Re: chaotic pendulum

On Thu, 28 Feb 2008 08:57:34 -0800 (PST), connie wrote:

I have a pendulum which is a stiff, V-shaped rod of uniform cross-
section, as below:



----------------------
/
/
/
/
\
\
\
\

The rod is free to swivel about the suspension point. The pendulum
exhibits very chaotic behavior, i.e., seemingly non-uniform period,
and somtimes rotating clockwise then counterclockwise. My mechanics
is rusty. Is this amenable to analysis? Any refs? Thanks.
CFE

The shape of the rod shouldn't matter that much, except for the effect it
has on the rod's moment of inertia about the pivot. What the bend will do
is give the rod a slightly different moment of inertia in one direction
than in the other.

Is the rod free to rotate about a vertical axis?

If we neglect rotations about a vertical axis, then for small oscillations,
the slight difference in moment of inertia would cause slightly different
oscillation frequencies in the two directions. A combination of these two
modes of oscillation could explain some of the behavior you describe.

But from experimenting a bit on my own -- I've got a pair of scissors
hanging from a short (compared to the scissors) piece of dental floss -- I
don't think we can just neglect spinning about a vertical axis. In fact, I
suspect that even if it doesn't start out spinning in that direction,
oscillations in the other two directions can start it spinning.

In any case, the problem is amenable to analysis, or at the very least we
should be able to come up with some differential equations to describe its
behavior. You shouldn't need Lagrangian mechanics, although you could use
it if you wanted to, just a calculation of the torque and the angular
acceleration it causes. I think the worst part may come from the
mathematics of combining 3-D rotations. Lagrangian mechanics might help
with this, but I'll have to work through the problem to see.

--
Jim E. Black (domain in headers)
How to filter out stupid arguments in 40tude Dialog:
!markread,ignore From "Name" +"<email address>"
[X] Watch/Ignore works on subthreads
.

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