Bursting speed of flywheels (and overspun ball bearings)

Machinery Handbook has long had a section of flywheels. In the 20th
edition, on page 346, states that all that matters is surface speed at
the periphery, and the tensile strength of the steel, and gives a
formula: V=Sqrt[10*s], where V is surface speed in feet per second, and
s is tensile strength in pounds per square inch.
Let us assume that s= 300,000 psi, the cited strength of ball bearing
race steel. Sqrt[10*
300000]= 1732 fps.
A bearing 1.75 inches in diameter will have a circumference of
(1.75)(3.1416)/12= 0.4561 feet, so 1732 fps implies 3,781 rps, or
226,832 rpm.
The speed of sound is about 300 meters per second at sea level, or about
900 feet per second, so the surface speed of the outer race is 1732/900=
1.92 times the speed of sound at sea level.
If the airjet is at the speed of sound, and is impinging on the balls,
the outer race will go twice the speed of sound.
If the bearing has ten balls, the siren tone will be at 3,781*10= 37,810
Hz, well into the ultrasonic, as people have observed.
The guy that did the experiment showing a max speed of ~20,000 rpm for
whatever reason did not achieve full speed, as 20,000 rpm isn't nearly
enough, and yet people have no problem causing bearings to burst from
overspeed.
Basically, it all fits together. Then it bursts.
Joe Gwinn
Reply to
Joseph Gwinn
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Think of rotation as the average motion and resonant acoustic vibration as the instantaneous motion. The stress from the maximum vibration excursion can be much greater than that from the rotation.
Reply to
Fred R
Joseph,
The balls of the bearing whizzing around do not carry any hoop stress, but produce a radial outward force on the outer bearing ring analogous to the internal pressure of a pressure vessel.
Since you are into the arithmetic and I am too lazy to figure it out, what is the fraction of the outer ring hoop stress due to the orbiting balls?
Just curious.
Wolfgang
Reply to
wfhabicher
I'd love to use a high speed camera to take a picture of the burst. I wonder if a light beam could be used for this. I'm picturing (sp?) 3 mirrors, a laser pointer, and a photodetector of some sort to trigger the shutter. Maybe use a very bright halogen light to illuminate the area. Use the mirrors to make a box shaped area with the laser. Maybe use more than three mirrors to make a cube shaped area. I guess if the bearings are exploding at 25000 rpm then if my math is right the pieces will be moving at about 218 feet per second. With a 1/1000 shutter speed it looks like the parts would travel about 2.6 inches. Maybe a better solution is to leave the shutter open and use a flashlamp instead. Hmm. ERS
Reply to
Eric R Snow
None, I think, because the flywheel theory depended only on the surface speed at the rim of the flywheel, and not at all on what was inside.
Joe Gwinn
Reply to
Joseph Gwinn
From Machinery handbook - 22nd edition. (page 226) "The bending stresses in the rim of a flywheel may exceed the centrifugal (hoop tension) stress predicted by the simple formula s = V(squared) divided by 10 by a considerable amount. See relevant section for further edification.
Have fun. Ken.
Reply to
Ken Davey
I think the guy claiming 20,000 rpm max is wrong, and the actual rotational rate is ten times that. So, it would take a microsecond light pulse to stop the pieces, which are moving at Mach 2. This is high-powered rifle bullet speed.
An easier way to measure the speed of the fragments is two curtains of fine wire - measure the time delay between disruption of the inner curtain and the outer curtain. One can also use disruption of a curtain to trigger a flash.
Joe Gwinn
Reply to
Joseph Gwinn
That's if you try to change the axis of rotation while the flywheel is madly rotating.
Joe Gwinn
Reply to
Joseph Gwinn
The formula in MH is an approximation that works for steel and materials with similar specific gravity. The real formula is:
stress = (density / gravity) * radius^2 * angular velocity^2
or
angular velocity = sqrt((stress * gravity) /
(density * radius^2))
where angular velocity is in radians/s and density in weight/unit volume.
