# Can we make a floating metal ball?

Okay, so I was watching the Mythbusters make a lead balloon, I got curious. How far away are we from being able to make a floating sphere? As in a hollow metal vacuum sphere that is bouyant in the atmosphere? Now ignore any problems about how so make it, just assume it’s created perfectly. And we can ignore any kind of internal skelton support or honeycomb stuff, cause that gets really complicated.

So I started calculating using sea level air and Titanium on a ball with an inside radius of 10cm.

Here are my calculations(Please don’t take points off for sig. digits ).

Volume of sphere 4/3(pi)r^3

Density of air. 1.2kg/m^3
Density of Titanium 4507Kg/m3

10 cm ball = .0012Kg air mass
.0012Kg of Titanium = .000000266 m^3

Thickness of metal : 4/3(3.1416)(X)^3 - 4/3(3.1416)(.1)^3. = .000000266
: 4.188X^3 - .004188 = =.000000266
: x^3 = .00100000635
: X=.10000212M^3

So, my sphere would have a wall thickness of .00000212m or .000212cm. And theoretically it would float in the air if it could withstand the crushing of one atmosphere of pressure at that thickness. Now, that is pretty damn thin, and it won’t work feasibly, But I’m curious exactly how far away it is. Material strength engineering was never my strong point, can anyone tell me the approximate crush restistance of a 10cm titanium sphere with a wall thickness of .000212cm?

Thanks.

Could synthetic carbon molecule materials do this?

I’m thinking Zeppelins.

Actually, Zeppelins were mostly metal. Even the outer skin was fabric impregnated with aluminum powder and nitrate dope.

I think the OP’s point is to make something that floats without being filled with hydrogen or helium. Otherwise all the material has to do is contain the gas, which is pretty easy (in theory–I haven’t tried it!)

Yeah, exactly just a thought exercise, but I want a vacuum.

And hopefully a hard rigid ball. I have no idea how rigid those arships were. Like could you put your weight into it without it moving?

since it’s metal, are we allowed to use electricity or magnetism to make it float?

Not sure what you mean by put your weight on it. I gather there was an aluminum deck or catwalk anywhere you would walk. The outer skin was stretched taught on an aluminum frame. The actual gas bags inside were just bags.

Limp airships are pretty rigid too, when filled. They’ve got a “ballonet” of regular air inside the envelope to maintain pressure when the amount of lifting gas is varied to adjust buoyancy.

They didn’t have ridig walls though, my ball should theoretically be something you could bounce on the floor. without deformation.

Hmmm. If your calculations are correct, you’re talking about a sheet thickness only about 10 times that of gold leaf, the most ductile material there is–using a much less ductile material. I really doubt you can get titanium that thin. I’d suggest trying two things:
[ul]
[li]Try a lighter metal. Aluminum, for instance, would yield a thicker wall that could end up being more rigid structurally, even though it’s got a lower tensile strength.[/li][li]Try a larger ball. Even with titanium, I think you’ll find your wall thickness will increase as the ball gets larger. That’s because the volume of the sphere (and metal wall) increases by a cube function while the surface area only increases by a square.[/li][/ul]

Make the sphere big enough, and vacuum or special light gases are no longer needed:

Somehow, I think there would be severe FAA restrictions on a floating quarter-mile sphere.

From Roark’s Formulas for Stress and Strain, 7th Ed (the only version I have at home), the hoop stress from an evenly distributed pressure load in a thin-walled spherical pressure vessel is:S = qR/2twhere q is the pressure (14.7psia or 101.4kPa for an evacuated vessel at standard temperature and pressure), R is the radius to the shell midplane, and t is the wall thickness. This gives a stress in your vessel of S = 101.4kPa * 5cm / 2*0.000212cm = 1195MPA or 173.4ksi. This is within the range of high-end titanium alloys like ASTM B265, Grade 5 (which is, conincidentially, used in applications like turbine rotors and thin-walled pressure vessels).

However, you have some additional issues here; for one, the formula assumes a constant stress that would be provided by internal pressure resulting in tensile hoop stress, whereas when applying pressure externally on a thin-walled body you are going to have to cope with buckling phenomena, which are non-linear for any real-world situation and are thus much harder to predict; any slight variation in thickness, or a geometric discontinuity (like a weld-bead or slight wall thickening) will concentrate compressive stresses resulting in failure. Another issue is manufacture; this is thinner than the thinnest foil; any attempt to press or forge a structure at this thickness would leave residual stresses which would again result in local failures. The only way to practically make such a structure would be via some deposition process where the material is layed down a few atoms of thickness at a time. As for being able to bounce it like a ball, forget it; it would be far too fragile to survive an impact. The analysis of this type of structure under impact is extremely complicated–you not only need the strength but also other mechanical response properties, and information on the structural response of the surface it is bouncing off–but it’s a no-brainer that any impact would result in local deformation which would subsequently cause the sphere to crack like an under-cooked egg yolk.

