As I mentioned a couple of days ago, it appears that the solution to the Big Bertha 2.0 problem is replace the 1" square aluminum tube with an channel. This will support the tube on both sides, instead of leaving it only supported at the point where the screws go through both parts.
I'm having a little trouble finding the formula for computing the deflection on a channel. I was thinking of using a 4" wide aluminum channel. Obviously, the wider the channel is, the more support the sides of the channel create on the tube, and the less strain it puts on the screws that hold the tube and channel together. Of course, the wider the channel is, the heavier it is, too.
I have an intuitive sense that a channel is less stiff than square aluminum tube of the same dimensions because the channel is missing the top of the square. I also intuit that stiffness declines as the verticals of the channel get shorter. The ideal channel, from a longitudinal stiffness perspective, has the vertical as tall as the width of the channel. Obviously, the only way that I could use a channel like that for this application would be if the channel was 20.4" wide, and the verticals were 10.2" tall. Of course, that would be a very heavy channel.
The thicker the channel, the more stiff it is longitudinally--but the heavier it is, too. My guess is that to get a channel of approximately the same stiffness as the 1" square tube (which was 1/8" wall) I will need a 4" wide channel that is 1/4" thick. I haven't done the trigonometry yet to figure out how tall the verticals will have to be to get the telescope tube assembly to fit to the bottom of the channel, but a 4" wide channel is going to have relatively short verticals. Perhaps I would be better off with a wider channel of a thinner material? The wider channel means taller verticals, and thus more stiffness with a thinner material. There's no magic solution on this--whatever I build is going to be heavier than the 1" square aluminum tube, I know that.
Anyway, the net result is that I need a formula for computing how much deformation is created by applying force N on material with Young's modulus E, and length L. I believe that I could even use the current formulas that I have if I knew how to compute the moment of inertia for a channel with dimensions for the thickness, width of the base, and height of the verticals. Or perhaps there is some way to use the moment of inertia calculation for an I-beam, and modify it for a channel, which is essentially a I-beam that has been turned on its side and chopped? From that example, it looks like a channel's moment of inertia (where T=thickness, W=width of the base, and V=height of the verticals) would be computed by adding the moment of inertia for each of the verticals (TV^3/12) and for the section between the verticals (W-(2*T)*((T^3)/12)
I=2*(TV^3/12) + (W-(2*T))*((T^3)/12)
Did I miss something here?
UPDATE: Here's the formula for moment of inertia of a channel. It doesn't seem to be that simple!
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