Maximum Energy stored in a wheel

[HimynameisJUL]

Simple problem I've been thinking about,

How much energy can you store in a wheel of uniform mass density?

There must be some kind of maximum energy you can store in the wheel.

Anyone got any ideas?

Thanks!

 

[Whovian]

How would the energy be stored?

 

[mikeph]

I don't see why there would be a limit, are you talking about the relativistic limit on the speed of an outer point limiting the rotational kinetic energy?

I think you could continue adding a torque and in a similar way as the linear case, the wheel would continue to gain angular momentum without exceeding c > v = rw. Unless it broke apart, which is a case of elastics.

 

[Bob S]

The moment of inertia of solid wheel of radius r and the stored rotational energy is
$$I=\frac{1}{2}mr^2 \text {, }E=\frac{1}{2}I\omega^2$$
The maximum stored energy is limited by the radial centrifugal forces and circumferential stresses at high RPM.

 

[AlephZero]

In a rotating solid wheel, there is a tensile stress in the axial direction, for the same reason that swinging a stone on the end of a string creates tension in the string.

The details of the stress distribution in a wheel are a bit more complicated than for a string (there is also stress in the circumferential direction), but the basic idea is the same for both.

All materials have a limit to the amount of tensile stress they can support before they break. If you spin a disk fast enough, it will break for the same reason that the string will break if you whirl the stone fast enough.

 

[HimynameisJUL]

Thank you AlephZero! Is there any way to calculate this? How is the tensile stress given?

 

[AlephZero]

For a general shaped wheel you would do it numerically (e.g. with a finite element analysis program). Here's the math for a simple shape (equations 27 and 28 are the "answer"):
http://en.wikiversity.org/wiki/Introduction_to_Elasticity/Spinning_disk

 

[HimynameisJUL]

Thank you so much!

 

[Khashishi]

Look up flywheel disintegration. It can be pretty dangerous to spin up a flywheel too fast.

 

[Lsos]

I've wondered if it would be possible to spin a flywheel faster than it's material limitations imply?

If for example we counteracted the tensile stresses with some kind of bearing structure around the flywheel, perhaps even using magnets (in a similar fashion that plasma is confined in a Tokamak).

Could we then say that there really is no maximum energy we could store? Or at least that it's much greater than what is limited by a simple stress equation?

 

[mikeph]

If you span it from the outside rather than an inner axis? Like a modified washing machine drum? I guess if you chose a suitably elastic material the stresses would tend to compress rather than expand the material, which might prevent cracking.

I don't know what would happen, maybe you could get it spinning higher, but then the question becomes "how do you spin the drum that fast without it breaking?

 

[mfb]

You would just replace flywheel material by another material.
It is interesting that the maximal velocity of the outer parts does not depend on the radius, just on the geometry and the material. If I remember correctly, it is something like ~4km/s for a solid disk and some strong material. As the stored energy per mass is given by this speed and geometry, too, you can calculate a limit on the energy density you can reach with flywheels.

 

[Bob S]

Look up flywheel disintegration. It can be pretty dangerous to spin up a flywheel too fast.

Here is a specific example. At the University of California (Berkeley) there was a large weak-focusing proton synchrotron called the Bevatron.. See http://alumni.berkeley.edu/news/california-magazine/november-december-2008-stars-berkeley/particles-dust

The power supply for the Bevatron magnets was two 3000 HP induction motors driving two alternators, with ignitrons to supply a pulsed dc current to the machine. Because of the very large pulsed load, each motor was coupled to the alternators through large 65 ton flywheels, about 8 or 10 feet diameter. See photo at http://imglib.lbl.gov/ImgLib/COLLECTIONS/BERKELEY-LAB/images/96602956.lowres.jpeg During the pulse (every 8 or 10 seconds), the flywheels slowed down from about 1200 RPM (synchronous speed) to about 900 RPM. In 1962, cracks emanating from the keyways were discovered in both flywheels. Engineers estimated that if a flywheel disintegrated, a ~15 ton fragment could land on the Berkeley campus. So the Bevatron was shut down for an extended period while the two flywheels were replaced.

 

[ashishsinghal]

I don't think there should be a limit in an ideal world. If we consider the values of tensile strength of the material or relativity then it is different.

 

[Whovian]

Also, if we have a strong enough wheel, we still need to take the increase in mass predicted by relativity into account. I don't know if such wheels exist, and I doubt it, but I feel like randomly throwing something into the discussion.

 

[Bob S]

If we had a heavy metallic wheel spinning at relativistic speeds, it would have to be aligned with the Earth's axis of rotation, or else the gyroscopic precession force would cause the Earth to wobble on its axis. Even today, heavy rotating machinery should be aligned with the Earth's magnetic field to minimize eddy currents.

 

[Lsos]

If you span it from the outside rather than an inner axis? Like a modified washing machine drum? I guess if you chose a suitably elastic material the stresses would tend to compress rather than expand the material, which might prevent cracking.

I don't know what would happen, maybe you could get it spinning higher, but then the question becomes "how do you spin the drum that fast without it breaking?

I'm thinking spanning it from the outside, but having the outside be stationary. The flywheel would be spinning on the inside with the tendency to fly apart, while the outside would be squeezing it in, counteracting this tendency. I'm sure there's some issues involved, I just don't know what they are.

 

[Bob S]

The stresses on a spinning flywheel are primarily radial and circumferential tension (not compression), and limit rotor speeds to much lower than relativistic velocities. Recall the stuxnet virus that infected the Iranian uranium enrichment P-1 centrifuges and were destroyed because they spun out of control. See http://itmakessenseblog.com/2011/01/16/israel-software-worm-shuts-down-iran%E2%80%99s-centrifuges-and-causes-delay-in-iran-nuclear-bomb-program/

 

[K^2]

The moment of inertia of solid wheel of radius r and the stored rotational energy is
$$I=\frac{1}{2}mr^2 \text {, }E=\frac{1}{2}I\omega^2$$
The maximum stored energy is limited by the radial centrifugal forces and circumferential stresses at high RPM.

Note that if you combine these, the energy depends on mass and velocity of the outer edge. Furthermore, if you look at the equations for stress on the disk, you'll notice that disk will disintegrate pretty much at the same outer velocity regardless of radius. So the maximum energy stored in the disk would depend only on material used and mass of the disk.