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Subject: Re: Is a Lehman geometry rolling pivot inherently unstable?
From: Charles Patton charles.r.patton@........
Date: Sat, 05 Jul 2008 16:07:54 -0700

ChrisAtUpw@....... wrote:
> In a message dated 2008/07/05, charles.r.patton@........ writes:
>> Discussion:
>> Assume:
>> 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.
> Hi Charles,
>        Can you define the systems that you are considering more clearly, 
> please?
>        I get a confused word picture in my mind.
>        There are two types of crossed flexure suspension. In one type 
> two flat strips / straight wires are clamped at right angles - the 
> classic crossed foil suspension. As they flex, the suspension point 
> moves to one side along an ellipse about the fixed member.
>        In the other sort, you have figure of 8 foil / wire loops rolling 
> on cylinders and the flexure point moves in a circle around the fixed pivot.
>        Remember that the fixed clamp / vertical support rod is nearer to 
> the mass and the moving clamp / rod on the arm is on the far side of the 
> support rod.
>        However, in both systems as the mass moves to one side, the plane 
> containing the centre of mass, the top suspension and the bottom flexure 
> makes an increasing angle to the vertical and hence, it is stable. 
>        We need to consider deflections of less then 5 degrees.
>        I can imagine amateur constructors 'getting into trouble' with 
> inadequately designed levelling systems. I provide spherical ends to the 
> levelling screws, either by fitting a SS dome nut or by gluing a SS 
> bearing into the end of the bolt (preferable). There is inevitably some 
> side to side 'slop' in screw threads. I provide a wavy spring washer and 
> a locknut on the top side of the frame to keep the levelling screws in 
> fairly high tension at all times. The levelling screw and the fixed 
> thread should both be made of the same metal to mimise expansion 
> effects. You definitely do need a smooth hard flat surface on the ground 
> for the spherical ends to rest on. I use 2" - 3" square x 1/8" thick SS 
> plates.
>        Regards,
>        Chris Chapman

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.

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