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"Indeed, the rage of theorists to make constitutions a vehicle for the conveyance of their own crude, and visionary aphorisms of government, requires to be guarded against with the most unceasing vigilance."
     -- Joseph Story
     Commentaries on the Constitution of the United States
     Book III, § 1857.

Thursday, October 30, 2003

Steven Den Beste on practical problems with a space elevator. Three widely separated anchors won't solve the oscillation problem; they'll make it worse.

Tie three strings of equal length to a weight, and then fix the other ends to three equidistant points on the ceiling. The weight will hang placidly in the middle of the room; everything is fine, as long as nothing moves.

Detach any two of the strings from the weight, and you have a system where the weight can be anywhere within a sphere (well, hemisphere, really -- can't forget the ceiling). As long as both its position and velocity are on the surface of the sphere, it will move more or less smoothly, swinging underneath its anchor until the oscillations damp out, at which point it will hang placidly once more. Now, though, lift it a few feet straight up and let it go again: It will fall straight down until the string jerks it back, then it will bounce a few times until those oscillations damp out (assuming the first bounce didn't rip the anchor out of the ceiling); once that settles down it will be swinging slowly back and forth.

Now reattach the other two strings. The volume the weight can occupy is now bound by the intersection of three spheres, a shape with definite edges and a point on the bottom (on the top, too, if the ceiling weren't in the way); as long as the system is not disturbed, the weight sits at that point on the bottom.

Now perturb one of the strings however you please, and then let it go. The other two strings remain straight, so the weight travels upward along one of the edges of the intersecting-spheres figure as you perturb the string; when you let go, gravity drags it right back down again, and it arrives quickly at its rest point with a significant momentum the vector of which is pointed in a direction the preturbed string prevents it from moving. There is no longer any way, no matter how the system is perturbed, for the weight to return to its rest position without bouncing really hard on the ends, not just of the one perturbed string, but all three of them.

Now flip the figure over and make it a triple-anchored space elevator. It will have exactly the same problem: There is no way to perturb it such that its return to its rest state does not inflict extraordinary stresses upon the system (the flip side of huge transient accelerations upon the weight itself), and the point at which the weight is expected to rest, and to which it will return after being perturbed, is also the point where it will experience the worst stresses if it is not already at rest.

I have an argument of sorts in mind to the effect that, given that the weight is in orbit, and that the strings have their own inertia, it would be impossible to perturb just one string while keeping the others taut, which would thus make the bouncing problem even worse. I would need a better grasp of orbital mechanics to say for sure, but the picture I have in mind is such that any perturbation on one string will wind up being amplified by the other two. Maybe Steven will have a go at it.

Also, unless all three of the strings are attached to the weight precisely at its center of mass, those bounces are going to turn into some hellacious angular momentum. A one-string system would have this same problem, though, and in even worse form: Either it will eventually twist that string enough to break it, or it will be the biggest yo-yo ever built.

-- posted by Clayton 10/30/2003 08:20:00 PM

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