You *must* account for the balls, which is why I've been using 1300FPS as the limit for 300 ksi steel rather than 1700FPS. Based on a SWAG that the balls weigh a bit less than the race I used a density of 0.5lb/in^3 in the formula, rather than steel's actual 0.28lb/in^3.
Ned Simmons
Reply to
Ned Simmons
No, that's if you're talking about a real flywheel with a thick rim and spokes. The formulas we've been bandying about are only strictly applicable to thin cylinders.
Ned Simmons
Reply to
Ned Simmons
It's not. Show me a reference to a blowgun that produces a supersonic air jet with shop air and I'll reconsider.
Okay, since this keeps coming up, despite what seems common sense to me, I set up a test myself. That's a 6204 bearing with the seals removed, the grease washed out, and relubed with a few drops of light spindle oil. There's a paint mark on the race and the ball cage.
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I spun the bearing up and measured the difference between the speed of the ball cage and the outer race at several speeds between 1400 and 5500 RPM. The difference in the angular velocity in all cases was 100~200RPM, i.e., the race was going 5-10% faster than the balls. Exactly as you'd expect in an unloaded bearing with internal clearance where the balls are free to slip relative to the races. As I've said at least twice before, this clearance will only increase with speed.
I checked this as well just to make sure the assumption that the frequency of the sound from the bearing does in fact correspond to the ball passing frequency. I got my teenage son, who's involved in composing synthesized music, to set up his laptop with an FFT to monitor the bearing siren tone. Agreement was within a few percent, probably as good as could be expected with me getting a strobe fix while asking him to read the frequency.
Because it's well known that, for the sort of bearing we're talking about, speeds in the few tens of thousands of RPMs are the lubrication limit for properly mounted bearings with elaborate mist lube systems. It's no surprise at all that a loose bearing that's just had all its lubrication removed would fail at somewhat higher speeds.
Ned Simmons
Reply to
Ned Simmons
I didn't say supersonic, I said sonic (as the upper limit). The airflow chokes in the orifice, being limited to the speed of sound, so this is the upper limit (unless one has a nozzle that looks like the back end of a rocket engine, with an expansion bell).
It appears to have eight balls. What's the OD?
I see one problem in the photos: That long thin copper tube will not achieve anything like the airspeed that a proper nozzle will achieve.
I would suggest using a piece of 3/8" tubing with a machined brass nozzle hard soldered into one end. The brass nozzle would have a 60-degree (included angle) cone inside, going from 3/8" to 0.014" diameter at the face.
I would put around the bearing a piece of heavy metal pipe lined on the inside with wooden staves, to stop the shrapnel. Even if you believe that the bearing won't burst.
These are very low rotational speeds. As the speed increases, won't centrifugal force pin the balls against the inside of the outer race, reducing or eliminating slippage?
Good. The "siren" theory is confirmed.
So, how do we explain the reports that the tone went ultrasonic just before the bearing exploded? With eight balls, this implies 20000/8= 2,500 rps, or 150,000 rpm, a factor faster than the 20,000 rpm discussed here.
Also unexplained is the essentially perfect symmetry of the explosions.
While I don't doubt that being run bare at such high speeds chews the bearing up pretty fast, the guy I was mentioning also used the long thin air tube, and so didn't achieve full airspeed.
Joe Gwinn
Reply to
Joseph Gwinn
Hmm. On second thought, I think you're right.
With balls rolling on the outer ring at very high speed, we should see some metal fatigue effects from the cyclic bending seen as the balls pass by.
Wonder if we are running through the fatigue life of the steel, which then cracks, precipitating the burst? One problem with this theory is that one would not expect this mechanism to lead to the essentially symmetrical explosions that have been universally reported.
Joe Gwinn
Reply to
Joseph Gwinn
My theory was that the balls are less dense than solid disk, and more or less equivalent to (moving) spokes), so the MH formula would apply.
Yes, MH is full of practical approximations, and they do say that steel is assumed.
What's "gravity", and how does it differ from "density"? This theory cannot depend on the presence of a planet or its gravitational field.
