## PSN-L Email List Message

**
Subject: Re: Is a Lehman geometry rolling pivot inherently unstable?**

From: Charles Patton charles.r.patton@........

Date: Mon, 07 Jul 2008 00:02:39 -0700

Brett, Chris,
Thanks to Chris for tossing the wrench in the works -- :-) -- I have now
spent the day trying to resurrect trig math I haven’t done seriously for
35 years. I think I have the formula for Chris’s bottom pivot, but I’m
still psyching out the upper pivot, so I haven’t started sims yet. What
I think at this point is that as the gate swings the lower point
trajectory tightens up (radius decreases) with swing while the upper
pivot flattens out (radius increases) which I would assume leads to
unstability – the bob going lower as it swings. But this seems to fly
in the face of Chris’s success in long period Lehman. The answer may lie
in the combination in that the gate is twisting as it swings so the
vertical position of the bob would play an important part of the
stability. There has to be some point that the beam twists about, and
if the bob is mounted above or below this point the twist could
compensate or increase the trajectory error of the bob.
The morning is spoken for, so I’ll try to get back on the problem in the
afternoon.
Now I’ll take a big leap of faith and put forth the formulas I think
describe the lower pivot. Assume a cylinder of radius R with a beam of
length K having a flat face resting on the cylinder. Angle T is the
angle of the contact point of cylinder/face (beam angle). The angle of
a line thru the center C of the cylinder and plumb bob P is equal to
angle T minus angle B. Line G = C to P. All angles are in radians.
Then:
B = T(R/K)
G = (R+K)/cosB
x & y are referenced from C
x = G(sin(T-B))
y = G(cos(T-B))
If there’s interest, I can try to do a standard proof deriving the
above. I didn’t get to the formulas above with a step-by-step written
proof, so it very well could be flawed. Good for discussion though.
Anyway, later.
Chas.
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