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So to increase force in a cable, the cable must be spooled, and energy applied to spool it. This energy is stored in the elastic strain of the cable material. A thinner cable stretches more per unit of force, and moves faster, so more energy must be expended to spool it.

Assume a cable with a density of $ \rho $ and a static tension $ F_T $, with a spool at $ l = 0 $ and an infinitely strong attachment at $ l = L $. For a step increase in force of $ \delta F $ away from the attachment, the strain wave propagates down the cable towards the attachment at the speed of sound $ V_C $, reflecting off it and propagating back, returning to the spool in time $ T_{RT} = L / 2 V_C $. The cable in front of the strain wave moves away from the attachment at velocity $ V_S $, and the force accelerates a new segment of length $ d l $ in time $ d t $. The segment length $ d l = V_C d t $, the segment mass $ d m = \rho d l = \rho V_C d t $ and the force is $ \delta F = \rho V_C V_S $. Assuming $ \delta F \ll F_T $, the power expended spooling the cable is $ P = F_T V_S = \delta F F_T / \rho V_C $. The total energy is $ E = 2 \delta F F_T / \rho {V_C}^2 $.
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The CTE simulation by [[ htp://etd.lib.fsu.edu/theses/available/etd-08262005-003434/.../Thesis.pdf | Prakash ]] is used here, but other simulations differ wildly. The CTE simulation by [[ http://etd.lib.fsu.edu/theses/available/etd-08262005-003434/.../Thesis.pdf | Prakash ]] is used here, but other simulations differ wildly.

Linear Cables

Stabilization and elevator cables on the launch loop are very long, and propagation delay is a big issue. In most systems people are familiar with, cables are short enough and forces change slowly enough that propagation delay is not a major issue. With a launch loop, forces can change rapidly (milliseconds) while the propagation delays are 10s of seconds.

For example, when an instantaneous force change is applied to one end of a very long cable, the end does not stretch a little, it moves, and keeps moving until the force has had time to propagate to a stationary attachment and back. For a 100km Kevlar 49 stabilization cable, that can be 22 seconds, in which time many meters of cable moves.

So to increase force in a cable, the cable must be spooled, and energy applied to spool it. This energy is stored in the elastic strain of the cable material. A thinner cable stretches more per unit of force, and moves faster, so more energy must be expended to spool it.

Assume a cable with a density of \rho and a static tension F_T , with a spool at l = 0 and an infinitely strong attachment at l = L . For a step increase in force of \delta F away from the attachment, the strain wave propagates down the cable towards the attachment at the speed of sound V_C , reflecting off it and propagating back, returning to the spool in time T_{RT} = L / 2 V_C . The cable in front of the strain wave moves away from the attachment at velocity V_S , and the force accelerates a new segment of length d l in time d t . The segment length d l = V_C d t , the segment mass d m = \rho d l = \rho V_C d t and the force is \delta F = \rho V_C V_S . Assuming \delta F \ll F_T , the power expended spooling the cable is P = F_T V_S = \delta F F_T / \rho V_C . The total energy is E = 2 \delta F F_T / \rho {V_C}^2 .

Material

density

elastic

strength

CTE

Vsound

Support

100km

Therm exp

notes

modulus

Length

Round Trip

100Km*100K

gm/cm3

GPa

GPa

um/m-K

km/s

km

seconds

m

Steel SAE980x

7.9

200

0.65

12

5.0

8.4

40

120

Pure Kevlar

1.44

124

3.62

-2.7

9.3

250

22

-27

Pure Spectra

0.97

168

2.58

-12

13.2

270

15

-120

continuous creep

Pure Diamond

3.52

1140

>60

1.2

18.0

1700

11

12

Pure nanotube

~1.4

~1000

~60

-9?

26.0

4400

8

-90

composites:

80% Kevlar

80% Spectra

80% nanotube

note: Nanotube properties are controversial. The CTE simulation by Prakash is used here, but other simulations differ wildly.

LinearCables (last edited 2009-08-14 20:56:24 by KeithLofstrom)