Simulated Gravity, Am I calculating it properly

[Nimbian]

I am trying to write a hard Science Fiction Novel.
Its been more then a few years since I took Physics in High school, but a Little research led me to a equation to determine how much simulated gravity some one might feel on a rotating station.

A_c=V²/r

The Station in question is O'Neill style rotating cylinder; with a Diameter of 20km I'm trying to determine how fast it should be rotating for it to have a 1g on the inside.

9.81 = V²/10km
10 km * 1000 = 10000m
9.81*10000 = 313.209²

Therefor the station should be rotating at 313.21 m/s? This seems way off to me

 

[The_Duck]

Your formula is right. It gives the velocity of the outer edge of the station. What sort of number were you expecting to get?

You might get a better sense for the result if you calculate the rotation rate, or equivalently how much time it takes to complete one rotation. The outer edge of the station traverses a circle of radius ten kilometers, which has a circumference of 2*pi*(10 kilometers). If you divide this circumference by the speed you found you'll obtain the time necessary to complete one rotation. To me, at least, this gives a better sense of how fast the thing needs to rotate.

 

[Nimbian]

Thanks for the quick reply.
Your right That that would give be a comparison. with a radius of 10km the station would have a circumference of 62.8km (2*3.14(pi)*10(km))
If said station was rotating 62.8km/h then I could work what the "Surface Gravity" would be on the interior of the station

A_c= 62.8²/10
A_c= 394.389 km/h²

394.389km/h² = 0.0304 m/s²
Which would mean the station has less then 1% Earth normal Gravity (as expressed as Acceleration)

And I Think I see my earlier mistake, I'm going to rewire the Formula tell me if I made a mistake

√(A_c*r)=V
Therefor
A= 9.81m/s² (the target acceleration) Converted to Km/h2 as Km/h is my target Unit (127137.6 km/h²)
r = 10km the Stations radius
V= should be the Rotational Velocity

√(127137.6 * 10) = 1127.553 km/h

So my station would be rotating around 18 (17.954) times an hour, for it to have Earth gravity. am I right?

 

[Nimbian]

I'm sorry I also just realized I posted this in the wrong forum section

 

[K^2]

All of these numbers are correct. The station needs to be spinning pretty rapidly to simulate Earth's gravity. A revolution will take just over 3 minutes. Which is one of the things that tells you that it's a challenging proposition. Anything more than 2 RPM tends to disagree with human inner ear. So your station will have an acceptable angular velocity.

The other part is structural, because you are effectively trying to build a suspension bridge that's suspended on itself. Furthermore, it's suspended 10km bellow the "anchoring point", so you have to factor in the weight of the support structure. Of course, artificial gravity drops linearly as you go towards the center, so it's an equivalent of building a suspended bridge 5km tall on Earth. We do have materials that can make it possible, but these aren't going to be your ordinary steel cables.

So yeah. Artificial gravity. It is appropriately tricky to achieve.

 

[mfb]

Well, it is a suspension bridge where you have a free anchor "5" km above the bridge. Vertical cables of 5km length at 1g are no problem, even with conventional steel, but the mass fraction required for stability would be significant (guess: something like 20%). Other materials are better by a factor of ~10, giving ~2% of the mass in cables.

However, I would expect that the cylinder can support itself - the required strength to density ratio depends on the outer velocity only, and wheels with 300m/s are possible (with a good safety margin).

 

[Nimbian]

Thanks for the Replies

And thanks for Identifying another problem this station faces, Namely how to stop the station from flying apart under the stresses of the rotation.

Solution likely some form of Carbon Nano tubes.
Light weight and high Tensile Strength

Mass production of such a material should be trivial when constructing a Cylinder Station like this one

 

[mfb]

You do not need carbon nanotubes - existing material would be able to support this cylinder, just with the cylinder itself (no cables in the central area).

r=10km and v=300m/s requires a specific strength of 2*10^5 Nm/kg or 200kN*m/kg.
Steel has a bit more, fibreglass already gives a safety margin of factor 5, Kevlar is better than steel by a factor of 10. This allows to use a big part of the mass for non-structural elements.

 

[K^2]

Huh. You are right. A cylinder supporting itself requires only 4x the specific strength, and you are not wasting all that structure in the center this way. I did not think about that earlier.

You definitely don't want to be working that close to ultimate strength with steel, but composite materials are definitely sufficient here.

 

[Nimbian]

Again thanks for the replies.

Maybe it would help if I described the the station in greater detail so I can work all the flaws out.

It is a Cylinder 20km in diameter, and 100km long. This station rests in the L1 position. We've already established that it rotates 17.954 Times a Hour.

Inside the station is a reproduction of the environments of Earth from pole to pole. with 628 km² there should be enough room for at least 6 Biomes. A Thin Tube lies at the center of the cylinder providing light, and channels Air to create air currents. Half way down the cylinder is structure that is roughly a sphere (1km Diameter) standing on eight "legs" that are evenly spaced around the circumference of the cylinder. (acts as a transportation hub, as well as the maintenance systems for "Sun Tube", and Command and control for the majority of the stations systems.)

