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Design of a Space Elevator. Rutgers Symposium on Lunar Settlements June 3-8, 2007 Seon Han Texas Tech University Lubbock, TX 79410. 1. Introduction. A space elevator is essentially a long cable that starts from Earth’s surface and extends beyond the geosynchronous (GEO) orbit. - PowerPoint PPT Presentation
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Design of a Space Elevator
Rutgers Symposium on Lunar SettlementsJune 3-8, 2007
Seon HanTexas Tech University
Lubbock, TX 79410
1. Introduction
• A space elevator is essentially a long cable that starts from Earth’s surface and extends beyond the geosynchronous (GEO) orbit.
• GEO is where the gravitational pull and the centripetal acceleration are balanced. GEO is has an altitude of 35,800 km.
• A portion below GEO tends to pull toward Earth, and the portion beyond GEO tends to pull away from Earth.
• A space elevator is a tension structure.
2. Benefits
• Space elevator can provide easier, safer, faster, and cheaper access to space.
– Currently, it costs approximately $10,000 per pound to launch items into orbit. It costs about $400 million to launch a satellite to GEO.
– Average rocket failure rate is 1 in one hundred. Out of total 117 space shuttles missions, 2 resulted in catastrophic failures (Challenger, 1986 and Columbia, 2003).
3. Contributors
• Russian visionaries – Tsiolkovsky, 1959– Artsutanov, 1960, 1969, 1970
• Isaacs et al., 1966
• Clark, 1978 Fountain of Paradise
• Pearson, 1975
• Edwards, 2000
4. Design Ideas from Previous Studies• Elevator should be placed near the
equator.– At the equator, the centripetal acceleration is
along the cable.
• The cross-sectional area must be tapered.
• The counter mass can be used to shorten the space elevator.
4.1 Design Ideas - Tapering
• Cable should be tapered so that the stress on the cable is uniform (Pearson, 1975)– Theoretically, the tower can be made of any material by simply
using a large enough taper ratio (Area at GEO/Area at surface).– Cross-sectional area is largest at GEO.– However, the taper ratio can be unrealistically large.
– Larger the specific strength, smaller the taper ratio. – For steel, it is for – For carbon nanotubes, the ratio can be as low as 1.5 at
0.776 ( / )/ o or gs oA A e
31.5 10/s oA A e 250 .MPa
150 .GPa
4.2 Design Ideas – Counter Mass
• A counter mass is used to shorten the length of the elevator
– Without a counter mass, the length of the elevator is 144,000 km.
– Tower length must extend beyond GEO (35,800 km).
– Heavier mass is needed for shorter tower.
– Lighter the counter mass, the lower stress level.
x 1 04E lev a to r len g h ( )km
0 5 1 0 1 50
2
4
6
8
1 0
poin
t mas
s/ c
able
mas
s
GE
O
= 1 5 0 G P a
= 5 0 G P a
144,
000
km
= 1 0 0 G P a
Mass ratio vs. elevator length
4.3. Design Parameters
• Constant stress level
• Counter mass and elevator length
• Cross-sectional area
• Constraints from dynamics– Resonance– Response due to moving load
5. Dynamic Model• From linear analysis
• Eigenvalue problem
2
2
2
2,,
, ,
0, 0
0pL tL t
u r t u r tEA r A r
r r t
with
u t
u uM EA
t r
2
2
2
2,,
, ,
0, 0
0pL tL t
v r t v r tA r A r
r r t
with
v t
v vM A
t r
2
2
2 2
0
/ or /
p
L
Md dA d dUA U AU with U L A
dx dx dx dx
E
5.1 Natural Periods – Longitudinal motion
31300 /
1
kg m
E TPa
4L en g th o f th e E lev a to r ( )km
N
atur
al P
erio
d of
the
Lon
gitu
dina
l Mot
ion
()
hr
= 1 0 0 G P a
4 6 8 1 0 1 2 1 4 1 60
5
1 0
1 5
= 8 0 G P a= 6 0 G P a
M o d e 1
M o d e 2
M o d e 3
x 10
x 1 04
L en g th o f th e E lev a to r ( )km
Fun
dam
enta
l Per
iod
of
Tra
nsve
rse
Mot
ion
()
hr
0
10
20
30
40
50
60
= 1 0 0 G P a= 8 0 G P a= 6 0 G P a
M ode 1
M ode 2
M ode 34 6 8 1 0 1 2 1 4 1 6
5.2 Natural Periods – Transverse motion
6. Forces on the Space Elevator
• Tidal Forces– The period of the tidal force is approximately 12.5
hours due to the Moon and 12 hours due to the Sun.– The tidal force due to the sun is less than 50% of that
due to the Moon.– The forcing strength varies along the cable. Further
away from Earth, the larger the tidal force.
