Design of a Space Elevator

1. Design of a Space Elevator Rutgers Symposium on Lunar Settlements June 3-8, 2007 Seon Han Texas Tech University Lubbock, TX 79410

2. 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.

3. 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).

4. 3. Contributors • Russian visionaries • Tsiolkovsky, 1959 • Artsutanov, 1960, 1969, 1970 • Isaacs et al., 1966 • Clark, 1978 Fountain of Paradise • Pearson, 1975 • Edwards, 2000

5. 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.

6. 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

7. 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. Mass ratio vs. elevator length

8. 4.3. Design Parameters • Constant stress level • Counter mass and elevator length • Cross-sectional area • Constraints from dynamics • Resonance • Response due to moving load

9. 5. Dynamic Model • From linear analysis • Eigenvalue problem

10. 5.1 Natural Periods – Longitudinal motion

11. 5.2 Natural Periods – Transverse motion

12. 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.

13. 6.1 Tidal Forces due to the Moon • 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.

14. 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.

15. Solar tidal forces at the surface, GEO, and end point Periods: 12 hr, 8 hr, 6hr, 24 hr Periods: 24 hr, 12 hr, 8 hr, Periods: 12 hr, 8 hr, 6 hr, 24 hr

16. Lunar tidal forces along the elevator in three directions

17. 7. Dynamic Responses – Constant Stress Model •  = 100 GPa •  = 1300 kg/m3, E=1 TPa • Ao = 6.1129e-008

18. 7.1. Dynamic Response in the Radial Direction

19. Dynamic Response in the Radial Direction – Spectral Density

20. 7.2 Dynamic Response in the S-N Direction

21. Dynamic Response in the S-N Direction –Spectral Density

22. 7.3 Dynamic Response in the E-W Direction

23. Dynamic Response in the E-W Direction – Spectral Density

24. 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

25. Natural Periods of the Longitudinal Motion Periods are very similar to those of the constant stress model

26. Natual Periods of the Transverse Motion

27. Dynamic Response in the Radial Direction

28. Dynamic Response in the S-N direction

29. Dynamic Response in the E-W Direction

30. 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.

31. 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.

32. 10. Additional Thoughts • Another mass at GEO for better stability • Subharmonic resonance • Effects of elevator car