DIYbanter - View Single Post - Bursting speed of flywheels (and overspun ball bearings)

In article ,
says...
In article ,
Ned Simmons wrote:

Yes, MH is full of practical approximations, and they do say that steel
is assumed.

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.

What's "gravity", and how does it differ from "density"? This theory
cannot depend on the presence of a planet or its gravitational field.

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.

Where are you getting these better formulas? I'd like to read up on it.

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.

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.

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.

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