Name: 太空电梯可能吗（Are Space Elevators Possible?）
Uploaded: 2021-06-09T11:38:37.000Z
Duration: 15 min
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This episode of Real Engineering is brought to you by Brilliant, a problem solving website

that teaches you to think like an engineer.

Space elevators are one of those technologies that sci-fi nerds, like me, obsess over. They

straddle the line between outlandish impossibility and genuine engineering potential. It's

a technology which could cross the divide of science fiction to science reality, if

we somehow improved on existing technologies.

It's the kind thought experiment and lofty engineering challenge that could drive the

development of future technologies. Necessity is the mother of invention after all.

Before jumping into the technologies that need to emerge to facilitate space elevators,

let's first explore what a space elevator actually is.

A space elevator is exactly that, a giant elevator shaft that we can climb to reach

space. Eliminating our dependence on rocket fuel to reach orbit and hopefully, in the

This isn't your typical construction that relies on the compressive strength of a material

Our buildings are largely restricted in height as a result of the compressive strength of

our building materials. The higher we build the more weight is piled onto the foundations

of the building. We can counteract this by widening the base of the construction, to

spread the weight over a larger area and then taper the building as it rises to reduce the

weight being added as we add more floors. The most obvious examples of this are the

pyramids, but even the burj khalifa uses the same principle, being widest at its base and

gradually narrowing to it's seemingly impossible height.

We can build higher with current materials if we widen the base, but that becomes uneconomic

pretty quickly as the base would take up an unreasonable amount of space.

So how would a space elevator solve this problem? By counter-balancing the weight of the structure

by pulling upwards. We can do this thanks to centrifugal force.

Imagine a tether ball swinging around a poll. At a certain angular velocity the string will

be held straight and taut against the poll, because centrifugal force, an apparent force

that appears in a rotating reference frame, pulls outwards.

Now, the problem is, the whole point of tetherball is to wrap the string around the poll, if

the string can't rotate around the centre of spin it will simply wrap itself around

We are essentially trying to recreate this dynamic, but on an astronomical scale and

to do that we have to work with earth's natural rotation.

So, our structure will need to be located on the equator. Let's imagine a base located

From here we are going to draw a straight line out into space. For now, this is just

a line, no structure exists. But, any structure that is constructed will need to exist along

this line. If it is not insync with earth's rotation the tether will curve and break,

or in some sort of cartoon world wrap around the earth like our tetherball example.

Our orbit will also need to be circular, rather than elliptical, as an elliptical orbit would

require a tether capable of constantly changing length without breaking.

We can find an orbit that will achieve this with some simple algebra. To remain in a steady

circular orbit we need our centrifugal force to equal the gravitational force. [1]

Centrifugal force is defined by this equation. Where ms is the mass of the satellite, w (omega)

is the angular velocity and r is the distance to the centre of the earth.

While the force due to gravity is defined by this equation. Where G is the gravitational

constant and mp is the mass of the planet. The mass of the satellite cancels out while

we manipulate the equations to get a value for r, our orbital radius. [Reference Image

Now we have an equation with all known values, which we can solve for by inputting the values

for earth, and we find a value of 42, 168 kilometres. This is the distance from the

centre of the planet, so this will be about 36 thousand kilometres above the surface of

Okay, so this gives us a starting point for our construction. We are going to put some

form of massive satellite into this orbit and begin the construction process. Building

up from the planet is not an option, we need to build down.

Now this is where things get tricky. If we extend our tether directly down to earth,

we will shift our centre of mass and disrupt our orbit. To counter this we are going to

extend our tether in both directions. This keeps our centre of mass in geostationary

orbit and so maintains our circular orbit. If we place a counter weight on our far end,

we won't have to have equal lengths of tether on either side, to balance our load, and this

counterweight could be a useful platform for operations, so let's do that.

Now, something interesting happens when we start to extend our tethers out. Since this

is our neutral point, where gravitational force and centripetal force equal, any material

extended toward earth will experience more gravitational force, while any material extending

away from earth will experience more centrifugal force. This creates tension in our tether,

which will reach its maximum at our neutral point at geostationary orbit, as everything

below it is pulling towards the earth and everything above it is pulling outwards towards

We can calculate the max tension in a cable with a uniform cross section with this equation.

Where G is the gravitational constant, M is the mass of the earth, rho is the density

of our material of choice, R is earth's radius and Rg is the radius of geostationary

orbit. There is an explanation of how this was derived in this paper, which you can find

by matching the reference number appearing on screen now, to the reference list in the

All of these numbers are fixed, bar one. The density of the material we chose. If we chose

to build this cable out of steel, with a density of 7,900 kg/m^3. Our maximum tensile stress

would be 382 gigapascals. That's 240 times the ultimate tensile strength of steel. In

So can we solve this problem? Steel is one of the strongest materials we have. We certainly

don't have a material 240 times stronger. But we do have less dense materials, which

will reduce the tensile stress we have to endure. On top of this, we don't have to

Our tensile stress approaches 0 at it's endpoints, but material at these points have

the highest effect on our stress as gravitational force and centrifugal forces increase as we

move further from our geostationary orbit neutral point. So it makes sense to minimise

materials at the end points and maximise it where it's needed most.

This will result in an improved design called the tapered tower.

So this brings us to a new question. How can we calculate the area needed at any point

along the tether. Our previous paper has the answer once again. This is the equation they

derived. Here As is the area of the tether we chose at earth's surface. This starting

value will largely depend on design considerations that we can't possibly know right now, but

we are going to want to minimise it, because this right here is an exponential function.

Meaning, our width is going to increase exponentially as we rise. It is imperative that we minimise

this value inside this bracket, and we only have two values we can control in this equation.

The density, which we want to minimise and stress value we are designing for, which he

is donated by T, which we want to maximise.

Normally, we wouldn't use the maximum stress a material can hold as the design stress.

That leaves zero margin for error. We should be designing in a safety factor. But for now,

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太空电梯可能吗（Are Space Elevators Possible?）

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