Brett,
Thanks for the trigger word, “cycloid.”  I had been thinking it too, but 
somehow your writing it got me thinking about a book I had stashed away, 
“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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