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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 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. A thinner tether has lower acoustic impedance - higher velocity per amount of force. |
Space Elevator - Singing 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 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.
A thinner tether has lower acoustic impedance - higher velocity per amount of force.