• Wind Forces– Three quarters of the atmosphere's mass is within 11
km or 0.03% to the distance to GEO.– Wind force typically has a peak period in the order of
100 s, which is well away from natural periods of the constant stress space elevator.
6.1 Tidal Forces due to the Moon
23
/ moonemmoon em em
em
GMF mass R s I
R
i
k
D
E arth
M o o n
M o o n
(a ) V ie w fro m th e n o rth (b ) S id e V iew
m
r
mr
R em
R em
D em
em
sem
C G M
N
S
I
i
• xyz is attached to Earth. x is in the radial, y in meridional, and z in equatorial directions.• Period associated with em is 27.3 days.
-1
0
1
2
3x 10
-5
-2
-1
0
1
2x 10
-5
0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0-2
-1
0
1
2x 10
-5
Tid
al a
ccel
erat
ions
( )
m/s
2
x
y
z
Tim e ( )hrs
Lunar tidal forces at the surface, GEO, and end point
Periods: F=12.5 hr, 8.3 hr, 6.2 hr, 25 hr, 13.6 days
Periods: F=25 hr, 27.3 days, 12.5 hr,8.3 hr,
Periods: F=12.5 hr, 8.3hr, 25 hr, 13.6 days.
Tid
al a
ccel
erat
ions
( )
m/s
2
x
y
z
Tim e ( )h rs
-1
0
1x 10
-5
-5
0
5x 10
-6
0 100 200 300 400 500 600 700-1
0
1x 10
-5
Periods: 12 hr, 8 hr, 6hr, 24 hr
Periods: 24 hr, 12 hr, 8 hr,
Periods: 12 hr, 8 hr, 6 hr, 24 hr
Solar tidal forces at the surface, GEO, and end point
0
-5
0
5x 1 0
-5
-5
0
5
5 1 0 1 5-5
0
5
R ad ia l d irec tio n
M erid io n a l d irec tio n
E q u a to ria l d irec tio n
x 1 0-5
x 1 0-5
x 1 04
Tid
al f
orce
per
uni
t mas
s (
)m
/s2
D istan ce fro m E a rth 's su rface ( )km
Lunar tidal forces along the elevator in three directions
7. Dynamic Responses – Constant Stress Model
= 100 GPa = 1300 kg/m3, E=1 TPa
• Ao = 6.1129e-008
7.1. Dynamic Response in the Radial DirectionR
adia
l Res
pons
e of
the
Cou
nter
mas
s (
)m
Tim e ( )hrs
0 50 100 150 200-50
0
50
100
0 50 100 150 200-50
0
50
0 50 100 150 200-100
0
100
0 50 100 150 200-200
0
200
0 50 100 150 200-200
0
200
400
0 50 100 150 200-500
0
500
L = 4 .3 6 e7 L = 6 .3 6 e7
L = 8 .3 6 e L = 1 0 .3 6 e 7
7 7
7
L = 1 2 .3 6 e L = 1 4 .3 6 e
Dynamic Response in the Radial Direction – Spectral Density
0 10 20 30 40 50-10
0
10
20
0 10 20 30 40 50-10
0
10
20
0 10 20 30 40 50-10
0
10
20
0 10 20 30 40 50-10
0
10
20
0 10 20 30 40 50-10
0
10
20
0 10 20 30 40 50-10
0
10
