# Centrifuge at hypersonic speeds?

I'm interesting in testing some aerospace propulsion ideas at actual hypersonic velocities, perhaps up to even orbital velocity. Of course
hypersonic wind tunnels are quite expensive and none even go up to orbital velocity. What I wanted to do was have a rotor move at rim velocity at this speed range. For current materials this would be far beyond their tensile strenth if you are using a uniform rotor. However, I was wondering if tapering might make it possible using materials in common use. Here's a post to sci.astro presenting the calculation for rotating spheres made of diamond:
Newsgroups: sci.physics From: snipped-for-privacy@cars3.uchicago.edu Date: Fri, 09 Jul 2004 19:56:04 GMT Local: Fri, Jul 9 2004 3:56 pm Subject: Re: What is the Fastest spinning man made Object? http://groups.google.com/group/sci.physics/msg/924fb8ea41244702
The example given there for a rotating sphere resulting in:
v_lim = sqrt(2*S/rho), where S is the yield strength and rho the density,
suggests that it might work, since this is larger than for a uniform rotating ring in which v_lim is:
v_lim = sqrt(S/rho).
So I'm thinking a sufficiently tapered rotor might be able be able to get orbital velocity rim speed using current materials (not diamond). I know that the highest speed flywheels are able to get a rim velocity in the range of 1,100 m/s using carbon fiber composites. But this is an anisomorphic material and might be difficult and expensive to produce in the right shape I need. So I'm thinking of using high strength aluminum, titanium, or steel alloys. I know that the theory for tethers proposed for the "space elevator" use exponential tethering. Then this tapering should also work if you were using a rotating rod-like structure. However, for my usage I need a disk-like or torus-like rotor. What's puzzling me is that with these kinds of shapes you need to worry about tangential stresses, not just radial ones. So my question is if you used the right tapering to take care of the radial stresses, would that be sufficient to take care of the tangential stresses as well?
Bob Clark
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Robert Clark wrote:

[snip rest of crap]
Have you ever been correct about anything?
idiot
The limiting equatorial velocity, for a rotating sphere ( velocity is independent of the size of the object) is given by
v_lim = sqrt(2*S/rho)
where S is the yield strength and rho the density. Now, 10 tonnes/mm^2 translates to about 10^11 Pa (100 gigapascals). Diamond density is about 3500 kg/m^3. Throwing this into the formula obtain
v_lim = 7600 m/s (approximately)
i.e. a tad less than 5 miles/sec.
It arises from forces acting on a surface element with area dA of the rotating body. For an infinitsimally thin surface element the only stresses acting are tangential since radial stresses go to zero on the surface. Said tangential stresses still yield a radial force component for a curved surface. It is a general result from the theory of elasticity (first by Laplace, that given a surface element with area dA and thickness dt, with tangential stresses present in the surface, the normal (to the surface) force acting on this element is
where R1, R2 are the main radii of curvature of the surface (at the point of evaluation) and S1, S2 are the stresses along the directions of the corresponding axes of curvature).
Take the ball rotating with an angular velocity w, and evaluate the forces acting on an equatorial surface of element. Do the evaluation in the rotating reference frame. The surface element is acted upon by two forces. First, there is the elastic force, described above, that simplifies to
since R1 = R2 = R, the radius of the sphere. This force is pulling the surface element inwards, towards the rotation axis. The second force is the centripetal one
dF_c = R*w^2*dm
where dm is the mass of the suface element, given by
where rho is the density. So, we get
This force acts ouwards, away from the rotation axis. Since the surface element remains stationary in the rotating frame (until the sphere pops), the two forces dF_s and dF_c must be equal. So, we've
Cancelling the common factors and reorganizing we get
(R*w)^2 = (S1 + S2)/rho
R*w is simply v, the velocity of the equatorial point. As for S1 and S2, neither of them can be larger than the tensile strength S. So, we get
v^2 <= 2*S/rho
with the = sign obtaining at the limit, before the material gives. This is the relationship used above to find the limiting velocity. Note that the radius cancels out of it.
The ratio S/rho, with the dimensions of energy/mass, is binding energy/unit mass of the material. With this, the result above can be simply interpreted as saying that the ball will hold together as long as the kinetic energy per unit mass (in any locality) is no larger than the binding energy per unit mass.
This provides a sanity check for any exaggerated claims. At absolute strength limit, given a single giant molecule (diamond) with no faults and flaws whatsoever, the binding energy per unit mass still doesn't exceed something of the order of eV/amu.
idiot
--
Uncle Al
http://www.mazepath.com/uncleal/
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Robert Clark wrote:

[snip rest of crap]

Go ahead, build it. We'll wait. Empirical physical reality is Uncle Al's ally - and a powerful ally it is.
--
Uncle Al
http://www.mazepath.com/uncleal/
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Ya know, Uncle Al, I bet he could learn more about wind tunnels. But I bet you can stop being such an asshole.
Anyone wanna bet?
Jonathan
s
/lajos.htm#a2
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On Sat, 15 Aug 2009 06:45:57 -0700 (PDT), Robert Clark

Think shock and expansion tubes. The later can go to super-orbital velocities, and are being used for re-entry tests of planetary bodies other than earth. See the NASA HyPulse facility at ATK-GASL, CALSPAN, T5 at GALCIT, the newer tube in Germany and the X2, X3 facilities in Australia. Shock and expansion tubes are routinely used for scramjet experiments. In fact NASA showed quite good compaisons between the Hyper-X Mach 10 flight test and ground test data obtained in HyPulse.
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