Acoustic Climber for Space Elevator

The current reference space elevator design assumes solar-powered climbers. This assumes that vast areas of solar panels can cantilever from the sides of a climber - in gravity - and provide megawatts of climb power, while being lightweight and affordable. The example pictured is a DLR solar sail, intended for microgravity, NOT an array of solar cells. Solar sails are ultrathin plastic films covered with just enough shiny metal to reflect light. Aluminum conductivity is 2.8e-8 ohm/meter; a film with 10 ohm per square resistivity (95% reflective) is 2.8e-9 meters thick - a few atomic layers. The density is 2700 kg per cubic meter, works out to 8 kilograms per square kilometer on top of the plastic. This is far less than actual satellite solar panels (300 W/m2, 1 kg/m2), which are designed for microgravity, not to deploy in a gravity field. These arrays are stabilized by gravitational gradients in orbit, and they are test-deployed on the ground hanging sideways from heavy structure.

Instead, a superstrong tether can carry megawatts of subkilohertz acoustic power, which can be impedance-matched and mechanically rectified (mad handwaving here) to produce climber thrust. The acoustic transmitters on the ground and at GEO node can provide 2 MW and 10 MW of climb power respectively, more by trading off climber mass, gravitational weight, and climber speed. Climbers will have a mechanical receiver that transforms vibration to rotary wheel motion.

A preliminary paper

That paper is old, a new one will be written with the material here. This is morphing into two banks of electrical motors separated by a quarter wave, with electrical power conversion at each motor group, and a power cable and stiff tether in between them.


Click image for larger view or download attachment:ClimberPickoff.png !LibreOffice source

A cheesy drawing of the two motor group "wheels". The grey "foot" makes contact to the tether when it passes by at maximum velocity. Alternating wheels have feet that turn in opposite directions, and the red and blue "free wheels" help maintain spacing but do not provide power. The power wheels turn at half the power vibration rate.


Analogy Between an Electronic Signal Cable and a Stiff Tether

All units MKS: meters, kilograms, seconds, volts, amperes (amps), radians

( FYI: if you don't think radians are a unit, you are turned around, and can't distinguish energy from torque. :-? )

Relationship between voltage and current in a uniform lossless electronic signal cable:

~~~~~~~~ { \Large {{ \partial V(x,t) } \over { \partial x }} } = - L { \Large { {\partial I(x,t) } \over { \partial t }} } ~~~~~~~~~~~ { \Large {{ \partial I(x,t) } \over { \partial x }} } = - C { \Large {{ \partial V(x,t) } \over { \partial t }} }

Wave equations for a uniform lossless electronic signal cable:

~~~ L C ~ { \Large {{ \partial^2 I(x,t) } \over { \partial t^2 }} } ~=~ { \Large {{ \partial^2 I(x,t) } \over { \partial x^2 }} } ~~~~~ L C ~ { \Large {{ \partial^2 V(x,t) } \over { \partial t^2 }} } ~=~ { \Large {{ \partial^2 V(x,t) } \over { \partial x^2 }}}

Sinusoidal solutions (many others are possible): ~~~I(x,t) = I_0 \sin( \omega t + k x ) amps ~~~~~ V(x,t) = V_0 \sin( \omega t + k x ) volts

Mechanical cable

Relationship between displacement, velocity, acceleration, tension, and strain in a mechanical cable:

~~~~~~~~ { \large e(x,t) = \Large {{ \partial \psi(x,t) } \over { \partial x }} }

~~~~~~~~ { \large f(x,t) = Y_c \Large { { \partial e(x,t) } \over { \partial x }} }~=~{ Y_c \Large {{ \partial^2 \psi(x,t) } \over { \partial x^2 }} }

~~~~~~~~ { \large v(x,t) = \Large {{ \partial \psi(x,t) } \over { \partial t }} }

~~~~~~~~ { \large a(x,t) = \Large {{ \partial^2 \psi(x,t) } \over { \partial t^2 }} }~=~{\LARGE{ 1 \over \rho }}~{\large f(x,t)} ~=~ { \Large { Y_c \over \rho } ~ {{ \partial^2 \psi(x,t) } \over { \partial x^2 }} }

Wave equations for a uniform lossless mechanical cable:

displacement: ~~{\Large{{\partial^2 \Psi(x,t)}\over{\partial t^2}} } = {\Large{Y_c \over \rho }~{{\partial^2 \Psi(x,t)}\over{\partial x^2 }} }~~~~ velocity: ~~{\Large{{\partial^2 v(x,t)}\over{\partial t^2}} } = {\Large{Y_c \over \rho}~{{\partial^2 v(x,t)}\over{ \partial x^2}} }~~~~ strain: ~~{\Large{{\partial^2 e(x,t)}\over{\partial t^2}} } = {\Large{Y_c \over \rho}~{{\partial^2 e(x,t)}\over{\partial x^2}} }

For sinusoidal waves,

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