20
PS
D (
)dB
P erio d ( )H rs
L = 4 .3 6 e7 L = 6 .3 6 e7
L = 8 .3 6 e L = 1 0 .3 6 e
L = 1 2 .3 6 e L = 1 4 .3 6 e
7 7
7 7
F 1N 1 , F 2
F 4
F 1
F 2
F 3 , N 1F 4
F 1 F 4
F 2
F 1
F 2
F 3 , N 1F 4
F 3 , N 1
F 1
F 2
F 3 , N 1F 4
F 1
F 2
F 3 , N 1 F 4
7.2 Dynamic Response in the S-N DirectionR
Mer
idio
nal
Res
pons
e of
the
Cou
nter
mas
s (
)m
Tim e ( )h rs
L = 4 .3 6 e7 L = 6 .3 6 e7
L = 8 .3 6 e L = 1 0 .3 6 e 7
7 7
7
L = 1 2 .3 6 e L = 1 4 .3 6 e
0 50 100 150 200-1000
0
1000
0 50 100 150 200-1000
0
1000
0 50 100 150 200-2000
0
2000
0 50 100 150 200-5000
0
5000
0 50 100 150 200
-5000
0
5000
0 50 100 150 200-1
0
1x 10
4
Dynamic Response in the S-N Direction –Spectral Density
PS
D (
)dB
P erio d ( )H rs
L = 4 .3 6 e7 L = 6 .3 6 e7
L = 8 .3 6 e L = 1 0 .3 6 e
L = 1 2 .3 6 e L = 1 4 .3 6 e
7
7 7
0 10 20 30 40 500
20
40
0 10 20 30 40 500
20
40
0 10 20 30 40 500
20
40
0 10 20 30 40 500
20
40
0 10 20 30 40 500
20
40
0 10 20 30 40 500
20
40
7
F 1 , N 1
N 1
F 1F 2
N 2
N 1
F 1F 2
F 4 , N 2
F 2F 3
N 1
F 1F 2
F 4 , N 2F 3
N 1 F 1F 2
F 4 , N 2
F 2F 3
N 1 F 1F 2
F 4 , N 2
F 2F 3
7.3 Dynamic Response in the E-W DirectionR
Mer
idio
nal
Res
pons
e of
the
Cou
nter
mas
s (
)m
Tim e ( )h rs
L = 4 .3 6 e7 L = 6 .3 6 e7
L = 8 .3 6 e L = 1 0 .3 6 e7
7 7
7
L = 1 2 .3 6 e L = 1 4 .3 6 e
-5000
4
0 50 100 150 200-50
0
50
0 50 100 150 200-500
0
500
0 50 100 150 200-1000
0
1000
0 50 100 150 200-2000
0
2000
0 50 100 150 200-5000
0
5000
0 50 100 150 200-5000
0
5000
Dynamic Response in the E-W Direction – Spectral Density
PS
D (
)dB
P erio d ( )H rs
L = 4 .3 6 e7 L = 6 .3 6 e7
L = 8 .3 6 e L = 1 0 .3 6 e
L = 1 2 .3 6 e L = 1 4 .3 6 e
7
7 7
7
N 1
F 1F 2
F 4 , N 2
F 3
0 10 20 30 40 500
20
40
0 10 20 30 40 500
20
40
0 10 20 30 40 500
20
40
0 10 20 30 40 500
20
40
0 10 20 30 40 500
20
40
0 10 20 30 40 500
20
40
N 2
F 1F 2
F 3
N 2
F 2
F 1
N 1 , F 3N 2F 2
F 1
N 1 , F 3N 2
F 2F 1
N 1N 2F 3
F 2F 1
N 1F 3
N 1F 1
F 2
F 3
N 2
F 1F 2
F 3
N 2
N 1F 1
F 2
F 3
N 2
F 1F 2
F 3
N 2
8. Design Variations
• Constant cross-sectional area– Maximum stress occurs
at GEO and is about 60 GPa
– The cross-sectional area is kept the same as the cross-sectional area at Earth’s surface of the constant stress model.
