# 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 RE-EXAMINED

Journal of The British Interplanetary Society, Vol 52, pp. 175-180,

1999.

http://www.aeiveos.com/~bradbury/Authors/Engineering/Landis-GA/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 self-weight

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*(1-Poisson'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 zenith-pointing. However, it

may be that by varying the cable lengths you could maintain the

parabolic shape.

Bob Clark

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