the "limit" length and the maximum stress by appropriate choice
of the cross section area function. Let us consider the cross section area
functions as the quadratic polynomial functions of the following type
S(x) = S_{0} 
é ê ë 
1 + 
ax
L

+ 
bx^{2}
L^{2}


ù ú û 
,  
(14) 
where a,b are arbitrary
constants.
Substituting (14) into (8), we
obtain the stress function depending upon parameters a,b. Since the stress
function expression is very cumbrous we do not present it
here.
By means of numerical analysis of this
expression the cable forms, namely, parameters a,b are determined, providing the
implementation of the following conditions
s(x) ³ 0, 
max x

s(x) 
® S


min

 
These numerical
results for two considered cases are the following:
cable "limit" length
L_{0 }= 18, (b = 1.04, a = 2.0), max_{x}s(x) = 0.61rg_{0}R_{0},
cable "limit" length
L_{0 }= 16.0, (b = 2.2, a = 4.23), max_{x}s(x) = 0.52rg_{0}R_{0}.
Let us now consider the cable
with a "counterweight" of mass M_{0} attached to its outward end. In
this case, we have the following boundary condition
s(L) = 
M_{0}g_{0}g(L)
S_{0}

,  
g(L) = 
é ê ë 

(1 + L)
288

 
1
(1 + L)^{2}


ù ú û 
.  
(15) 
The stress function satisfying the
boundary condition (15) is the following
s(x) = rg_{0}R_{0}(x) 

(x) = 
(L  x)[(1 + L)(1 + x)(2 + L + x)  576]
576(1 + L)(1 + x)

+ 
M_{0}L
M


é ê ë 

(1 + L)
288

 
1
(1 + L)^{2}


ù ú û 
,  
(16) 
where M_{0 }= rLR_{0}S_{0} is the cable mass.
When the
"counterweight" of mass M_{0} is attached to the cable outward end, it
is possible to consider a design of uniformly tensioned cable when mechanical
stress is constant along the cable length. Such a design for a ponderable rod
was considered first by S. Timoshenko [2].
Seeking in equation (1) solution
s(x) = const we come to the following solution for the
cable crosssection area function.
S_{*}(x) = S_{0}exp 
é ê ë 

rg_{0}R_{0}
s_{*}


L ó õ x

g(x)dx 
ù ú û 
,  
(17) 
where
s_{* }= 
M_{0}g_{0}g(L)
S_{0}

,  
g(x) is the function
defined by (4), S_{0} is the cable crosssection area at the outward
end, where the mass M_{0} is attached. For a cable with "limit" length
L_{0 }= 22.4 function S(x) reaches to its maximum value S^{*} at
point x » 5.60.
The maximum value of the (17) can be
written as
On
the other hand we have
where s_{0 }= 0.76rg_{0}R_{0} is the maximum stress of the of
non tapered cable. From (18), (19) follows that decrease of the maximum stress
leads to exponential increase of S_{*}. E.g. when b = 2, we have S_{* }= 7.39S_{0}. For a
continuous cable of a circular cross section, it means that the cable radius
increase 2.7 times.
Table 2
"Limit" lengths and stress maximum values in depend of the
"counterweight" mass
M/M_{0} 
100 
50 
20 
10 
5 
2 
1 
L_{0} 
22.3 
22.07 
21.0 
20.48 
19.2 
16.2 
13.2 
s_{0} 
0.76 
0.74 
0.66 
0.63 
0.53 
13.2 
0.22 
The "counterweight" decreases slightly the "limit" length and stress
maximum values, when M >> M_{0}. In the contrary, the
"counterweight" decreases essentially the "limit" length and stress maximum
values, when the cable mass is compared with "counterweight" mass.
Let us
now consider the elastic stability problem in the non tapered cable when its
length is less than "limit" length, L < L_{0}. In this case the
compression stresses arise localized near cable base and due to it, the cable
may become unstable.
The cable elastic
stability equation and appropriate boundary conditions can be written as

d^{4}W
dx^{4}

 l 
d
dx


é ê ë 

_{0}

(x) 
dW
dx


ù ú û 
= 0;  
(20) 
W_{x=0 }= 0,

dW
dx


ê ê ê 
x=0

= 0,  

d^{2}W
dx^{2}


ê ê ê 
x=L

= 0, 
d^{3}W
dx^{3}


ê ê ê 
x=L

=0,  
(21) 
where the dimensionless function
(x) is defined as in (6), EI is
the bending rigidity of the cable material, W(x) transverse displacement of the
elastic cable, l º rg_{0}R_{0}^{3}S_{0}(EI)^{1}. Using notation y(x) = [dW/dx], integrating equation
(20) and taking into consideration the boundary conditions we come to the
following equation and boundary conditions

d^{2}y
dx^{2}

0  l 
_{0}

(x)y = 0,  
(22) 
y_{x=0 }= 0,

dy
dx


ê ê ê 
x=L

= 0.  
(23) 
Equation (22) with conditions (23)
is the selfadjoint boundary value problem determining eigenvalues l. Since the function
(x) changes its sign in the interval x
Î [0,L], the eigenvalues can be both positive and
negative ones. The minimum positive eigenvalue l_{0} corresponds to the critical load behind which
the cable becomes unstable.
Let function
(x) change its sign from a negative
value to a positive one at point x_{0}.
Then for minimum positive
eigenvalue, l_{0} the following inequality is
valid
l_{0} <

x_{0} ó õ 0

(y^{¢}_{0})^{2}dx 
x_{0} ó õ 0

y_{0}^{2}g(x)dx  
,  
where function
y_{0}(x) is any admissible function satisfying to the conditions
y_{0}(0) = y_{0}(x_{0}) = 0.
Let us consider the cable
with L = 22.0. For this cable x_{0 }= 0.04; and taking
y_{0}(x) = x(x  x_{0}) we come to the
inequality l_{0} < 6326. Based on this
inequality we can postulate that for a rod with circular cross section of radius
r when r < r_{0} the elastic rod under consideration is unstable,
where
r_{0 }= 0.025 

Therefore, the
solution of the selfadjoint boundary value problem (2223) even in this case
(L = 22.0) results in the instability of the cable for any values of elastic and
geometrical parameters having practical and science forecasting meanings.
Conclusions.
Based on
the solution of the one dimension equation of the elasticity theory qualitative
and quantitative results are obtained related to strength and stability problems
of a space elevator cable/ribbon. It is assumed that the cable is subjected to
action of the Earth gravity inward force, defined by Newton gravity law and the
centrifugal outward force, due to the Earth's daily spinning. The numerical data
related to the strength characteristics of new modern materials such as carbon
nanotubs are omitted, since these data are in [1]. We confined ourselves to
results related to the tapered cable design strength problems. Based on the
solution of a one dimension elasticity equation solution of the tensile strength
does not determine an absolute value of cable crosssection area. This cross
section value parameter may be determined while considering the two dimensional
problems or dynamics problems, especially elastic wave propagation along a
tensioned cable.
On the other hand, there is no doubt that the future objects
of the investigation should be constructions of elastic closed shells (pipes)
type, which are of interest from an applied point of view. For such
constructions, the new mechanical problems should be considered, which take into
account circular, transversal stresses and displacements arising in the shells.
Among these problems we can list the dynamic interaction of closed shells
subject to external media, including electromagnetic, temperature and atmosphere
fields actions.
Institute of Mechanics of NAS RA
Ëèòåðàòóðà
1. B. C. Edvards, E. A.
Westling  The Space Elevator. Houston. USA.
2002.
2. S. Timoshenko  The Strength of Materials. Fizmatgiz. Moscow. 1965.