How to Use Buckingham's PI-Theorem for Rolling Cylinders?

[BruceSpringste]

Hello! I am doing an experiment on rolling cylinders with both an inner diameter and outer diameter. I.E. they are not solid. I have to determine the time it takes for a cylinder to roll down an inclined slope. I need to do a dimensional analysis with Buckinghams PI-Theorem but I am stuck and need a little help.

Parameters: Width (w) of cylinder, mass (m) of cylinder, inner diameter of cylinder (d), outer diameter (D), length of slope (l), angle of slope (v) and gravity (g).

t=(d,D,w,m,l,v,g)
We define a lengthscale L₀ = d, massscale M₀=m and time scale T₀=(L₀/g)^1/2

Any thoughts on the next step?

 

[Delphi51]

Are you thinking of using conservation of energy or a forces/acceleration/velocity approach? It would be interesting to try it both ways.

 

[Stephen Tashi]

Any thoughts on the next step?

You should do better than making time t a function of all the fundamental parameters. Instead of

$t=f(d,D,w,m,l,v,g)$

you should show some functions "inside" of $f$

One way is to think of sets of parameters that can be varied while keeping crucial quantities the same. Another way is visualize solving the problem "the long way". Think of situations where you'd use some parameters to compute some intermediate quantities and not need to use parameters again.

For example, changing the parameters $d$ and $W$ doesn't seem to matter as long as we keep the cyclinders moment of inertia constant.

Use $g(d,D,w,m)$ as the moment of intertia within the function and eliminate $d,w$ from the argument list.

$t = f( g(d,D,w,m) ,D,m,l,v,g).$

The forces on the cylinder are determined by $v,g,m$ (There is also a frictional force. I'm assuming it's sufficient to make the cylinder roll without slipping.) In terms of dynamics, the only role I see for $v,g$ in the problem is to compute the forces. So forces are functions of the form $h(v,g,m)$.

$t = f( g(d,D,w,m), h(v,g,m), D,m,l )$

 

[Chestermiller]

Mass m is not a parameter, because it is the only one with units of mass on your list. w is not a parameter because it is a 2D problem. So,

$$t\sqrt{\frac{g}{l}}=f(\frac{d}{D}, v, \frac{D}{l})$$

Chet

 

[BruceSpringste]

Mass m is not a parameter, because it is the only one with units of mass on your list. w is not a parameter because it is a 2D problem. So,

$$t\sqrt{\frac{g}{l}}=f(\frac{d}{D}, v, \frac{D}{l})$$

Chet

I kind of got to the same conclusion. However I had $$(\frac{d}{l}, v, \frac{D}{l})$$How did you come to that conclusion?
Also how do you scale with l? Shouldnt time be scaled with $$t\sqrt{\frac{g}{sin(v)*l}}$$.
Since g acts vertically I mean.

 

[BruceSpringste]

You should do better than making time t a function of all the fundamental parameters. Instead of

$t=f(d,D,w,m,l,v,g)$

you should show some functions "inside" of $f$

One way is to think of sets of parameters that can be varied while keeping crucial quantities the same. Another way is visualize solving the problem "the long way". Think of situations where you'd use some parameters to compute some intermediate quantities and not need to use parameters again.

For example, changing the parameters $d$ and $W$ doesn't seem to matter as long as we keep the cyclinders moment of inertia constant.

Use $g(d,D,w,m)$ as the moment of intertia within the function and eliminate $d,w$ from the argument list.

$t = f( g(d,D,w,m) ,D,m,l,v,g).$

The forces on the cylinder are determined by $v,g,m$ (There is also a frictional force. I'm assuming it's sufficient to make the cylinder roll without slipping.) In terms of dynamics, the only role I see for $v,g$ in the problem is to compute the forces. So forces are functions of the form $h(v,g,m)$.

$t = f( g(d,D,w,m), h(v,g,m), D,m,l )$

You're not using Buckinghams I am guessing? Also the inner diameter (d) does change the moment of inertia?

 

[Chestermiller]

I kind of got to the same conclusion. However I had $$(\frac{d}{l}, v, \frac{D}{l})$$How did you come to that conclusion?

The two representations are equivalent. But using d/D appealed to me more aestheticly.

Also how do you scale with l? Shouldnt time be scaled with $$t\sqrt{\frac{g}{sin(v)*l}}$$.
Since g acts vertically I mean.

That makes use of what you know about the physics of the problem. According to my understanding, you were only allowed to use the Buckingham Pi approach exclusively.

Chet

 

[BruceSpringste]

The two representations are equivalent. But using d/D appealed to me more aestheticly.That makes use of what you know about the physics of the problem. According to my understanding, you were only allowed to use the Buckingham Pi approach exclusively.

Chet

Thank you, I think I understand your line of thought now!

Edit: Altough the fact that you wrote d/D bothers me a bit. It appealed to you aestheticly. Imagine I could have used physics when I were done with the analysis. Could I have come to the conclusion d/D?

 

[Chestermiller]

Thank you, I think I understand your line of thought now!

Edit: Altough the fact that you wrote d/D bothers me a bit. It appealed to you aestheticly. Imagine I could have used physics when I were done with the analysis. Could I have come to the conclusion d/D?

No. It's mathematical. If d/l and D/l are dimensionless groups in your problem, then their quotient must also be an acceptable dimensionless group. But, if you include their quotient as one of your dimensionless groups, then you need to drop either d/l or D/l.

Chet

 

[BruceSpringste]

No. It's mathematical. If d/l and D/l are dimensionless groups in your problem, then their quotient must also be an acceptable dimensionless group. But, if you include their quotient as one of your dimensionless groups, then you need to drop either d/l or D/l.

Chet

Yes I understand that. But how come you didn't write D/d instead? I thought maybe you had some physical reasoning behind the choice of d/D instead of D/d.

 

[Chestermiller]

Yes I understand that. But how come you didn't write D/d instead? I thought maybe you had some physical reasoning behind the choice of d/D instead of D/d.

No. Either one is fine.

Chet