I'm currently figuring between a thickness of 100 - 200m. Most of the population, lives within the shell. as well as most of the farming is carried out there.

As it stands now is there any major problems with the design? (keeping in mind I'm aiming for a Hard science answers where ever possible.)

 

[sophiecentaur]

I imagine that the best sort of support structure would be along the same lines as a bicycle wheel, with a rigid, disc-like hub and non-radial spokes. That takes some pretty massive stresses.

 

[mfb]

Why do you need 100-200m thickness of the cylinder? This requires ~1000km^3 of material.

Hmm, I forgot the air pressure, which gives additional 2*10^9N/m or ~1000kg/m^2 of Kevlar (about 1m, the Kevlar required to account for this Kevlar and so on is negligible)

Things I would expect, with guessed numbers:
~20cm - 5m soil only where things should grow
~50cm solid floor, depending on the purpose of the area
~5m personal rapid transit system, water, electricity and data management system
~10cm insulation (water/air/temperature)
~3m+ Kevlar or some similar material, mixed with additional thin insulation layers as security. Add sensors to detect problems like gas leaks or structural issues.
~Photovoltaic cells as electricity source.

Everything else is built above the solid floor. 1m of Kevlar can support ~10000kg/m^2. Heavy objects which do not require gravity are built in the center (everything storage-related which is not required on a daily basis), if there is a need for large buildings with gravity, make some rings with additional kevlar outside or use cables to reduce the stress.

From the inside, everything up to the insulation is accessible, roboters can access the outside, and if something in the kevlar goes wrong I have no idea how to fix this.Some elevators to reach the central area for maintenance and so on. A space port at one or both sides of the central axis.

 

[Nimbian]

Thanks again for the replies.

Sophiecentur, I actually did not consider the "Bicycle wheel" Model, which actually makes more sense considering the orientation of the Gravity. although the "Spokes" that were around the central sphere were non-radial.

mfb, The main reason for the thickness of the shell amounts to Politics of the universe, which I have been attempting to avoid discussing since I can talk for hours on the subject.

However the civilization that built the station is essentially showing off. there are other reasons but that amounts to the primary reason. The second reason is space necessary for high impact armoring, radiation shielding, and other similar defenses to protect the inhabitants from a Hostile universe, little which has to do with the environment of space.

Farming has to take place below the interior because the builders didn't want to interfere with the Biomes. (which amount to ecological preserves). I actually see the builders Hollowing out a stellar mass (like a asteroid) that has a similar shape to their intended structure, and shaping that. the primary reason for that is for rotational stability, Stresses introduced from one part of the station being heavier then another, and such mass can easily be removed to allow post-construction expansion.

But your right in thinking that most people (and the Humans in the story) would rather live above "Ground". Perhaps Archeology style towers that cross the center?

As to power generation I had an Idea but I don't know if its viable.
Would the spinning station that is 20km in diameter be able to generate electricity from interaction with the Earth Magnetic field?

 

[mfb]

Magnetic field at L1? Problably not. In addition, you would reduce the angular velocity and would need some way to speed it up again. In the best case, you can use it as energy storage.

I actually see the builders Hollowing out a stellar mass (like a asteroid) that has a similar shape to their intended structure, and shaping that.

Ok, this will require a lot of support structure, as the asteroid cannot get any significant stress.

Towers that cross the center are easy to build, they have "gravity" gradually going down from 10m/s^2 to 0 and back to 10m/s^2 in the other direction.

 

[Nimbian]

Thanks for the reply mfb

a quick look tells me that the Earth's Magnetic field only extents to around 36,000km
While the station would be at around 192,000 km. (Earths center)

Any further Questions about this particular station seem to me to be Engineering questions more then Physics questions, so I guess in the future I'll post in the Engineering area.

Thanks for all the help posters.

 

[sophiecentaur]

As to power generation I had an Idea but I don't know if its viable.
Would the spinning station that is 20km in diameter be able to generate electricity from interaction with the Earth Magnetic field?

If you were to get significant total energy out of this generator system, it would just slow the rotation of the station down because the energy has to come from somewhere. In a station of that size there would be hectares of area on which to collect solar energy - the station could be arranged to receive a similar amount of energy on its 'farmland' as you can get on Earth if it orbits at the same distance from the Sun. But what would be the purpose of the station?

 

[Nimbian]

Thanks for the Reply.

The problem with Photoelectric is that the station would for a a significant part of the its orbit be in Earth's Shadow then in the Moon's Shadow, due to its L1 placement. Due it its unreliability (also vulnerability to attack, again Hostile universe) it would not be the primary source of power. it would certainly be used as a supplemental source of power however.

The Purpose of the station is three fold.
To show the economic and technical strength of the builders, The Eden Confederacy
To Act as a Genetic repository for the majority of life from Earth, (Earth had been Nuked, Glassed or however you want to call it, and was no longer suited for human habitation)
To act as a reminder and Promise that the Eden Confederacy will never again falter in protecting humanity and its Colonies. (more story based politics I'll post if requested)

Currently at the time of the story the Station is used primarily as a tourist spot, where "pilgrims" come to see humidity's home world, and as the last peace of neural territory left.