Chris

At 09:08 PM 2/17/2008 -0800, you wrote:
>Randall,
>I understand the folded pendulums you mention, but I want to touch on=20
>several related subjects.  Back of the napkin pendulum length for 10 secs=
=20
>is about 1000 inches.  A one inch swing would be a =BD milli-inch=20
>rise.  This gives me a bit of feel/insight on possible error=20
>mechanisms.  It strikes me that one general problem with flexures is that=
=20
>they are not a pivot in the sense of having a known axis like a bearing=20
>does.  I haven=92t totally worked out the ramifications, but I=92m sure=
 this=20
>is the reason many amateurs have problems taking Lehman style instruments=
=20
>to long periods.  Even if they=92re not using flexures, pivot points are a=
=20
>round point that also may or may not have a constant point of rotation,=20
>depending whether it is rotating in a pocket or rolling on the surface of=
=20
>its pivot support, so the length may well be getting shorter as it rotates=
=20
>and a shorter length on the beam equates to the weight dropping, not=20
>rising as is necessary for stability and so the distance to un-stability=20
>is around =BD a milli-inch.

Both those issues were of great interest to pendulum clock makers.  The=20
latter was studed by no less of an authority than Pierre-Simon LaPlace who=
=20
came to two conclusions.  First, a (very) small radius would be better than=
=20
a knife-edge.  Second, it might even be possible to consider a roller.  He=
=20
studied the geometry and concluded that the deviation from pendulum arc=20
circularity was a small fraction of the edge radius.  That and very=20
thorough analyses of flexure suspensions, including effective pivot point=20
and nonlinear losses are covered in detail in the most excellent book  by=20
A. L. Rawlings "The Science of Clocks & Watches  3rd edition, 1993"  ISBN 0=
=20
950 9621 3 9  which is a revised and annotated version of the 1948=20
edition.  See:=20
http://www.ubr.com/clocks/clocks-and-time-horological-books/clocks-and-time-=
new-books-and-reviews/the-science-of-clocks-amp-watches.aspx=20
It may not be in print but I have seen them, used, priced from $35 to $67=20
through Amazon and Barnes & Noble.  Anyone who is serious about suspension=
=20
design should have this book.

>So the way I perceive it, a big problem is having a system where the axis=
=20
>of rotation remains constant, quite accurately.  Unfortunately the only=20
>solutions I keep coming back to are bearing style things.  So then the=20
>question becomes, =93Can a bearing be made that has low loss?=94  But a=20
>concurrent question is do I really need a very low amount of loss?  I know=
=20
>recent discussions have experimented with crossed pivots of extremely low=
=20
>loss.  Why?  The immediate next step will be to add a damper to get to=20
>something close to critical damping.   My understanding is that the only=20
>reason to have low loss is to be able to use lots of feedback to lengthen=
=20
>the period.  But if the period can be achieved directly, and it includes=20
>some damping, so what?  In my mind, the important item is=20
>hysteresis/stiction.   As bearings and bearing surfaces can easily be=20
>ground to a ten-thousandth or even better, 10 or 20 second period=20
>structures should be in reach.

For displacement-to-force feedback and possibly for other configurations, I=
=20
believe you are exactly right.  The main reason for having low pivot loss=20
is to make it 'easy' for the feedback to do its job, resulting in higher=20
loop gain.  In general the pivot losses in such an instrument should have=20
very little effect on the instrument performance.  Consider that the STS-1=
=20
used bearings which I believe had a relatively poor hysteresis spec., yet=20
its performance was considered to be pretty good.

>Back to possible structures.  The structure I originally presented is=20
>probably not possible geometrically.  But one that is obviously possible=20
>is as follows.  Imagine a hollow cylinder (like a pipe) that has been=20
>centerless ground to be round.  Now take a high density rod like lead or=20
>tungsten and center it down the axis of the cylinder with fine adjustment=
=20
>screws so you can offset the center of gravity by a fraction of a=20
>thousandth.  (The hollow cylinder construction is to reduce the rotational=
=20
>moment of inertia.)  Now place this cylinder on a surface plate (again a=20
>commonly available object that can be obtained flat to fractions of a=20
>ten-thousandth.) that is level better than a ten-thousandth per inch.  Use=
=20
>very fine steel (a few thousandths) wire as Rollamite bands.  The cylinder=
=20
>should roll to center the mass down. So lets assume a three inch dia.=20
>pipe.  That=92s roughly 10 inches circumference, or 2.5 inches to 90=20
>degrees, and raising the mass by the amount of the off-center that could=20
>be easily set to 1 mill.  Easily greater than 10 seconds rotation=20
>period?  Once you have that structure in mind, chop off =BE of the cylinder=
=20
>not in contact with the surface plate.  As long as the center of mass is=20
>below the center of rotation this has become an upside down pendulum that=
=20
>is stable on the surface place and the rotational inertia has been reduced=
=20
>to a minimum.  The position sensor is placed to monitor the mass at the=20
>=91top=92 of this pendulum.
>Just some more idle musings.


Regards,
Brett


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