Switching to another material isn’t really an option; at this thickness even very high strength exotic materials aren’t going to be sufficiently resillient to local impact to make such a structure stable. Making the sphere larger, however, will have a substantial improvement, as volume increases (and composite density thus decreases) as a cube in proportion to the radius, while thickness to maintain idential stresses increases in square proportion to radius. If you could manufacture a sphere large enough it would presumably be possible to evacuate it and make it lighter than the air it would displace, giving it a degree of buoyancy as dictacted via Archimedes principle. I’ll leave it as an exercise to the o.p. to run through the studies to figure out the thickness and radius, but it would be enormous.

Airships, by the way, work by enclosing a lower mass gas in tension-stabilized structure (i.e. a balloon) in which the gas is just slighty in excess of atmospheric pressure. The tensile strength of the structure therefore need not be enormous but merely enough to maintain a spheroidal shape, and because it is always in tension there are no skin buckling issues as with an evacuated pressure vessel as described by the o.p., so the mechanics of the two situations are very different.

BTW, Buckminster Fuller once envisioned tension-stabilized geodesic (of course) floating cities a la Airship One which would maintain altitude via a slight temperature (and thus pressure) differential in the enclosed volume of air. Such a structure would have a very large minimum size but would scale upward and be thermodynamically stable owing to the large thermal mass of air. Doing this on a small scale, however, would be unstable and mechanically impossible, hence why airships use helium and balloons use propane burners to heat air.

And now I see upon review that Squink and mwbrooks have addressed the issues in the last two paragraphs. Oh well. Q.E.D., it depends upon how well armed your floating city was, and also what kind of rock bands they were able to attract in exchange for publicity. You would definitely need to lay in a good store of booze, cheese, and crackers, and an exceptional sound system.

Stranger

As long as it’s just restrictions, that’s OK. Prohibitions might cause us trouble though.

The real question is: What if we put it on a treadmill? :dnr:

I swear to God we did this thread a couple years ago; I remember doing some buckling calculations. I seem to recall the problem, as you would expect, boiled down to finding a material with a high enough stiffness-to-density ratio. I can’t for the lide of me find the earlier thread, though.

I’ve got text with a pretty good section on buckling of thin-walled pressure vessels at work; if ther’s still interest in this threadon Monday I’ll see if I can dig it out.

You’d want to use Aluminum Honeycomb Panels (or maybe titanium panels, specially made). They will have a higher strength to weight ratio compared with a solid skin of the same material. Probably not feasible for a 10 cm ball, but as you get to tens or hundreds of meters or larger, that’s what you’ll want.

As it happens, I have a coupon of titanium alloy honeycomb on my desk at work. It’s generally used for aerospace structures like fairings or fins where strength against compressive aerodynamic loads is critical. Your structure would definitely have to be much larger than 10cm to contain a sufficient vacuum that would allow the structure to be buoyant in air, but honeycomb is ideal for increasing buckling strength.

Stranger

Here we go. Here’s the thread I was thinking of earlier, and here’s another one yet. Although I think there was a third one even earlier.

Anyway, the volume of a sphere is 4/3[symbol]p[/symbol]r[sup]3[/sup] and the volume of the shell around it is 4[symbol]p[/symbol]r[sup]2[/sup]t. The buoyancy of this sphere is the volume times the density of air (1.3 kg/m[sup]3[/sup]), and the weight is the shell volume times a material density, [symbol]r[/symbol]. The sphere will float if 4/3[symbol]p[/symbol]r[sup]3/sup > 4[symbol]p[/symbol]r[sup]2[/sup]t[symbol]r[/symbol]. Simplifying, that’s

[symbol]r[/symbol] < 0.43(r/t)

Likewise, from my earlier post, since this is an external pressure vessel, buckling is probably the limiting factor. If so, my reference says the critical buckling pressure is somewhere between P = 0.37E(t/r)[sup]2[/sup] and P = 1.21E(t/r)[sup]2[/sup] depending on manufacturing precision. Atmospheric pressure is 100,000 Pa, so to withstand buckling, (0.37 to 1.21)E(t/r)[sup]2[/sup] > 100,000. Rearranging,

E > (83K to 270K)(r/t)[sup]2[/sup]

The r/t ratio can be anything, but of course we want it to satisfy both inequalities above. Combining, we can drop out the r/t ratio to get E > (83K to 270K)([symbol]r[/symbol]/0.43)[sup]2[/sup] or

E/[symbol]r[/symbol][sup]2[/sup] > (450K to 1.46M)

So we need a material with a stiffness divided by density squared of right around a million. From this graph, you can see that the best materials (wood, as a matter of fact, as well as diamond) have
E/[symbol]r[/symbol][sup]2[/sup] ratios of about 0.1. Not even close, assuming I’ve done my math right.

Is a substance with such a stiffness-to-density ratio possible?

Aerogels might be suitable for increasing the compressive strength of a thin metal sphere. With a density of 1mg/cm[sup]3[/sup] in vacuum, you could make a pretty thick, sturdy shell to support the metal layer, without losing much buoyancy.

I’m an EE, not an ME, but don’t those equations have to be approximations that fail for large r? We can certainly make Ibeams that work for r = 0, or even r < 0.

The equations are specifically for the buckling of a spherical pressure vessel, taking only pressure difference into account. They’re unitless, so scale shouldn’t be a factor (except for the obvious question of practicality).