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RE: [ATM] .707*Radius (long)
It has been suggested in other posts that the difference between the
classical
.707*Radius position of the 3 point supports and the PLOP calculation might
be
due to variation in the thickness of the mirror (sag) or differences in the
mirror thickness ratio - thick 6:1 (classical std) or thin 10:1 (modern
example).
The following table represents 24 runs of PLOP (v2.1.6) in 4 sets of mirror
specs
(columns) and 6 sets of optimization settings (row groups). Each set of 3
values
vertically represents the RMS, P-V, and fractional radius results of one
PLOP run.
A 200 mm diameter mirror with a 39mm obstruction is used in all cases while
using
the 4 combinations of 2 thicknesses (6:1 and 10:1) with 2 radial thickness
variations (F20 and F4).
Note that when the P-V error was used for optimization, in some cases, the
results
are not the same if the radius optimizes upward or downward. This may
indicate
multiple P-V error minimums that I cannot explain. PLOP recommends that P-V
error
not be used for the optimization. Therefore, results for "Use P-V error -
Yes"
will not be analyzed. Using RMS error (P-V error not checked) did reach the
same
results, up or down. So I will concentrate on those results (1st and 4th
row
groups).
Comparing Col 1 with Col 2 and Col 3 with Col 4, we see that thick glass is
stiffer (lower error values) than thin glass, as expected. Also comparing
Col 3 with Col 1 and Col 4 with Col 2, we see that the mirror with the
smaller
radial thickness variation is stiffer (lower error values) than the mirror
with
the larger radial thickness variation, also as expected. The radius is
consistent across these changes (near .38 for Refocus On and .64 for Refocus
Off). Therefore the .707 discrepancy is not due to differences in mirror
specs.
But what is Refocus On and Refocus Off requesting?
Refocus On is requesting that PLOP calculate the support points for the
lowest
RMS error and state how much it differs from the requested focal length.
Refocus Off is requesting that PLOP calculate the support points for the
mirror
shape of the requested focal length and state the RMS error there.
The errors differ between Refocus On and Refocus Off by a factor of only
about
2.3 which does not appear to be that significant. This supports the claim
previously posted that anywhere between .4 and .6 and maybe more is likely
acceptable.
But neither of these values is the .707 value. Where does that come from?
See
after the table.
200 mm dia, 39 mm obs Results Col 1 Col 2 Col 3
Col 4
Thickness variation Larger (F4,fast)
Smaller (F20,slow)
Thickness ratio 6:1 10:1 6:1
10:1
33mm 20mm
33mm 20mm
Refocus On and RMSerr 1.39E-6 3.42E-6 1.26E-6 2.78E-6
Use P-V error - No P-Verr 6.75E-6 1.53E-5 6.04E-6 1.24E-5
Radius .396685 .409583 .387163 .387565
Refocus On and RMSerr 1.66E-6 3.52E-6 1.57E-6 2.81E-6
Use P-V error - Yes P-Verr 6.76E-6 1.50E-5 6.25E-6 1.21E-5
High start Radius - .7 Radius .483384 .367908 .487043 .362275
Refocus On and RMSerr 1.40E-6 3.52E-6 1.26E-6 2.81E-6
Use P-V error - Yes P-Verr 6.28E-6 1.50E-5 5.74E-6 1.21E-5
Low start Radius - .3 Radius .377738 .367908 .374485 .362275
Refocus Off and RMSerr 3.20E-6 8.11E-6 2.96E-6 7.05E-6
Use P-V error - No P-Verr 1.43E-5 3.50E-5 1.32E-5 3.04E-5
Radius .63992 .653429 .632607 .641338
Refocus Off and RMSerr 3.20E-6 8.23E-6 2.96E-6 7.09E-6
Use P-V error - Yes P-Verr 1.42E-5 3.48E-5 1.32E-5 3.02E-5
High start Radius - .7 Radius .634722 .635962 .63253 .629919
Refocus Off and RMSerr 3.20E-6 8.23E-6 3.90E-6 7.09E-6
Use P-V error - Yes P-Verr 1.42E-5 3.48E-5 1.57E-5 3.02E-5
Low start Radius - .3 Radius .634722 .635962 .531862 .629919
John Hindle proposes .707 (=1/sqrt(2)) in Ingalls' ATM book in the chapter
Mechanical Flotation of Mirrors (Chap B.8 in the revised ATM book 2) as the
radius at which the area (and the weight, neglecting sag) inside the radius
is
equal to that outside the radius. He calls this the Radius of Equilibrium,
but
he does not support it with any derivation in that article. He states that
each
point must support equal areas (therefore weights), and I assume that we all
agree
on this for this type of cell.
But it should be noted that any 3 points at the same radius separated by 120
deg
will satisfy this condition, whether the radius is 1.0, 0.5, or any other
between
0 and 1, due to angular symmetry. The 1/2 inside, 1/2 outside sounds
reasonable,
but it is only an assumption. An equal argument can be made for the center
of
gravity of the 120 deg sector.
Area of a circle of radius 1 is Pi, for 120 deg sector, A(s)=Pi/3.
Draw the bisector radius (60 deg) of the sector from the apex (circle
center) to
the circumference. This balances the sector in the angular direction, so
the CG
is somewhere on this line. Integrate the moment along that radial to find
the CG:
r=2/A(s) [sqrt(3)*Integral(r^2 dr) + Integral(r*sqrt(1-r^2) dr)]
r=0->.5 r=.5->1
which evaluates to r=sqrt(3)/Pi or approx .551
This is the balance point of the sector, independent of the other 2 sectors.
But the sectors are each rigidly connected along their 2 radial faces. PLOP
uses
FEA to resolve the forces and bending moments with the result that the best
support point moves inward to about .4, likely due to the strong radial
banding
effect that exists in disks acting like tight steel bands wrapped around the
edge
that resist gravity's tendency to bend the edge down and outward.
These effects are small, but simple CG analysis shows that .551 is better
than .707
and it requires structural analysis to give the best location, which is why
we use
PLOP.
Clear skies,
got to go to my astronomy club meeting, now.
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