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 .

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