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Subject: Is a Lehman geometry rolling pivot inherently unstable?
From: Charles R Patton charles.r.patton@........
Date: Sat, 05 Jul 2008 09:23:51 -0700

Hi Brett,

I tried to send this yesterday, but think I messed up the addressing, 
but it gave me a chance to correct a couple of errors.

Question: Is a Lehman geometry rolling pivot inherently unstable?

1) That the Lehman is constructed in a typical “garden gate” fashion 
with a horizontal main beam with rolling pivot and a suspension wire to 
the pivot bearing.
2) At the point of the rolling pivot, the wire does not bend. I.e., the 
wire/pivot may be considered rigid in that area. This constraint will 
hold true if the wire rigidity is greater than the torque required to 
roll the pivot. Something I believe is a reasonable constraint/assumption.

So, if the Lehman is adjusted to a long period, a very small change in 
the geometry will lead to big changes in period/stability. In 
particular, with the suspension wire version it seems to me that as the 
beam moves from a centered position, the effective length of the 
suspension wire increases due to the movement of the pivot contact point 
around the diameter of the pivot rod along with the contact point moving 
sideways along the line of contact. Making it simple to do the mind 
experiment imagine going through 90 degrees. The wire lengthens by ½ the 
rod diameter. The contact point moves sideways by ¼ x pi x dia or approx 
0.78 dia. So as the pivot rotates, the wire length starts to lengthen by 
0.5 dia, which is the condition for stability. Now the lower beam pivot 
does the same thing but acts in the direction to shorten it by 0.5 dia. 
The sideways motions do not cancel but lead to increased rotation of the 
gate. By definition, the wire is at an angle to the beam, so that means 
that in all real constructions, the lower beam shortening effect is 
larger than the suspension wire effect (cos(wire/beam angle) x 0.5), but 
both effects lead to a lowering of the bob as it moves sideways. Energy 
constraints say that the bob wants to go to the lowest potential energy 
(flopping). This does not consider the additional effects that the gate 
is twisting due to the rolling pivots. So is this a possible explanation 
of the difficulty many people describe in trying to adjust a Lehman? I 
pose this to you since you’ve always been good with the mechanical 
simulations to see if I’m way off in this conjecture.

An added question is this:
What is the effect of the moving effective pivot point in a flexture 
pivot? Since a standard rectangular shim (or rod/wire) flexure point 
cannot not have a fixed point due to the stress/strain relationships as 
the flexure bends and the weight shifts to a side load the point would 
move back (?) towards the upright on the upper wire/beam and probably 
towards the support (again shortening the beam and lowering the bob) on 
a lower tension type flexure.

Am I all wet --a welcome condition on a hot, hot 4th of July? (I started 
this email on the 4^th and did some corrections on the 5th) Comments 

Charles R. Patton

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