[The following is an excerpt from a letter to Allen Meece]
[Updated 6 September 2002]
The net work required to raise the elevator along the tether without
friction is equal to the integral of F * dr over the entire tether,
where F is the force vector applied to the
elevator and dr is the vector differential of the radius with
respect to the origin of the tether's curve. If the force is constant and
the tether never reverses direction, then the net work is:
SW = F s
where s is the displacement of the platform relative to the
ground station. It is roughly equal to the net work for motion along the y
axis plus the net work for motion along the x axis.
SW = Fx x + m g h
where Fx x is the integral with respect to x of the
horizontal component of the force.
On the other hand, the integral of power over time is equal to:
W = F L
where L is the length of the tether. (Note that this formula implies
constant velocity along L as well.)
The length of a catenary curve is
L = a sinh( x / a )
where a is the characteristic ratio of vertical and horizontal tension
forces and x is the x coordinate of the end of the catenary. The function
of such a curve would be:
y = a cosh( x / a ) - a
The displacement of the end of a catenary curve is:
s = sqrt( x^2 + y^2 )
It is possible to show that L > s. This means that the integral of
the power applied to climb to the top of the tether is always greater than the
net work done by getting there. It is this energy value that we must
ultimately use to determine the amount of energy required by the elevator to
climb the tether.
The net work is the absolute minimum energy required to move the elevator
to its target position using the given force. We are concerned with the
actual energy required, which is the integral of the applied power over
time. For simplicity, I'm assuming that the force is applied by a motor --
this leaves F and dr parallel at all
times, which is not always the case for a lifting body elevator.
So, the amount of energy needed to reach to top of the tether using a
motorized elevator is directly related to the length of the
tether. Logically, ascending less tether requires less energy.
As far as the elevator motor is concerned, the shorter we can make the tether,
the better.
The tether can be shortened by increasing the
platform's positive bouyancy.
Strictly speaking, what I've said here applies only to elevators propelled
by force parallel to the tether (a motor, etc.), but the first paragraph
applies to all cases with constant force, including lifting bodies.
