Brett,
It would seem me, too, that the figure-8 pivot is very similar to a ball 
on a plate except that the figure-8 pivot would have about twice the 
effect due to the point of contact moving along a second curved surface 
  adding to the effective pivot point movement.  One quick observation 
-- you say, "...traced out a circle, accurate to +/- 0.00043" or 43 
microinches per degree."
But 0.00043" is 430 microinches, not 43 microinches and in my brief 
conjecture, 100 microinches is enough to lead to failure of the swing 
trajectory in a 20 second period Lehman.
Regards,
Chas.

Brett Nordgren wrote:
> Charles,
> 
> Just a quick comment relating to my studies of the geometry of the 
> 'figure-8' pivot, which may also apply to the rolling geometry (or 
> not).  What I found was that, although the axis of rotation moved 
> somewhat as the hinge rotated, a point on the beam actually described 
> quite an accurate circular arc.  In my extreme example, using 1" dia 
> rollers and a 5" boom, the center of rotation moved about 0.009" per 
> degree of rotation, however, over +/- 5 degrees, the end of the beam 
> traced out a circle, accurate to +/- 0.00043" or 43 microinches per degree.
> 
> After looking at it for awhile it became apparent that, as the center of 
> rotation changed, the instantantaneous radius of curvature also changed 
> to largely compensate, resulting in the nearly circular locus.  When I 
> get a free moment, I'll see if a similar analysis gives similar results 
> for your rolling pivot.
> 
> Regards,
> Brett
> 
> At 04:07 PM 7/5/2008 -0700, you wrote:
> 
>> Chris,
>> Admittedly my description is brief.  I left unsaid that everything is 
>> perfectly rigid and properly set up.  The pivot types I'm really 
>> interested in are ones used on a a Lehman geometry where rolling pivot 
>> types are used, such as a carbide ball in the end of the beam rolling 
>> on a hard plate and the same for the suspension wire/beam to the upper 
>> pivot point on the upright.  Not the crossed flexure (Bendix bearings, 
>> or single flexure -- although I asked the question does this apply to 
>> them, too?)  What I'm trying to think about is not spurious effects 
>> such as compression (Rockwell hardness), and lack of sufficient 
>> rigidity in the structure, but rather the effect that a rolling 
>> geometry inherently has.  If one thinks about what happens when a 
>> properly adjusted Lehman gate swings, the plumb bob is taking a 
>> flatter and flatter trajectory as the period is increased.  As I 
>> recall, a 10 second period pendulum will lift about 1/2 a thousandth 
>> inch per 1 inch swing.  So if the geometry causes the bob to drop 1/2 
>> a thousandth per inch of swing then it cancels what other wise would 
>> be a stable adjustment.  The problem is that a 20 second pendulum 
>> would be more like 0.0001"/inch.  Even a very small ball-point pen 
>> ball is perhaps 0.04" in dia --so 1/2 diameter is 200 times 0.0001".  
>> So a very small swing starts to have an accelerating change of length 
>> leading to a total flatting of the bob trajectory and then "flopping" 
>> of the pendulum bob.  This swing may come from slight floor tilt and 
>> what might otherwise be stable is not because the geometry leads to 
>> unstability.  The only saving grace I see is that the twist that also 
>> takes place is in the direction of increasing stability (lifting the 
>> bob) but in my mind it is secondary in effect to the primary drop due 
>> to the effective pivot point change leading to an apparent pivot 
>> point/beam/bob length change.
>>
>> Now to discuss the flat flexure problem. When rolling points are used, 
>> the lower point is in compression on a plate and the upper is in 
>> compression on the opposite side such that the both lead to dropping 
>> the bob as it swings. Now if flat foil flexures are used, the upper 
>> flexure is on the bob side and initially bends at some point, but as 
>> the bob moves sideways, doesn't the flexure point move towards the 
>> upright?  If so the upper supension is effectively getting longer, 
>> lowering the bob during a swing, again unstable.  The lower beam 
>> flexure is in tension on the back (further away from the bob) side of 
>> the upright.  This time it's not clear to me which way the bend point 
>> moves, if at all, however my suspicion is that the bend point moves 
>> toward the upright which also leads to an effective shortening of the 
>> bottom beam, again the unstable lowering of the bob.
>>
>> Much of this came about as I conjectured why so many anecdotal stories 
>> about the difficulty of adjusting Lehmans in the long period realm.  
>> So the thought experiment described above.
>> Regards,
>> Chas.
> 
> 
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