Randall,
I understand the folded pendulums you mention, but I want to touch on 
several related subjects.  Back of the napkin pendulum length for 10 
secs is about 1000 inches.  A one inch swing would be a ½ milli-inch 
rise.  This gives me a bit of feel/insight on possible error mechanisms. 
  It strikes me that one general problem with flexures is that they are 
not a pivot in the sense of having a known axis like a bearing does.  I 
haven’t totally worked out the ramifications, but I’m sure this is the 
reason many amateurs have problems taking Lehman style instruments to 
long periods.  Even if they’re not using flexures, pivot points are a 
round point that also may or may not have a constant point of rotation, 
depending whether it is rotating in a pocket or rolling on the surface 
of its pivot support, so the length may well be getting shorter as it 
rotates and a shorter length on the beam equates to the weight dropping, 
not rising as is necessary for stability and so the distance to 
un-stability is around ½ a milli-inch.

So the way I perceive it, a big problem is having a system where the 
axis of rotation remains constant, quite accurately.  Unfortunately the 
only solutions I keep coming back to are bearing style things.  So then 
the question becomes, “Can a bearing be made that has low loss?”  But a 
concurrent question is do I really need a very low amount of loss?  I 
know recent discussions have experimented with crossed pivots of 
extremely low loss.  Why?  The immediate next step will be to add a 
damper to get to something close to critical damping.   My understanding 
is that the only reason to have low loss is to be able to use lots of 
feedback to lengthen the period.  But if the period can be achieved 
directly, and it includes some damping, so what?  In my mind, the 
important item is hysteresis/stiction.   As bearings and bearing 
surfaces can easily be ground to a ten-thousandth or even better, 10 or 
20 second period structures should be in reach.

Back to possible structures.  The structure I originally presented is 
probably not possible geometrically.  But one that is obviously possible 
is as follows.  Imagine a hollow cylinder (like a pipe) that has been 
centerless ground to be round.  Now take a high density rod like lead or 
tungsten and center it down the axis of the cylinder with fine 
adjustment screws so you can offset the center of gravity by a fraction 
of a thousandth.  (The hollow cylinder construction is to reduce the 
rotational moment of inertia.)  Now place this cylinder on a surface 
plate (again a commonly available object that can be obtained flat to 
fractions of a ten-thousandth.) that is level better than a 
ten-thousandth per inch.  Use very fine steel (a few thousandths) wire 
as Rollamite bands.  The cylinder should roll to center the mass down. 
So lets assume a three inch dia. pipe.  That’s roughly 10 inches 
circumference, or 2.5 inches to 90 degrees, and raising the mass by the 
amount of the off-center that could be easily set to 1 mill.  Easily 
greater than 10 seconds rotation period?  Once you have that structure 
in mind, chop off ¾ of the cylinder not in contact with the surface 
plate.  As long as the center of mass is below the center of rotation 
this has become an upside down pendulum that is stable on the surface 
place and the rotational inertia has been reduced to a minimum.  The 
position sensor is placed to monitor the mass at the ‘top’ of this pendulum.
Just some more idle musings.
Regards,
Charles R. Patton


Randall Peters wrote:
> Charles,
>     In effect, what you have described, is to take advantage of the same property that is used by the folded pendulum, which
> comprises both a `regular' pendulum and also an 'inverted pendulum.  Separated from each other and connected by a rigid
> horizontal boom, their relative influence ('restoring' from the one, and 'destoring' from the other) is determined by how close
> the inertial mass is placed to one or the other.
>     Because the folded pendulum can be made to have a very long period, upper valuve being limited by mesoanelastic complexity,
> it appears clear then, that the feedback drive of the primary pendulum by an inverted secondary one is capable (for ideal
> meaterials) of very long period indeed, and therefore very great sensitivity.  Moreover, since the adverse effects of material
> problems can be essentially eliminated by means of the feedback, I see this as a really attractive idea to try and demonstrate!
> Are there any takers?  (meaning folks like Brett who know how to make control systems work right).
>     Randall
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