Wikipedia recommends aiming laser beams at the climber, or by electric wires powered from the base, or by nuclear powered cars, or by solar power. I've got a better solution: as cargo goes up, have even heavier cargo go down (mined from asteroids etc). Use the weight differential to power both movements. That totally solves the problem of providing power, and gives the elevator extra power it can use to damp oscillations etc. Have cars cycle up then down again.
In the atmosphere, wind resistance is a problem. This problem drops off rapidly, mainly in the first few km, and disappearing entirely after 100 km. Geosychronous orbit is 36000 km up. So: have the cars connected in a train at the surface to minimize air resistance, going at 100 km/h again to reduce air resistance, then after air resistance is negligible (something like 5 km up) separate the cars and start accelerating at 1G. If cars can get up to 10000 km/h, passengers will reach geosynchronous orbit in 4 hours and will only need to bring lunch. Full speed is reached in a few minutes, so the total weight of cargo is only 1/100 what it would be if it were packed as close together as it was on earth's surface. The cargo can be 20x the average cross-sectional weight of the elevator at earth's surface and still be 1/6th of the overall weight.
Since cars have to go up and down at the same time, they can't be centered on the elevator. However, if they are not centered on the elevator, they will impart a sideways wave to the elevator as they travel. A way around this is to have a pair of cars going up on opposite sides of the elevator, and a pair of cars going down 90 degrees to the up cars.
Cars don't want to physically touch the elevator, and we want to transfer power from down cars to up cars. That means electromagnetic braking down and electromagnetic propulsion (rail gun) up. Since we have 2 cars up passing 2 cars down, have at least 4 tracks. The tracks are supported by the elevator, they are not necessarily the strong cables themselves.
Cars would be on the outside surface of the elevator, and their escape plan would be to push off and fall.
There is no known material strong enough to make the cables. I don't have any solution to that. But if one were created ...
In order to avoid vibrations, increase stiffness, and increase survivability from failures, the elevator shouldn't be one solid cable. It should be a mesh of interlinked cables. Hoytethers seem close to right: a hollow tube formed by many cables, all mostly vertical, with each joint linked to two or more joints above and below. A nested tube would make it easy to link to 6 joints above and below rather than just 3. The tube would allow cars outside the tube to smoothly shift around known damage. The tube would be about 20 meters in diameter, with the nested tube about 16 meters in diameter, and joints about 1000 meters apart vertically. For some reason the Hoytether has secondary cables that are normally unstressed ... I would have all cables stressed all the time, but have either reduced stress per cable or increased load to take advantage of those extra cables. The cables should be held apart by rigid triangles, damping oscillations, with short compressed horizontal spacers and long vertical cables in tension. There will still be oscillations, but this limits their maximum frequency. Since the elevator has excess power, that power can be used to push and pull on cables dynamically to quickly cancel waves. A car moving at 10000 km/h covers 3 km per second, so waves with length under a km would be experienced as jolts rather than swaying.
There should be a cable maintenance that can reach and replace broken cables. If it takes simultaneous cable failures in n particular cables to cause elevator failure, and maintenance replaces failed cables in 1/m the time betweem mean failures, the chance of random failure due to those cables is about m-n per time unit, where a time unit is how long it takes to repair a broken cable. However, failures come in multiple sizes. Assuming proper maintenance, elevator failure would typically come from large events that cause too many simultaneous cable failures. (This is very similar to losing 3-replica or erasure-encoded files in data centers.)
Another problem is containing broken cables. I don't have a full solution for this. Cables are stiff and strong, similar to rebar. Still, hits from space debris can easily vaporize a spot of a cable, leaving two ends swinging. If they swing into the path of a car the car gets skewered. Cars pass roughly once a second, so there is little time to clean up. As long as the broken cables are inside the elevator instead of sticking outside, they are OK. A just-broken cable will be broken in only one spot. If you lash the end of one cable to the start of the next, both segments of the cable will want to stay vertical. The falling section wants to keep hanging down, it just needs oscillations damped. The rising section wants to lean over. Elastic cords between cables can persuade the rising section to stay out of trouble, except immediately after impact.
An alternative is to have to separate elevators, one for up and one for down, with cars centered on each cable. Having the car centered on the cable means you only need one up-car and down-car, not a pair of up-cars and a pair of down-cars. It prevents up-cars from running into down-cars. But it makes the cables thinner, so harder to damp waves. The two cables may attract each other electrically and gravitationally. Power has to be beamed from the down cable to the up cable. You'd have to transfer cars from one cable to the other at the top and bottom. I prefer the one giant tube elevator approach, because the huge tube damps waves and is big enough that up-cars won't run into down-cars already.
Another alternative is to have a really big balloon 20km up, and to tether it to the earth by taut sloped Zylon tethers, and to launch by rail guns along the earth and then up the tethers. This can give you a long railgun with a much higher release point than any mountain. The higher release point doesn't mean much in terms of getting to orbit, but it does get you mostly out of the atmosphere. My back-of-the-envelope says a 100km railgun accelerating at 2G up to a 20km balloon will take 101 seconds, and reach a speed of 2000m/s (a small fraction of earth's 11200m/s escape velocity). If the angle is 30 degrees up, freefall would take another 101 second before the upward velocity was 0, rising an additional 50km, for 70km total. If you launch a rocket rather than just a car, it could take over the task of accelerating after release. Perhaps it could accelerate slower and more efficiently than if it were launched from the ground, allowing more payload per rocket. 700 seconds at a rocket acceleration of 4/3 G still at 30 degrees would about reach escape velocity before you stop gaining altitude. As I've said elsewhere, a good way to make a really large lighter-than-air balloon is to fill it with H2, surrounded by a shell of CH4, surrounded by a shell of N2, because neighboring layers are not mutually flammable, and all layers are cheap and lighter than air and not poisonous. If you're not launching humans, a 50km railgun up to a 20km high balloon accelerating cargo at 100G should reach escape velocity at the point of release. You'd have to use a vacuum tube for the rail or the air resistance would destroy it, and even at 20km the air resistance might destroy it.
Bob Predicts the Future
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