– Ao=6.1129e-008 m2
L en g th o f th e E lev a to r ( )km
Stre
ss (
MP
a)0 5 1 0 1 5
4
0
1 0
2 0
3 0
4 0
5 0
6 0
7 0
x 1 0
Natural Periods of the Longitudinal Motion
L en g th o f th e E lev a to r ( )km
Nat
ural
Per
iod
of th
e L
ongi
tudi
nal M
otio
n (h
rs)
4x 1 0
4 6 8 10 12 14 160
2
4
6
8
1 0
1 2
1 4
1 6
Periods are very similar to those of the constant stress model
Natual Periods of the Transverse Motion
L en g th o f th e E lev a to r ( )km
Nat
ural
Per
iod
of th
e T
rans
vers
e M
otio
n (h
rs)
4x 1 0
4 6 8 1 0 1 2 1 4 1 60
2 0
4 0
6 0
8 0
1 0 0
1 2 0
1 4 0
1 6 0
1 8 0
2 0 0
Dynamic Response in the Radial Direction
Rad
ial R
espo
nse
of th
e C
ount
erm
ass
()
m
Tim e ( )hrs
L = 4 .3 6 e7 L = 6 .3 6 e7
L = 8 .3 6 e L = 1 0 .3 6 e 7
7 7
7
L = 1 2 .3 6 e L = 1 4 .3 6 e
0 50 100 150 200-50
0
50
0 50 100 150 200-50
0
50
0 50 100 150 200-200
0
200
0 50 100 150 200-200
0
200
400
0 50 100 150 200-500
0
500
0 50 100 150 200-500
0
500
1000
Dynamic Response in the S-N directionM
erid
iona
l R
espo
nse
of th
e C
ount
erm
ass
()
m
Tim e ( )h rs
L = 4 .3 6 e7 L = 6 .3 6 e7
L = 8 .3 6 e L = 1 0 .3 6 e 7
7 7
7
L = 1 2 .3 6 e L = 1 4 .3 6 e
0 50 100 150 200-200
0
200
0 50 100 150 200-2000
0
2000
0 50 100 150 200-4000
-2000
0
2000
0 50 100 150 200-5000
0
5000
0 50 100 150 200-10000
-5000
0
5000
0 50 100 150 200-5
0
5x 10
5
Dynamic Response in the E-W Direction
Equ
ator
ial
Res
pons
e of
the
Cou
nter
mas
s (
)m
Tim e ( )hrs
L = 4 .3 6 e7 L = 6 .3 6 e7
L = 8 .3 6 e L = 1 0 .3 6 e 7
7 7
7
L = 1 2 .3 6 e L = 1 4 .3 6 e
0 50 100 150 200-50
0
50
0 50 100 150 200-200
0
200
0 50 100 150 200-500
0
500
0 50 100 150 200-1000
0
1000
0 50 100 150 200
-2000
-10000
1000
0 50 100 150 200-10000
-5000
0
5000
9. Summary of Results
• Longer the tower, the tower experiences larger dynamic motion. Tidal force increases as we move away from the Earth.
• The primary periods in the responses are N1 and F1.
Summary of Results - continued
• When constant cross-sectional area is used, the maximum stress of about 60 GPa occurs at GEO.
• When constant area is used, the fundamental period of the transverse motion is further away from the tidal forcing frequencies. As a result, the transverse motion along the constant longitude (north south direction) is reduced about 5 times.
10. Additional Thoughts
• Another mass at GEO for better stability
• Subharmonic resonance
• Effects of elevator car
