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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