[ATM] RE: Flex / Potato Chip / Astigmatism question

[tkrajci at san.osd.mil (Tom Krajci)]

>From: Ian McKernan <mckernan@murdoch.edu.au>

>Hi all!  Am working on an 8" flex mirror at the moment, but was playing

>around with FLEX.exe and trying some different numbers.  Dreaming of a
20 
>inch flexed in the far future.

>Anyhow, according to FLEX.exe, a 20 inch diam f6.0 sphere, with a flat 
>back, 0.95" thick (since i'd be lucky to even find 1" thick glass here
in 
>W.A.).... 

>Problem is that one of the outputs is "Gravity Effect" which = 274% for

>this mirror.  According to the program documentation:

>" MAX GRAVITY EFFECT.  THIS IS THE EFFECT OF CHANGING
>  ALTITUDE.  100% GRAVITY EFFECT WOULD JUST REACH THE
>  THRESHOLD OF VISIBLE ASTIGMATISM AT ZERO DEGREES ALTITUDE.
>  GRAVITY EFFECT IS FACTORED BY THE COSINE OF ALTITUDE.
>  THUS THERE IS ZERO ASTIGMATIC GRAVITY EFFECT AT ZENITH."

>I assume that this is a "potato chip" effect where the mirror tries to
fold 
>over under its own weight.

There is another problem.  The mirror will sag under it's own weight
when facing near vertical...especially with a support scheme for 'flex
mirrors' with an edge support and a central 'puller'.  (In other words
the rest of the mirror is unsupported...only the very center and very
edge are supported with a typical flex/pull scheme.)  (Yes, all mirror's
sag under their own weight under gravity, but with a flex/pull
scheme...you have large distances and areas of unsupported
glass...compared to a whiffletree mirror cell.)

PLOP can be used to model this gravity deformation for a mirror facing
straight up...if you make this assumption:  for small deformations, you
can add the resultant flexures of various forces on the mirror and get
results that match, pretty well, with the real world.  In other words,
you can calculate the changing of shape (due to pulling on the mirror)
with FLEX.exe, and calculate the gravity induced sag with PLOP...and add
the two together...and still have a reasonably accurate model of mirror
behavior.

Let's assume the mirror is in zero gravity...and we flex/pull it from a
sphere to a paraboloid.  So far so good...FLEX.exe says it will work
just fine.  Now let's add gravity to the problem with the help of PLOP.

Open up PLOP and cut/paste the section below into the "Edit as text"
section of the PLOP Cell Designer window.  Then run PLOP.....

;six inch mirror, 1 inch thick, central and edge supports
diameter 150
thickness 24
density 2.23e-06
modulus 6400
poisson 0.2
focal-length 900
n-mesh-rings 15
rel-support-radii 0 0.99
num-support 1 36
support-angle 0 0
basis-ring-size 3
basis-ring-min 0
obstruction-diam 30

My results are that for a pyrex, six inch, .95 inch thick mirror...you
get a visible RMS error of 2.38559e-07 millimeters (refocus ON).  That's
so small that I don't think anyone will notice it...and it's 10 times
smaller than the tolerance I used in the past for my PLOP designed
mirror cells...I call that 'good enough.'  (Plate glass is a bit
softer...so your distortion due to gravity is about 10% larger...not a
very big difference between plate or pyrex.)

In other words gravity flexure is not a factor for this size/thickness
mirror.  Flex/pulling produces the overwhelming majority of the change
in the mirror's shape...which is what we want.

Now let's try a 20 inch, .95 inch thick mirror.  Load this in PLOP:

;20 inch mirror, 1 inch thick, central and edge supports
diameter 500
thickness 24
density 2.23e-06
modulus 6400
poisson 0.2
focal-length 3000
n-mesh-rings 19
rel-support-radii 0 0.99
num-support 1 36
support-angle 0 0
basis-ring-size 3
basis-ring-min 0
obstruction-diam 50

My results are that for a pyrex 20 inch, .95 inch thick mirror...you get
a visible RMS error of 4.84307e-05 millimeters (refocus ON).  That's a
lot of error...and it's 20 times *larger* than the tolerance I used in
the past for my PLOP designed mirror cells...I call that 'unacceptably
bad.' 

(From some more quick fiddling around with PLOP...with .95 inch thick
pyrex...somewhere around 10 inches in diameter...a larger diameter
starts giving you more 'gravity distortion' than you would want in a
flex/pull cell that has a central puller and an edge ring.)

But this begs the question for those that want to try flexing a big,
thin spherical mirror:  Can a flex/pull mirror cell be made that uses an
annular/edge support, but with a whiffletree support that provides many
support points distributed across the mirror's back?  Has anyone done
this already?  (This idea interests me a bit...it may be a way to make
reasonably good, large, thin mirrors...which may simplify
testing/fabricating the optic...but makes the cell design and
implementation a bit more difficult.  Is this idea worth more study?)

One key question:  was my assumption above "for small deformations, you
can add the shifts/resultants of various forces on the mirror" a valid
one?

Am I making any errors in analysis here?  Did I mis-use PLOP?  Do you
get the same results as I describe above?

I hope this helps.

Tom Krajci
Tashkent, Uzbekistan


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