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= Pulley Elevators =
== for the Launch Loop and the Space Elevator ==

== Under Construction ==

A pulley elevator is a simple-appearing way to provide both support and lift in a space elevator or a launch loop.  The vehicle can be passive, containing no more than some kind of automatically actuated clamp so that it can release from one moving elevator cable and clamp onto another.

Real life is never that simple.  The main drawback for an elevator cable is that if it is tall enough and the cables are closely spaced, then they can easily come close enough to abrade each other.  The design must prevent this.  The other problem is the same as all long cables, where force changes are rapid compared to the speed-of-sound propagation time;  the analysis is complicated, and involves not just the static stretch of the cable, but the inertia and mass flow rate.

Megastructure cables and elevators face yet another complication; their interaction with a nonlinear gravitational field in a rotating frame.  Very long cables may move in ways not anticipated by a designer focused on elasticity and strength of materials, particularly movements radially from the expected straight-line path.   Electrostatic repulsion may also play a role.

=== Parameters for two pulley elevators ===

||             || Launch Loop ||   SE1000    || unit   ||
||  Height     ||   68        ||    1000     || km     ||
|| pulley diam ||   40        ||     40 m    || meter  ||
||  speed      ||   400       ||             || m/s    ||
|| altitude0   ||    4        ||      0      || km     ||
|| altitude1   ||   72        ||   1000      || km     ||
|| g0          || 9.79        || 9.81        || m/s^2^ ||
|| g1          || 9.59        || 7.33        || m/s^2^ ||
|| Tension 0   || 200000      ||             || Newton ||
|| Tension 1   || 420000      ||             || Newton ||
|| Material    || Kevlar 49   ||    CNT      ||        ||
|| Density     || 1440        ||  1300       || kg/m3  ||
|| Modulus     ||  127        ||   630       || GPa    ||
|| strength    ||  3.6        ||   150       || GPa    ||
|| safety      ||  2.0        ||             ||        ||
|| area        ||  2.33       ||             || cm^2^  ||
|| diameter    ||  1.72       ||   ---       || cm     ||
|| width       ||  ---        ||             || meter  ||
|| thickness   ||  ---        ||             || micron ||
|| mass        || 46000       ||             || kg     ||
|| vib. period ||  142        ||             || second ||
|| sound speed ||  9400       ||  22000      || m/s    ||
|| round trip  ||  14.5       ||  91         || second ||

The cables are assumed to be thin enough that flexure can be ignored;  large, relatively low-rotation-rate pulleys are assumed to make sure of this.  The forces on the cables will be gravity, the second spatial derivative of tension, the second time derivative of axial motion, pulley deflection forces where the cable makes contact, and electrostatic forces.  The vibration period of the cables will be slightly shorter than the transit time from pulley to pulley, so there will probably be significant lateral vibration.
 
=== Limiting Case - a moving cable without external tension ===

If a cable is moving without pulleys, external tensions or forces, it will naturally form into a circle.  This is one limiting case.  It will probably vibrate and form lobes.

A cirular cable has the lowest energy per angular momentum, as the angular momentum is proportional to the area enclosed by the cable.


More later

=== Limiting Case - a slowly-moving cable around pulleys =

A very slowly moving cable without significant flexure forces or kinetic energy storage will hang straight between pulleys.  Because the cable has weight and is very long, it will have more tension at the top than at the bottom, and a lower mass-per-length because it is stretched.  Since the mass flow rate must be constant in steady state for every point along the path of the cable (i.e., it is not accumulating anywhere), the stretched cable at the top is moving somewhat faster than the cable near the bottom, and the bottom pulley turns slightly more slowly than the pulley at the top.  Note that the situation is reversed if the cable is moving faster than it's speed of sound, but that is not relevant to this discussion.


More Later

=== Electrostatic Repulsion ===

'''The launch loop''' uses round cables for the elevator.  The electric field $ E_0 $ will be highest at the edge of the cable:

$ E_0 ~ = ~ 2 \pi \epsilon_0 ~ \lambda / r_{cable} $

Where $ \lambda $ is the charge density in Colombs/meter and $ \epsilon_0 = $ 8.85 E-12 farad/meter is the permittivity of vacuum (and air, to a good approximation).  Alternately, for a given maximum electric field (say, 300KV/m, 10% of the breakdown field of air) then $ \lambda $ is given by:

$ \lambda ~ = ~ r_{cable} ~ E_{max} / 2 \pi \epsilon_0 $

The actual voltage on the cables will be on the order of 10 kilovolts, but will be dependent on the geometry of the bottom pulley in relation to ground.  For example, the field will be 26 volts per centimeter one meter away from the cable; if there are grounding structures at this distance from the pulley, the voltage will be 12KV.  Both cables have the same voltage, and will have the same charge, so they will repel each other.  The repulsion force per length $ f $ is given by the charge density times the electric field at the spacing distance $ r_{space} $ :

$ f ~ = ~ \lambda ~ \dot ~ 2 \pi \epsilon_0 ~ \lambda / r_{space} $

Or, defined in terms of the maximum electric field:

$ f ~ = ~  2 \pi \epsilon_0 ~ ( r_{cable}^2 / r_{space} ) ~ E_{max}^2 $

For the numbers above, with $ r_{cable} $ = 8.6E-3 meters, $ r_{space} $ = 40 meters, and E_max = 300KV/meter , the force per meter is 10 microNewton / meter .  On the other hand, if perturbation forces push the cables within 2 centimeters of each other (center to center, nearly touching), the force per meter increases to 20 milliNewtons/meter .

If the tension was uniform along the length of the cable, then a uniform force-per-length f will deflect each cable in a parabola.  In actual fact, the tension is higher at the top than at the bottom, so the deflection is less, and the deflection bulges towards the bottom.  That is a difficult calculation to make  (hint! hint! - somebody please make a differential equation and solve it!), so the following analysis averages the top and bottom tension and assumes a parabola.  

The total force along the entire length of the cable is $ F ~ = ~ f L $  Half the force at each end deflects the cable at an angle of $ \alpha ~ = ~ 0.5 F / T $, where T is the cable tension.  That angle times one quarter of the length is the total deflection:

$ deflection ~ = ~ 0.125 F L / T ~ = ~ 0.125 f L^2 / T ~ = ~ 0.25 \pi \epsilon_0 ~ L^2 ~ r_{cable}^2 ~ E_{max}^2 / T ~ r_{space} $

For a 40 meter spacing and an average tension of 310 kN, the deflection is 18 millimeters.  If the spacing is 2 cm, the deflection is 36 meters (increasing the spacing and reducing the force, obviously).  So, with small perturbations over a long distance, the cables will resist being pushed together.  The electrostatic forces are NOT strong compared to wind gusts affecting the cables differentially, however.

More Later

'''The space elevator''' uses wide and very thin ribbons of carbon nanotube for the cable.  The limiting fields will we quite high at the edges

More Later

=== The Steady State General Case - moving cables around pulleys, without velocity change ===

More Later

=== The Dynamic General Case, with velocity change ===

More Later