14-05-2012, 03:44 PM
Space Elevator
tapered towers.pdf (Size: 364.18 KB / Downloads: 57)
Introduction
A “space elevator” can be thought of as a
taut, rigid structure that stretches from the
Earth’s surface up to a point where its centre
of gravity is at a height of 35,000 km [1], the
orbital height of a geostationary satellite.
Attached at the top of the structure is a
counterweight to keep the structure taut. Due
to the relative ease of ascending such a tower
compared to conventional launch methods,
the concept has been viewed as a potential
launch mechanism.
A Free-Standing Cylindrical Cable
Consider the case of a cylindrical cable of
constant cross-sectional area in orbit. As
shown in Figure 1, the main forces that act on
an element of the cable in equilibrium are the
tension in the cable, the centripetal force
pushing it away from the Earth and the cable’s
weight pulling it towards the Earth. Here, the
tension can be defined as: AdT, where A is the
cable cross sectional area and T is the force per
unit area acting on the cable.
A cross section of an
element of the elevator
cable showing the forces
acting on it. Here, FW is the
combined weight of the
cable at that point whilst FC
is the centripetal force acting
on the cable at that point.
Using standard formulae [2] describing
gravitation and centripetal acceleration, it can
be shown that equating the forces in
equilibrium gives
where ω and M are the rotational velocity and
mass of the Earth, and r is the length of the
cable from the centre of the Earth to its peak
H. Here ρ is also the density of the cable.
Rearranging and substituting the appropriate
equivalent for ω gives a differential equation
that describes the rate of change in tension as
a function of radial distance from Earth
It can be demonstrated that the tension is at a
maximum at Rg, (radius of geostationary orbit).
It is also known that the tension in the
cable is zero at either end, so the boundary
conditions for this equation are T®=0 (where
Conclusions
Constructing a space elevator would be a
mammoth undertaking, due to the length it
would have to be. The magnitude of the
tension in a cylindrical model makes it
impractical. Tapering the cable would offer the
advantage of having a uniform tension
throughout the elevator, with the length being
the same as the cylindrical one. However, steel
would still be unsuitable due to the large taper
ratio required.
tapered towers.pdf (Size: 364.18 KB / Downloads: 57)
Introduction
A “space elevator” can be thought of as a
taut, rigid structure that stretches from the
Earth’s surface up to a point where its centre
of gravity is at a height of 35,000 km [1], the
orbital height of a geostationary satellite.
Attached at the top of the structure is a
counterweight to keep the structure taut. Due
to the relative ease of ascending such a tower
compared to conventional launch methods,
the concept has been viewed as a potential
launch mechanism.
A Free-Standing Cylindrical Cable
Consider the case of a cylindrical cable of
constant cross-sectional area in orbit. As
shown in Figure 1, the main forces that act on
an element of the cable in equilibrium are the
tension in the cable, the centripetal force
pushing it away from the Earth and the cable’s
weight pulling it towards the Earth. Here, the
tension can be defined as: AdT, where A is the
cable cross sectional area and T is the force per
unit area acting on the cable.
A cross section of an
element of the elevator
cable showing the forces
acting on it. Here, FW is the
combined weight of the
cable at that point whilst FC
is the centripetal force acting
on the cable at that point.
Using standard formulae [2] describing
gravitation and centripetal acceleration, it can
be shown that equating the forces in
equilibrium gives
where ω and M are the rotational velocity and
mass of the Earth, and r is the length of the
cable from the centre of the Earth to its peak
H. Here ρ is also the density of the cable.
Rearranging and substituting the appropriate
equivalent for ω gives a differential equation
that describes the rate of change in tension as
a function of radial distance from Earth
It can be demonstrated that the tension is at a
maximum at Rg, (radius of geostationary orbit).
It is also known that the tension in the
cable is zero at either end, so the boundary
conditions for this equation are T®=0 (where
Conclusions
Constructing a space elevator would be a
mammoth undertaking, due to the length it
would have to be. The magnitude of the
tension in a cylindrical model makes it
impractical. Tapering the cable would offer the
advantage of having a uniform tension
throughout the elevator, with the length being
the same as the cylindrical one. However, steel
would still be unsuitable due to the large taper
ratio required.