Re: Instantaneous axis of rotation?

On 5 May 2006 17:03:07 -0700, "Edward Green"
<spamspamspam3@xxxxxxxxxxx> wrote:

John C. Polasek wrote:

You have to know about the inertia tensor a 3x3 matrix that when
diagonalized has 3 different moments of inertia J11 J22 J33. The
natural spin, probably minimum energy is about any of these three
axes. If you cand identify the inertia tensor, you can't say much
about it. I will say the mathematics of rotation and momentum
(involving Euler angles as best choice) is just about insoluble. See
Goldstein Mechanics.

Just a quick note of disagreement (which are always so much easier to
write than letters of appreciation and statements of awe): the
instantaneous axis of rotation is a kinematical concept, and doesn't
care about the moment of inertia -- or the center of mass for that
matter, though that seems to be a popular association.

(Adding disputatious teasers is another easy way to overcome writers
block ;-).
Au contraire my friend. In a free body, spinning, it will spin about
its center of mass that is characterized by the inertia tensor. Every
lump has 3-axis inertia tensor and if it is spinning about the
centroid on a principle axis it will not wobble. If it is spinning off
the proper center then we use the translation theorem J' = J + Mr^2
where r is the offset. Then in time it will adjust to least energy,
but spinning up as J' approaches J. That's if it's near the principle
axis.
If it starts spinning off one of the 3 axes, it will wobble as could
be seen by the momentum resulting from
Mom = A'JA*W
where J is similarity transformed by A' and A rotation matrices so
that the single arbitrary rotation rate W produces Mom with components
on all three axes and not colinear with W.
The kinematical axis on a free body will find itself and its quite
dynamic, not kinematic.

John Polasek
http://www.dualspace.net
.

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