Ball rolling within a rolling cylinder

[ezfzx]

TL;DR Summary

Cylinders rolling down inclines are a common demo. But how do you model the movement of a sphere rolling within a rolling cylinder?

Cylinders rolling down inclines are a common demo.
But how do you model the movement of a sphere rolling within a rolling cylinder?
I teaching a physics class and this question came up and my dynamics math is a little rusty.
But I haven't found anything like this in any book or online.
There's plenty about stuff rolling down inclines, and balls oscillating back and forth inside hollow stationary cylinders.
But put a ball bearing inside a hollow soup can, and everyone's suddenly stumped.

Discuss. :)

 

[berkeman]

Summary:: Cylinders rolling down inclines are a common demo. But how do you model the movement of a sphere rolling within a rolling cylinder?

Cylinders rolling down inclines are a common demo.
But how do you model the movement of a sphere rolling within a rolling cylinder?
I teaching a physics class and this question came up and my dynamics math is a little rusty.
But I haven't found anything like this in any book or online.
There's plenty about stuff rolling down inclines, and balls oscillating back and forth inside hollow stationary cylinders.
But put a ball bearing inside a hollow soup can, and everyone's suddenly stumped.

Discuss. :)

LOL, no reason to be stumped. How big of a ball bearing? How big of a soup can? What is the coefficient of friction? What are the initial conditions?

Either the ball bearing rolls slowly at the bottom of the can as it rolls down the incline, or the conditions are right to make the ball bearing get stucked (that's a technical term) to the wall of the can as it rolls rapidly down the incline.

What are your calculations so far?

 

[Baluncore]

But put a ball bearing inside a hollow soup can, and everyone's suddenly stumped.

Stumped? Just write out all the differential equations and integrate the system.

Either the ball bearing rolls slowly at the bottom of the can as it rolls down the incline, or the conditions are right to make the ball bearing get stucked (that's a technical term) to the wall of the can as it rolls rapidly down the incline.

Or as the cylinder starts to move, the ball oscillates ahead of and behind the low point inside the cylinder. If that oscillation amplitude increases, the ball can enter an internal circulating path.

Do you have a reference to the definition of that term “stucked” ?

On the subject of technical terms, the correct term for the ball is not “ball bearing”, it is “bearing ball”.

 

[berkeman]

Do you have a reference to the definition of that term “stucked” ?

I do, but I losted it. Let me try to look harder...

 

[ezfzx]

OK, so let's assume, not "stucked".
And soup can is soup can size. Can assume I = kmr², where k is very nearly 1.
Bearing about an inch. Can assume I = kmr², where k is very nearly 2/5.
Assume rolls without slipping.
Don't really need numbers, per se.
Just equations, especially if they can help me model/animated the behavior.

Two approaches: the "high" road ... diff.eqns. (very very rusty), or the "low" road, computer iterations in very small increments to draw motion path.

And having watched this thing it does indeed begin with an oscillation, but as the can rolls faster, the oscillation seems to quiet down.
Wish I had a camera that could follow it down the ramp while watching inside the can.

Kinda get stuck on those first few instances of movement, as the can is translated and rotating ... what's the ball do? Whether is stays at the bottom point of the can or rolls backward into an oscillation sort of depends on how quick the start is, relative to the inertias involved, I'd imagine.

 

[Baluncore]

If the ball was heavy and the can very thin, the ball would effectively roll down the ramp.
If the ball was small and the can very heavy, the can would effectively roll down the ramp.

Whether is stays at the bottom point of the can or rolls backward into an oscillation sort of depends on how quick the start is, relative to the inertias involved, I'd imagine.

Initially, the point of contact of the can with the ramp is uphill from the centre of mass of the can and the ball. The ball will begin by lagging the can, as angular energy must be transferred to the ball before it can roll to follow the can's internal minimum.

From Fourier's viewpoint; the frequency of the ball's internal 'pendulum like' oscillation will be driven by the frequency components of the cylinder's release and acceleration. Unfortunately, the ball is affected by, and disturbs the flight of the cylinder.

 

[ezfzx]

Yep, I was afraid of that. When the system moves initially, you can see the can jostle, jerking slightly forward, then back, as the ball throws it off balance initially. But, after a short time it settles into an equilibrium.
Given time, I could model it on the computer without diff.eqns. ... it wouldn't be precise, but it would be very convincing. (In fact, as an exercise to the engineering students: "Watch this video and tell me if it's accurate." ... that would be a great assignment, for engineering students and comp.sci. majors.)
Unfortunately, my physics class is almost always for non-physics majors, so no d.e., and I've let my high level math skills erode. And I don't really have time to re-learn all of diff.eqns. for one problem.

Help me set up the equations, and I can go from there.
If an article comes out of it, I'll give you all a credit.
(Or give me a credit for giving you the idea!)

 

[Baluncore]

(Or give me a credit for giving you the idea!)

Give all credit to PF for making the interesting group excursion possible.
Post your equations in this thread as code, then we will see what develops if/when others contribute.

 

[ezfzx]

No equations available. Not confident with how I'd set it up.

 

[Baluncore]

Not confident with how I'd set it up.

Start by defining the parameters for the simulation. The angle of the ramp and little g. Cylinder ID, OD, and material density. Ball diameter and material density.

