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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

  • process, lower the cost of space travel.

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

  • to remain standing.

  • 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

  • the poll.

  • 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

  • in the middle of the Atlantic ocean.

  • 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

  • 1]

  • 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

  • the planet at the equator.

  • 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

  • space.

  • 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

  • description.

  • 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

  • other words, steel can’t do the job.

  • 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

  • have a uniform cross section tether.

  • 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,

  • I’m just gonna go with it and say this thing isn’t gonna be safe and I’m designing

  • it riiiiight on the edge of breaking. So, yeah….bear that in mind.

  • Remember that strength and density material selection diagram from our last video? Let’s

  • refer to that again to pick a couple of materials to analyse this structure with.

  • Steel is cheap and well understood, so let’s start there with a high grade high strength

  • alloy like 350 maraging steel.

  • This steel has an ultimate tensile strength that can range from 1.1 GPa to 2.4 GPa with

  • a density of 8,200 kilogram per meter cubed.

  • This paper quotes a steel with a UTS of 5 GPa and a density of 7,900 kilogram per meter

  • cubed. I don’t know what aliens they got their data from, but this is beyond the realm

  • of reality. We will use steel, but with more realistic material properties.

  • Then we will pick some better existing materials. They wisely picked Kevlar, which is a widely

  • available high strength fibre we could easily form into a tether.

  • We are going to throw two existing materials into the mix too. Titanium, which as we discovered

  • in our last video has excellent specific strength qualities, and carbon fibre composites, which

  • have even better specific strength qualities and would be used today if the SR-71 was redesigned.

  • Using these material properties. We can calculate the taper ratio, which will be the ratio of

  • the area of the tether at the bottom of our elevator to the area of the tether at its

  • widest at geostationary orbit.

  • I’m going to assume a circular area 5 millimeters in diameter at the base. By multiplying the

  • cross sectional area at the bottom by the taper ratio we find the cables widest point.

  • For steel, this taper ratio is so huge that our cable at it’s widest point will be this

  • number, whatever that is. For reference the width of the known universe is 8.8 by ten

  • to the power of 26 meters wide. Even dividing the diameter of this cable by the width of

  • the known universe yields this number, which I still can’t comprehend. Titanium is marginally

  • better.

  • Now Kevlar and Carbon Fibre are looking a lot better. They will have a circular diameter

  • of 80 meters and 170 metres respectively still not quite feasible.

  • The amount of material required to build something like this would outstrip any cost savings

  • it could possibly supply. And that’s just assuming the fibres could even be formed into

  • this shape without losing a significant portion of their maximum tensile strength, which is

  • a big assumption. Footage: I don’t know...

  • So I think it’s safe to say that right now space elevators are possible in the sense

  • the physics of how they work is based in reality, we just don’t have a material capable of

  • making it feasible.

  • Especially when you consider we are analysing this at ultimate tensile strength in reality

  • we should be using a value below our yield strength, as above that value our material

  • will begin necking where the cross sectional area actually decreases as the material elongates.

  • We aren’t even considering strain here.

  • One future tech that a lot of people are hyping up for future use in space elevators is carbon

  • nanotubes. Whose strength is off the charts with some studies quoting ultimate tensile

  • stress values as high as 130 Gigapascals and a low density of 1300 kg per metre cubed.

  • At that value the taper ratio is just 1.6.

  • If this material could be manufactured on a massive scale, it would revolutionise life

  • on earth, but we would still have to solve a huge number of engineering challenges.

  • Eliminating vibrations and waves propagating through the tether is a huge challenge. Powering

  • a climber and dealing with the adverse weather of the lower atmosphere and dodging space

  • debris in orbit are all massive challenges, before even starting on the most fundamental

  • problem of all. Manufacturing carbon nanotubes.

  • We will explore these problems and potential solutions in future videos. One on how carbon

  • nanotubes are made, why they are so strong, and what needs to happen to take them from

  • the laboratory to regular life. And then, we will revisit this subject with a design

  • investigation for an actual space elevator using this new theoretical material.

  • During the research of this video I noticed several mistakes in the paper I referenced,

  • small mistakes that anyone could make and easily overlooked. I only noticed them because

  • I applied their methods myself and noticed inconsistencies. Their rounding was so aggressive

  • that their results were off an inconceivably large number thanks to the exponential function

  • in their equation and I noticed their material properties for steel was incorrect because

  • I recreated their calculations for titanium and noticed it was worse, despite the equation

  • being entirely determined by specific strength.

  • This is the power of applying knowledge yourself, rather than being a passive observer. You

  • begin to understand things on a fundamental deeper level and this is why I love Brilliant

  • and believe they are the perfect compliment to my channel.

  • I found myself struggling to remember core parts of calculus while following the derivations

  • in the paper, as it’s been years since I had to integrate anything. So, I have committed

  • myself to completing Brilliants course on calculus and advanced mathematics to brush

  • up on my maths skills which have dulled with the passage of time. I can complete this course

  • gradually in my spare time, keeping notes and using their mobile app to steal spare

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  • This is just one of many courses on Brilliant that will improve my ability to understand

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  • As always, thanks for watching and thank you to all my patreon supporters. If you would

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This episode of Real Engineering is brought to you by Brilliant, a problem solving website

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Are Space Elevators Possible?

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    joey joey に公開 2021 年 06 月 09 日
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