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RE: [ATM] .707*Radius (long)
Very good description of what you were explaining. I
understood it completely (which is scary) up to the
numerical part where my public school education starts
complaining. I'm sure your numbers are correct so I'm
concluding that the support points can be anywhere
along the 3 radial lines between r.4 and r.6 ... is
that correct?
Ken Hunter
--- Donald Good <donald.good@comcast.net> wrote:
> 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
>
=== message truncated ===
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