Calculating Force of Bouncing Ball Bearing

[KHardin]

At work I have been given the task of finding force of ball bearing hitting a pick up finger in a ball nut. If you have no idea what that is then to simplify the scenario I am trying to model a ball bearing hitting a steel plate and the force that ball exerts on the plate.

Back to physics 101 I know that F=ma=m(dv)/dt

To simplify some more we can say a=a=∆v/∆t
where ∆v is the change in velocity and ∆t is the time the ball is in contact with the surface of the plate.

I know the mass of the bearing and initial velocity, I can assume an elastic collision so the exit velocity is the same as the entrance velocity. The big problem is determining how long the ball is in contact with the surface.

Here is my first attempt at the calculation.

I converted the 'ball' to a cylinder with the same mass and same height but different diameter for simplicity. I also assumed that there was no radial expansion of the cylinder (ignoring Poissons ratio)

The speed of sound in a solid is σ= (E^.5)/ ρ
Where E is Young’s Modulus for the material and ρ is the density

∆t=2L/ σ


From there it is easy to calculate force using F=m*∆v/∆t


Source for these equations http://www.jw-stumpel.nl/bounce.html"




The problem that I am encountering is the resulting force from these set of initial conditions.
diameter of ball=.141"
∆t=1.44E-5 sec
v=103 in/sec
m=1.27E-5 slugs
E=30500 ksi =3.05E7 psi
density=.282 lb/in^3

the result is a force of 181 pounds. I know this can not be the case because FEA analysis of this force on the mating part will cause a failure. Since these nuts can run for millions of inches of travel without failure this can't be an accurate result.


Is there a flaw in my calculations or assumptions or am I even close to the correct path?

 

[corey2157]

I know this question is 7 years old...

Without considering the method you used to arrive at the answer, I think you have used flawed logic in your dismissal of the result. A force of 181 pounds may well cause that part to fail is that force is applied for a long enough amount of time. But consider the part in question has a mass of its own, and inertia. The part you think would fail at 181 pounds of force... what distance would you expect it to deflect in 1.44E-5 sec with said force? You could first look at the deflection just as a function of mass, time and force. Then, using the FEM, see if the part would fail under that amount of deflection, and if not, have FEM give you the resorting force. My guess is you will see that the part will not fail under such conditions. I've never used FEM, but perhaps it does not handle momentary shock loads to well?