PSN-L Email List Message

Subject: Re: Pivot paper discussion
From: Charles Patton charles.r.patton@........
Date: Tue, 21 Oct 2008 19:59:26 -0700

Randy Pratt asked:
 > Ball on plate:
 > Columns A to C use lolocus and lolocrad in formulas.  Where are these
 > defined?
“lolocus” is cell C38 holding an input value.(highlight –select- a cell 
and if they have a text name associated it will show on the left in the 
toolbar where a cell location is normally shown.)  The name change 
happened because I was trying to clean up the visual aspect of the sheet 
and didn’t go through and fix the formulas, too.
“lolocrad” is cell C34 and doesn’t follow what I just said as it has two 
names defined.  Ah well the hazards of editing.  But double-clicking on 
a formula shows all the cells making up the formula, so you can see 
which cells are named what.

 > Is this analysis for a single pivot or for a Lehman with 2 pivots as 
 > on the main page?
It actually applies to both the upper and lower pivots, even though it 
might not seem so at first glance.  Realize that if we have an arm 
attached to the ball, the arm could extend up and to the left.  I.e., 
start at P and continue through C and call the new end PP.  C to PP 
would be the upper suspension and can have the same locus center point 
and as found for the ball on plate as illustrated (with the same locus 
errors.)  One can also enter negative values for P and get the results 
for a particular beam length in the upper pivot case.

 > I see references to twist in the revision history so is
 > this twist resulting from upper and lower pivot travel in opposite
 > direction?  If so would not the support structure height play a role 
as well
 > as angle of the support wire to the boom?
That revision was early in the discussion with Brett and Chris.  I had 
taken a simplistic approach to the path based on the contact point. 
Latter I agreed with Brett on the approach to finding the locus center 
as is embodied in the current XLS calculations.  That If a locus center 
point exists that makes the locus a circular arc, then the twist doesn’t 
exist.  But even if twist exists, you choose your bob center-of-gravity 
roughly half way in the Z axis between the upper and lower pivot points, 
and its effect  should drop out of the considerations.  Either way, I 
stopped considering it.

 > Why was the locus chosen as a point behind the plate rather than the 
 > point? Reason - In a Lehman, translation of a rolling pivot will 
result in
 > unsymetric forces on the boom.  The compression in the boom and the 
 > in the support wire will move out of the vertical plane.  A horizontal
 > friction force must be present to prevent slippage of the ball and this
 > force is also a moment around the mass.  These forces would have to be
 > analysed from the contact point. Larger angles and larger balls would
 > increase this effect.
Please see my answer to Bob McClure on 10/16, posted to the PSN list. 
The “center point” is chosen to give the “ best fit” to the calculated 
locus.  It is not chosen for any other purpose than to answer 
mathematically whether there is a close fit circular arc to the locus 
that has been calculated.  If there is, then it can be said that the bob 
with regards to the vertical beam travels a circular arc therefore a 
pendulum fit can be realized.  You may be right about secondary effects 
of the offset mass.  I’ll have to think about that further, but I’ll 
have to do that later.  I’ll post this to PSN, and maybe Chris or Brett 
can tackle that aspect.
 > An upper pivot does not travel a semicircle around the circumference in
 > relation to a near horizontal swing. The semicircle is tilted by the 
 > wire angle in relation to the near vertical swing axis.  Do you agree?
No, because a ball always has circular cross section and tilt only 
matters to the extent that it changes the length of CP.  So the 
hypothetical center of the arc may not be where the the physical contact 
point is, but as you adjust the period of the swing, you will actually 
be adjusting these centers in space to be almost in a vertical line, 
with the upper just a bit forward to give the bob a pendulum arc that is 
lowest in the center of the swing.
This was the whole point of doing all this analysis – trying to 
understand just what the weight/bob was experiencing and what 
arrangements of pivots had the best circular fit.  Additionally what 
shows up is the ball on plate inferior to plate on ball with regard to 
the side slipping you touch on.

 > Construction could change this I guess with rigid upper boom structure.
 > Randy

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