## PSN-L Email List Message

Subject: Re: Is a Lehman geometry rolling pivot inherently unstable?
From: Charles Patton charles.r.patton@........
Date: Sun, 06 Jul 2008 10:48:57 -0700

```Brett,
Thanks for the trigger word, “cycloid.”  I had been thinking it too, but
“Technology Mathematics Handbook” by Jan J. Tuma.  Just the thing for a
discussion like this.  Our problem can be defined as class of cycloids
called “prolate cycloids.”

Give a circle with center C of radius R rolling on a contact line, and a
Point P of K*R length (C to P), and A equals angle of CP to the normal
to the contact line, then the graph of P is:
X = R(A – KsinA)   Y = R(1 – KcosA)
This is a cycloid with loops on the end where the cusps would be if a
pure cycloid were graphed. (For a pure cycloid just set K=1)
A point moving around a point (i.e. what we really want) is:
X = RsinA         Y = RcosA

What immediately comes out of this is that just simple observation of
the prolate cycloid curve is that the upper pivot and lower pivot are
tracing different curves because they are effectively 180 degrees out of
phase in the equation so right away balance has to be changing.  Now
which way?

I think I’ll post this and continue with sims in Excel a bit later.  But
just some food for thought.  Also an important consideration is that
these will yield curves in the plane of the rotation, but they have to
be combined in a perpendicular plane to fully establish the final effect
on the bob trajectory.  I.e., the bob is a vertex on a triangle (and one
not necessarily a right triangle) formed by the upright, beam and
suspension wire.

In fact, the thought that the bob support does not have to be
constrained to a right triangle may provide the way out of the possible
geometry problem.  More food for thought.
Regards,
Chas.

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