Then compute once the dependent parameters of cylinder and ball, such as masses, radii and moments of inertia.

Then define the state variables that change during the 2D simulation. Some will be scalar; like energy, or angular velocity. Others will be vectors; positions, forces, velocities and accelerations. You must define a frame of reference.

Then simulate by playing numerical ping-pong with the ball and the cylinder. Compute the force of ball on cylinder, and then cylinder on ramp. The cylinder then accelerates, moves and integrates energy, and so applies a force to the ball. Repeat as required.

The initial condition must be clearly defined. I prefer the cylinder being held on the ramp, while the ball rests at the bottom of the cylinder. An alternative would be the ball held, pressing the cylinder against the ramp.

I guess we are assuming a perfect ramp, with continuous rolling frictional contact between the ramp, the cylinder, and the ball. In reality, it may be possible for the ball to lose frictional contact, and possibly fall in a parabolic trajectory from high on the inner wall. That exception, a non-positive contact force, should be detected during the simulation, and signalled as an alarm.

 

[ezfzx]

Thank you, but the computer part I can do ... just time consuming.
Iterating the application of forces, the resulting accelerations, velocities, position changes ...
A little tricky with the ball constrained to the inside of the moving can, which responding to the ball, which is responding to the can, etc.
The diff.eqn (and maybe it's solution) is what I'm asking for.

 

[Baluncore]

I am sure there is a physics engine that can handle the simulation.
https://en.wikipedia.org/wiki/Physics_engine

 

[ezfzx]

I'm sure there is too ... but what's the fun in that?
So I guess a diff.eqn. is too difficult?

 

[ezfzx]

OK, so how does this compare to a cylinder rolling on a moving conveyor belt?
Under what conditions is the cylinder moving toward or away from the direction the belt is moving, or is it rolling in place?

 

[Baluncore]

Having a ball, rolling in a cylinder, resting on a conveyor is more complex again.

OK, so how does this compare to a cylinder rolling on a moving conveyor belt?

A ball rolling in a cylinder is captive, while a flat conveyor has no minimum to hold a ball or cylinder.

Under what conditions is the cylinder moving toward or away from the direction the belt is moving, or is it rolling in place?

If the conveyor sagged and so had a minimum, a rolling article could settle at the minimum, but would not be captive and could escape the minimum.
Real articles that rest on a steep conveyor can repeatably stick, then roll to the bottom. That is a cyclic oscillation. The same happens with aggregate in a concrete mixer.

 

[ezfzx]

Well, interesting, but I think there was a misunderstanding.

In order to work on the "ball rolling inside cylinder which is rolling down an incline" problem, sometimes it helps to step back to a similar simpler problems.

So, instead of a ball, we have a solid cylinder, and instead of the inside of a can moving beneath the ball, we have a flat conveyor. Imagine everything stationary, then the conveyor begins to accelerate slowly. What does the cylinder do?
The point of contact, P, with the conveyor is not slipping so the acceleration at the surface of the cylinder at P is the same as the acceleration of the conveyor, a_P.
But that isn't the accel of the cylinder's CM, because inertia is holding it back.
If I = k m r² for the cylinder, where k = 1/2, then it can be shown that ...
a_cm = ( k / (k+1) ) a_P, which is in the same direction the conveyor is moving, but slower.
For a cylinder, a_cm = ( k / (k+1) ) a_P = (1/3) a_P.
For a sphere, a_cm = (2/7) a_P.

Now, instead of a conveyor, you have a giant rotating drum, with something rolling around inside, maybe a ball.
So, when it's all turned off, the ball sits in the bottom of the drum (the minimum).
At some point, the drum begins to rotating with a slow angular acceleration.
The ball, whose CM is not keeping up "falls behind", but along an arc path "uphill".
The ball tries to climb the side, but rolls back down to and thru the minimum, beginning an oscillation, which may settle out as friction pulls the ball to a position slightly uphill of the low point, constantly rolling downward (like cement in a mixer).

Now, instead of a stationary drum, you have the drum rolling along with translational motion ... maybe flat, maybe downhill. If it's accelerating, it doesn't matter too much. The initial wobble of the ball counter-wobbles the drum/can at first. It's probably easier if it's just rolling downhill, because then we know the source of the acceleration.

So, I'm expecting an equation for this motion to have some built in sine/cosine someplace which decays away in favor of terms responding to speed.

 

[Baluncore]

In order to work on the "ball rolling inside cylinder which is rolling down an incline" problem, sometimes it helps to step back to a similar simpler problems.

The ball is simpler than a roller. The difference is that the roller has length as an additional parameter. The mass of the cylinder is relevant so it needs a length parameter, and from that gains the mass parameter.
The rolling ball can be assumed to be in contact with the cylinder internal wall. The contact between the ball and the cylinder does not slide, it rolls. If you ignore any frictional energy losses, there will be no damping of any oscillation. When the location of the ball is constrained to follow the one dimensional inner path, it becomes an easier 2D problem.

 

[ezfzx]

I was actually modelling it as a 2D problem.
And we cannot ignore friction. If the goal is to produce a simulation that looks like the real thing, friction is essential.
The only unrealistic part is the assumption that it rolls without slipping.
But I'm sure that if I can get a solution to "rolling without slipping", I can then add to it later to incorporate slipping.