Space Elevator - Whistling Tether, Racheting Climbers
The "speed of sound" in a material is the square root of the modulus divided by the density. The space elevator is hypothesized to use carbon nanotube material with a bulk density of 1300 kg/m³ and a Young's modulus Y of 1 TPa (kg/m-s²), resulting in a speed of sound of 27.7 km/s . Presuming the material has binders, fillers, and cross links increasing the bulk density to 1600 kg/m³, and the same modulus, the speed of sound reduces to 25 km/s. A gossamer structure reduces modulus proportional to density.
The longitudinal acoustic impedance (kg/s) for a cable is the force (kg m/s²) divided by the resulting velocity (m/s), or the linear density (kg/m) times the speed of sound (m/s). For the center of the space elevator, the mass is 0.08 kg/m, and the speed of sound is 25km/s, so the impedance is 2000 kg/s. A velocity of 50 meters per second is produced by a force of 100 kN.
What that means is that if we pull down on the end of the central tether with a force of 100 kN, the end moves down at 50 meters per second, and the wavefront moves upward at 25km/s, accelerating 200 kg more of the (8 kg/km) tether downward every second. Actually, the central tether does not have nearby ends - so if we pull down with 100 kN, that launches two wavefronts, up and down, each responsible for 50 kN and each adding 200 kg to the moving mass per second, totalling 400 kg/s moving at 25 m/s.
The central tether has a specific strength of 27 MYuri - at 0.08 kg/m, that is a maximum tensile force of 2.2 MN. if 10% is available for acoustic motive power, that is about 200 kN. The velocity would be 100 meters per second.
The spring constant is k = Y \times A and the area is ( 0.08 kg/m / 1600 kg/m³ ), so A = 5e-5 m², Y = 1 TPa, so k = 50 MN, that is, a 1% extension creates 500 KN of force. If the tether is strained to 4%, one meter becomes 1.04 meters, and the energy stored in that meter of tether is 0.5 * 2MN * 0.04m or 40 kJ/m. If the strain is increased to 4.4%, the energy stored is 0.5 * 2.2MN * 0.044m, or 48.4 kJ/m. If the strain is relaxed to 3.6%, the energy stored is 0.5 * 1.8MN * 0.036m or 32.4 kJ/m . The difference between the extremes is 16 kJ/m peak to peak.
What if the tether was oscillating at 5 kHz, with a base tension of 2 MN, and varying between 2.2MN and 1.8MN? The energy swaps between velocity and tension, with the average energy being 8 kJ/m. If this is a travelling wave moving at 25 km/s, the wave carries 2 MJ/s of energy.
If this energy can be coupled to a 4000 kg vehicle in a 5 m/s gravity field, it will lift it at 100 m/s. If a lightweight ratcheting or solenoid-driven "grabber" can grab the peaks of the wave going past (mad handwaving here), a climber can lift very fast, with no onboard propulsion or power conversion system. The grabber could be a linear series of grips that might engage and disengage by less than a centimeter, but the series is activated with a group velocity matching the forward wave of the tether oscillation.
A thinner tether has lower acoustic impedance - less force for a given amount of velocity. The taper ratio for the reference design is 6, so the tether at the ground is about 0.013 g/m . Less energy can be sent from the bottom, less energy in the upwardly propagating wave. A descending climber can run the grips with the opposite phase, grappling the downwards part of the wave, amplifying it.
But what if the first packages we send up are components of an oscillator driver platform at GEO, powered by sunlight? Solar photovoltaic perhaps, or maybe just a mechanical heat engine and radiator. That will drive the fattest part of the tether, driving waves of energy downwards. The initial climb out of the deep gravity well will be slow, but we can speed up the transfer faster and faster as the vehicle climbs. Also, although the tether is moving longitudinally, it will probably make a high pitched acoustic whistle in the atmosphere. Hence the "whistle".
MoreLater. I need to include the differential equations and estimate how much of the power wave can actually be cancelled, and what all this does to the taper ratio.