As I was walking that ribbon of highway I saw above me that endless skyway...
19 May 2010 Space elevator to low orbit?
23 May 2010 Space Elevator and The Dynamic Grapple!
8 June 2010 Hooking on when you're off GEO - an even more dynamic grapple.
10 June 2010 Acceleration Matching for Space Elevator Grappling.
1 Dec 2012 U-Fly-It Satellite and Space Elevator Simulator.
Space Elevator and The Dynamic Grapple!
23 May 2010
When you let go from a space elevator and you are anywhere between 66% and about 132% of the way up to geosynchronous orbit, you will go into an orbit around the Earth. If you don't have a reentry system, to get home you will have to grapple back on to the cable. The easiest way back on will be at GEO. You just continue to float next to it and grab back on, right? Yup, that's right. But, what if the elevator is swaying? Space elevators are expected to sway more or less constantly. There is almost no natural damping out there**. The counterweight will oscillate mostly east-west, but also north-south much like an upside down pendulum.
Things push on the cable and counterweight. Traffic moving up pushes west on the cable. Traffic moving down pushes east on the cable. As the Earth turns, the cable and counterweight swing toward, then away from the Moon and Sun, they fly through varying gravitational fields. This gives nudges to the system, both east-west and north-south. The main resonant frequency of the system (the sway frequency) will be purposely designed to mismatch the pumping frequencies of the Moon and Sun. The canonical design (by Edwards) has a main sway period of about 7.8 hours. Engineers think about many ways to minimize sway. They have a lot of good ideas and those ideas will work, but there will likely always be some sway present. I expect the north-south component will have a slightly different frequency than the east-west component, setting up something of a Lissajous pattern.
Wait for it....Wait for it.....NOW!
The nifty thing about a "regular" rendezvous in space is that the two objects are subject to the almost exactly the same accelerations at the same time. They sidle up to each other with near-zero acceleration between them and latch on at their leisure. No hurry, take your time, approach slowly, don't bump into the other guy, easy does it now.
Okay then, 7.8 hours per cycle, 22236 miles high. What is the maximum sway angle expected and how fast will the GEO grapple point be skittering around? Could you "sidle-up". Could you grab it?
Most studies show the maximum sway angle to be "very small". The cable will still always be very nearly vertical. So what if the maximum off-vertical angle was 1/2 degree? [Lang, "Space Elevator Dynamic Response to In-Transit Climbers", figure 17, David D. Lang Associates, Seattle WA.] Compared to a perfectly stationary and perfectly vertical cable, at GEO-level the maximum deflection from vertical would be 194 miles. The average speed of the GEO point compared to a motionless cable would be about 100 mph! The maximum speed could be up to 156 mph. The speed and direction would also be constantly changing.
If the oscillation was only in the east-west plane, and if you could somehow park precisely at the peak deflection point and wait for it to arrive, the relative acceleration of cable there there would be 0.05 fps2. That means that 30 seconds before closest approach and zero speed, the cable would be approaching from 23 feet away at 1.5 fps. So, the cable would only be within a 25 foot "grabbing distance" for about 60 seconds. The acceleration doesn't sound like a lot, but you need to act fast, and you need to position yourself accurately ahead of time, which would be tricky if not impossible.
Another way to dock in this same planar-oscillation situation would be to match velocity with the cable grapple point at it's lowest acceleration. The lowest acceleration of a sinusoid is at the zero-deflection and the point of highest speed. In our 1/2 degree sway with a 7.8 hour period, the maximum speed (compared to no sway) is 229 fps. If we matched speed and position 7 minutes before the cable passed vertical there would be a window of 14 minutes where the speed differences would be less than 1 fps. However, if the speed difference wasn't followed with thrust, a gap of some hundreds of feet would open up during that time. Seems a little easier than position matching, though at the cost of delta-V.
The other two methods expect the sway to be almost completely in one (east-west) plane. I don't expect that to be the case. It will sway north-south too. The net effect is that it will seldom be stopped as in the position matching scenario, and it will seldom pass through vertical, as in the velocity matching scenario. Acceleration matching would find a window where there is a balance between excess velocity and acceleration, minimizing both as best as can be done. First there would be a matching of velocity and position, then the small additional accelerations would be matched with jets until there is a chance to grab on. Still tricky.
What to do about it.
GEO will be the easiest place to latch on, but it will still be trickier than the simple "sidle-up" maneuver we are used to. Add to that the cost of a mistake. We know from the Progress-Mir accident that collision mistakes will happen. On Mir, the maximum possible loss was the cost of the station, and possibly the crew if they weren't able to evacuate. A collision mistake grappling to a space elevator could result in severance of the cable, collapse, and fratricide of all other space elevators. Yikes!
So, I see a few possibilities for trade-off to solve this problem:
1) Develop methods and mechanisms to support a new dynamic rendezvous and grapple technique. Dynamic grapple systems might be able to handle positioning errors of hundreds of feet perhaps and significant differences in speed. The Fulton Recovery System exists and achieves a similar idea. Shoes on a power line might be another good idea to start from.
2) Develop methods of smoothly following and matching the accelerated motions of the GEO grapple point using thrust. Try not to spritz the cable with rocket exhaust.
3) Develop methods of nearly zeroing-out all sway to permit a traditional "sidle-up" rendezvous at GEO.
I favor emphasis on method (3). That would reduce or eliminate the needed "reach out and grab" radius of method (1)'s systems or the needed delta-V for method (2). It would also drive our ability to master the kind of control over the system that would be critical for other things as well.
Additionally, because of the threat to the cable posed by a collision, the cable should be shielded for some distance above and below the targeted grapple point. Shielding would protect against jet spray and Progress-Mir type collisions and it may extend a fair distance above and below the grapple point. At GEO, this grappling and shielding system would pose the least construction difficulty, from there it could fairly easily move up and down the cable to scheduled latchings-on.
So, no more easy going, sidling up, stately rendezvous. It's dynamic now!
**Some sideways energy will couple into stretching and dissipate as heat. How much will depend on design. I have no idea if it will be huge or miniscule (huge is better).