Where are you getting these better formulas? I'd like to read up on it.
If the balls weigh less than the race, the 0.5 lb/in^3 sounds wrong, as it's more than that of solid steel, 0.28 lbs/in^3. Perhaps some more explanation is in order.
Joe Gwinn
Reply to
Joseph Gwinn
Gravitational acceleration, to account for the fact that a pound mass exerts a pound force in a gravitational field of 386 in/s^2 and we're calculating the forces exerted by a lump of material in a rotating frame with a different acceleration.
I got the formula from "Roark's Formulas for Stress and Strain", but that's not the place to go for an explanation of the derivation. "Elements of Strength of Materials" (Timoshenko) has the derivation for the stress in a cylindrical pressure vessel, which is a very similar problem.
As an intuitive approach, you can think of the problem as two semicircular segments joined together. What's the force required at the joints to hold the two halves together? Divide that by cross section to get stress.
It's a fudge. As Wolfgang said, the balls are exerting a force on the race but don't increase it's strength. To account for the balls' additional mass I added it to the outer race's mass, but didn't change the cross section (strength) of the race. In other words, I decreased the strength to density ratio of the race to account for the loose balls.
Ned Simmons
Reply to
Ned Simmons
47mm - 1.85"
I have no interest in exploding a bearing . I just wanted to get it spinning fast enough to run the tests described.
Even at these speeds I don't imagine there's much slippage between the balls and the outer race - clearly there isn't. Where the balls *are* slipping is relative to the inner race, minimizing any speedup due to planetary action.
I can't. Maybe Eric's hearing is worse than he thinks. I wouldn't know how low my upper limit is in one ear (starts rolling off at a few kHz) if I didn't fail the hearing test in grammar school every year.
As I said before, it's easier to explain the lack of serious injuries if the available energy is much lower. Perhaps the bearings that exploded with bad consequences has selectively thinned the reporters .
I agree that higher speeds than 20 KRPM should be attainable, but the deterioration will limit the speed, and that may be why he couldn't go faster. As the bearing gets beat up it'll take more power to keep it spinning at a given rate.
Ned Simmons
Reply to
Ned Simmons
How fast did you get it to go?
So the balls and outer race spin together more or less as a unit? I guess I don't quite trust that so small a clearance will really always do the job. All it would take to get some real traction would be for the human holding the inner race to move slightly, twisting the axis of rotation a few degrees, causing the balls to come up against the sides of the groove in the inner race, pushing against the gyroscopic forces keeping the outer race from turning with the inner race.
The lack of lubrication will make for more traction, especially if the balls are galling with the races.
A test jig where the bearing is clamped to a bench would never see this effect.
I bet your Son can test your hearing limit.
Darwinism in action. But don't you think we would have heard the stories, if there were stories to be heard?
Will deterioration really be that much of a limit on an unloaded bearing, especially if it isn't in contact with the inner race all that much? And, the airjet has plenty of power.
Joe Gwinn
Reply to
Joseph Gwinn
Ah. The conversion from pounds (a unit of weight) to slugs (a unit of inertial mass) is partly buried in the formula. In the MH formula, all this disappears into a single unexplained constant.
Thanks. I think I have both books.
And as one increases the number of segments, the formula will approach the continuum case.
Ahh. OK.
Joe Gwinn
Reply to
Joseph Gwinn
About 6500 RPM. I didn't have the needle valve opened all the way, but I don't think it would have gone much faster with the small tube.
He tests many limits.
There's lots of power in the compressed air - transferring it to the bearing is another matter. Even air motors are notoriously inefficient.
Based on the numbers, and 25 years of machine design experience, I'm very confident that the bearings did not explode from centrifugal force alone. The reports of the noise going ultrasonic do give me pause, but not enough to make me believe that the bearings were spinning at 1300 FPM. I guess someone needs to volunteer to explode a few properly instrumented bearings. Not me, I'm off to Boston for college visits with the boy tomorrow.
Ned Simmons
Reply to
Ned Simmons

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