I Am Always Impressed How Smart My Readers Are

I mentioned here the engineering problem I was confronting on rebuilding Big Bertha, and the formulae provided by a reader for calculating deflection under load. I pulled together the appropriate data, and this is what I found.

First of all: I discovered that Young's modulus is expressed in Pascals (the metric unit of pressure). Because the Pascal is defined as one newton of force over a one square meter area, I needed to convert all my numbers from centimeter/gram/second to meter/kilogram/second.

Next, I used the equations to figure out the moment of inertia for a solid rectangle, rectangular tube, and a square tube. (I'm not considering the use of a round tube, simply because it is harder to get a solid connection between two pieces of metal if one of them is trying to roll.)

I plugged in dimensions for several commonly available 6061 aluminum shapes, and the moment of inertia came out like this:

section	moment of inertia	kilograms per meter
rectangular solid (2" x 0.5")	0.000000000009	1.74
rectangular tube (3" x 1" with .125" wall)	0.000000007	1.63
square tube (2" x 2" with .125 wall)	0.000000126	1.63

You can see what a difference a square tube makes relative to a rectangular tube of the same weight, or a rectangular solid that weighs slightly more!

There are two loads to consider for computing deflection: the point load (which assumes a weight that is concentrated at one point), and the load that the weight of the tube itself inflicts. When I plugged in those formulae, using a weight of 40 pounds for the mirror end of the telescope (which is by far the heaver load) and a distance of 40 inches (it will actually be a bit less, depending on the balance point), the square tube gave a deflection of .00069 meters for the point load, and .00056 for the beam load. I'm told that adding these together is probably sufficiently accurate for these purposes, so that comes to .00072 meters, or about .028".

That's not quite sufficient (especially because it will vary depending on whether the telescope is pointing horizontally or vertically), so I could either go to a larger tube, or plan on using two of them. Going to a 3" x 3" x .125" tube more than triples the moment of inertia, and knocks the deflection down to .008". I suspect that it may make more sense to use two or even four of the 2" square tubes instead. I don't know exactly how they would reinforce each other, but I suspect that two tubes would halve the point load per tube.

UPDATE: One advantage of using a single large tube to mount everything--it makes it easy to adjust distances. When you are doing astrophotography, you either need a lot of adjustment range in the focuser, or you have to move the mirror closer to the camera than would be needed for visual use (typically 2" to 2.5" closer). It would be fairly use to drill two sets of holes in the base tube: one set for mounting the diagonal/eyepiece/finder cage at astrophotography distance, and another set for visual distance. It might even be possible to do this on a sliding mechanism with set screws to lock everything into position.