Solving the Roll Problem

Solving the Roll Problem

I mentioned a few days ago that I was thinking of using a piece of aluminum channel to solve the problem of the round tube (especially the part holding the main mirror) from rotating off the square tube that mounts to the dovetail plate.

I've been doing the math, and I think I've got a reasonable solution. A channel definitely is stiff than a plate, but as the height of the verticals approaches zero, the more closely the channel approaches the stiffness (or lack of stiffness) of a plate. A channel that was 20.4" wide that captured both sides of the tube (as one person suggested) would be immensely stiff--and far too heavy. It would also be hard to find off the shelf!

So, I have resigned myself to a channel of a more reasonable width, probably just a bit wider than the dovetail plate, which is four inches wide. A 5" wide channel that is .5" thick needs to have verticals that are .5" high so that the tube fits into the bottom of the channel, where the bolts lock the tube to the channel. (The verticals prevent rotation.)

My first reaction was that a channel with such low shoulders isn't going to be a lot stiffer than a plate of the same width and thickness. Yet when I run the numbers, I get results that tell me that even with these low shoulders, it is really, really stiff! So much so that I don't trust my calculations.

When computing deflection of a beam with a central support, the formula appears to be:

deflection = FL^3/48*E*I

where F is the force, L is the length of the beam, E is Young's modulus for the beam, and I is the moment of inertia.

E is approximately 70 GPa for aluminum.

The moment of inertia for a plate is really simple: I=bh^3/12, where b is the width of the beam, and h is the height. Because the plate is different width and height,
the moment of inertia in the Y-axis is 0.00000002 and 0.00000217 in the X-axis. (No surprise: the plate is more resistant to bending on the 5" than the 1/2" dimension.)

Using this formula, a 5" wide by 1/2" thick aluminum plate with a 35 pound weight (156.07 Newtons) 18 inches (or .46 meters) from the end produces two results: a deflection in the Y-axis of 0.0002048 meters (.0081") and 0.0000020 meters (.0001") in the X-axis. The weight of the plate comes to 4.39 pounds.

At this point, only the real engineers, or masochists (which are somewhat the same thing) are still reading. Now we do the more complex set of equations for computing deflection for a channel.

Over at eFunda.com you can see this lovely set of equations for computing the moment of inertia for a square channel. I'm going to put my spreadsheet up at the end of this post for those who really want to check the math. (And if you do so--I'm grateful.)

In this case, I'm using a channel that is 5" wide, 1/4" thick, with 1/2" high verticals (the right size to cradle the 20.4" tube). The moment of inertia in the Y-axis is 0.00000022, and in the X-axis, 0.00004966. Using the same force of 156.07 Newtons, the same length .46 meters (18"), gives an X-axis deflection of 0.00000009 meters (0.00000352") and a Y-axis deflection of 0.00002037 meters (0.00080216"). As you can see, this is substantially stiffer than a much thicker plate--and a weight of only 2.85 pounds.

Anywhere, here's the spreadsheet. If you see any errors, let me know.

UPDATE: I found a couple of errors in the spreadsheet--which actually understated the stiffness of the channel. I have updated it as of 2/28/08 9:28 PM Mountain time.