# How to define forced non-centroidal rotation and conservation of angular momentum?

1. Oct 13, 2012

### diffusegrey

Hello all. I'm a newer engineer in rotor dynamics trying to better understand some things, so perhaps someone can provide some insight...

First, I'm trying to better understand the physics of forced, non-centroidal rotation (for a spinning cylinder in particular). Non-centroidal rotation requires some kind of constraint on a object to hold it to a certain axis, or else the object would naturally tend to rotate about its center of mass. Can this natural tendency (aka conservation of angular momentum) be represented via the principal of least action? Or is there another simple quantitative way of representing this, other than just declaring it so?

And at the same time, could the same principal or technique then be used to calculate the net force or energy required to constrain such a rotating cylinder to some other rotational axis at some given speed? And also, to find the total torque required to spin the off-center cylinder up to a given speed (versus if the cylinder were rotating about its center of mass axis)?

And, would the force required by the constraint be equivalent to the centrifugal force (m*e*omega^2) generated by the now-eccentric portion of the cylinder's mass?

2. Oct 13, 2012

### Staff: Mentor

Re: How to define forced non-centroidal rotation and conservation of angular momentum

You can always write an arbitrary motion as sum of a translation and a rotation around an arbitrary axis (not the direction, but the position). Without external forces, there is a single axis of rotation where the translation has a constant velocity, and this axis goes through the center of mass (can be shown with momentum conservation).

The force should be equal to a point-mass at the center of mass, rotating around your axis.

3. Oct 14, 2012

### diffusegrey

Re: How to define forced non-centroidal rotation and conservation of angular momentum

"You can always write an arbitrary motion as sum of a translation and a rotation around an arbitrary axis (not the direction, but the position)"

Thanks for the reply, but I’m not sure that clarifies it for me. Maybe I’ll ask another way…

How would you represent the natural tendency of a rotating object that is held in forced non-centroidal rotation to tend to return to centroidal rotation on its own if the holding force is removed (or overcome)? Obviously, it seems like it’s the removal of force, not the addition of force that facilitates this – is it? But then there is still a change in path or motion, which generally requires a force. How does one show, or calculate the “negative force” (for lack of a better word) that would return a forced non-centroidal rotating object to natural centroidal rotation when the holding force is removed?

And how would one show or model the net path of motion taken through such a transition? Is it an instantaneous shift in rotational center, in the instantaneous direction of the forced axis toward the CG? And since the CG point was originally in a forced rotational path, would the CG point continue in linear motion while equivalent rotation now proceeds around the CG?

And all this makes me think of another question: Does the physical manifestation of the conservation of angular momentum have a time constant or time lag in its action?

"The force should be equal to a point-mass at the center of mass, rotating around your axis."

Is this the same as the reactive centrifugal force generated by the eccentric cylinder? (that is, m*e*omega^2, where m is the total cylinder mass, e is the distance from the CG to the rotation axis.)

Thanks a lot.

4. Oct 14, 2012

### Staff: Mentor

Re: How to define forced non-centroidal rotation and conservation of angular momentum

Sorry, I don't see the problem. If you remove the fixed axis, I would switch the description of the system to avoid additional terms popping up in the calculation (fictional forces if you keep the original axis of rotation, or a rotating coordinate system if you take the axis through the center of mass in both cases).

It is an instantaneous shift of the coordinate system used to describe the motion. That is just useful to simplify calculations. The coordinate system is not a physical object.
You can do this shift due to the quoted part of my post.

?
Without external forces, angular momentum around every axis is conserved. It is the same for every point in time.

I think so.

5. Oct 14, 2012

### diffusegrey

Re: How to define forced non-centroidal rotation and conservation of angular momentum

"Does the physical manifestation of the conservation of angular momentum have a time constant or time lag in its action?

"?
Without external forces, angular momentum around every axis is conserved. It is the same for every point in time."

What I was wondering is this... For example, say you’re spinning around on your desk chair with your arms held close, and you then extend your arms outward, and your spin speed slows down as angular momentum is conserved. But does this slowing occur instantly and immediately in exact proportion for each infinitesimal outward move of your arms, or is there in real life some slight time lag within the rotational speed change?

6. Oct 14, 2012

### diffusegrey

Re: How to define forced non-centroidal rotation and conservation of angular momentum

"Sorry, I don't see the problem. If you remove the fixed axis, I would switch the description of the system to avoid additional terms popping up in the calculation (fictional forces if you keep the original axis of rotation, or a rotating coordinate system if you take the axis through the center of mass in both cases)."

For the other question I had...
What you said makes sense, in terms of calculating the rotational motion easily within each state - that is, with rotation at the initial axis, and then for rotation at the CG axis.

However, how would you track the motion of an individual particle within the object, in free space, as the objects switches its rotational axis as I described previously? While there is a clearly defined rotation at state 1 (non-centroidal axis) and state 2 (CG axis), in the process of switching between the two, it seems to me that a given constituent particle would take some non-simple path in space to transition between the 2 states. How would you identify this path of motion?

Thanks again.

7. Oct 15, 2012

### Staff: Mentor

Re: How to define forced non-centroidal rotation and conservation of angular momentum

As exactly as you can define "the rotation speed of your body", as your arms do not follow a perfect circular motion and your body is not completely rigid. And without time delay.

With two different descriptions for the different situations. We need coordinate transformations to get the position of the particle in the other frame... so what.
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