Telescope mirrors under tension.
From: Robert Clark (rgregoryclark_at_yahoo.com)
Date: 03/12/05
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Date: 12 Mar 2005 14:58:12 0800
In this article Geoffrey Landis proposes builing a space tower using
pressurized structures:
THE TSIOLKOVSKI TOWER REEXAMINED
Journal of The British Interplanetary Society, Vol 52, pp. 175180,
1999.
http://www.aeiveos.com/~bradbury/Authors/Engineering/LandisGA/TTTR.html
He notes that typically materials have higher failure or ultimate
strength in tension than in compression so this would allow higher
towers to be built by using pressurized towers that translate the
vertical compressional forces into tensional forces that tend to expand
the pressurized structure outwards. See section 4.3 and Fig. 3.
I wanted to use a similar idea for creating large telescope mirrors.
However, the formula for deflection of a mirror due to selfweight
depends on Young's modulus, not ultimate strength and Young's modulus
is about the same in tension and compression.
Still it may be possible to create larger mirrors because the
deflection formula also is dependent on density which will be much less
with a gas filled structure:
Deflection ~ Density*(1Poisson's ratio^2)/Young's Modulus
However, all three of these quantities will likely change for a two
material structure as with a pressurized mirror.
(BTW, perhaps someone can answer this for me, in the Landis article he
just calculates the tensional outward force that would need to be
supported, but surely the outside walls would still need to support
some vertical compression force. Is this just a minor component and can
be neglected?)
Some proposals for inflatable membrane mirrors for use with space
telescopes are given in this conference report:
Ultra Light Space Optics Challenge: Presentations.
http://origins.jpl.nasa.gov/meetings/ulsoc/presentations.html
Another possibility for a mirror under tension is given by this
surprising fact:
Parabolas and Bridges.
"If you hang a flexible chain loosely between two supports, the curve
formed by the chain looks like a parabola, but isn't. It is a catenary,
a more glamorous curve which can be represented algebraically by
hyperbolic functions [y = A (cosh kx  1)]. In this case, the vertical
load on the chain is uniform with respect to arc length. A whirling
skipping rope is another example of a catenary.
"The load on a suspension bridge is (approximately) uniform with
respect to the horizontal distance. In this case, the curve is a
parabola ..."
http://www.du.edu/~jcalvert/math/parabola.htm
Shape of a suspension bridge cable.
http://aemes.mae.ufl.edu/~uhk/SUSPCAB.jpg
Hanging With Galileo.
"Take a flexible chain of uniform linear mass density. Suspend it from
the two ends. What is the curve formed by the chain? Galileo Galilei
said that it was a parabola, and perhaps you made the same guess. This
time Galileo was not correct. The curve is called a catenary. However,
it is easy to see how he could arrive at this answer through casual
observation and incomplete deduction.
...
"We can get back to the chain solution later. First consider this
extension. What about the curve formed by the cables of a suspension
bridge? Is it too a catenary? No, it is a parabola. So, what gives? How
can this be a parabola while the other one is not?"
http://whistleralley.com/hanging/hanging.htm
Then to form a parabolic surface you could have the suspension cables
arranged in concentric circles hanging from the mirror supporting a
weight. In this case gravity would be working to *form* the surface
shape.
A problem is that just with liquid mirrors you might need to keep the
mirror horizontal so it would have to be zenithpointing. However, it
may be that by varying the cable lengths you could maintain the
parabolic shape.
Bob Clark
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