Monday, April 9, 2012

Big Bertha 3.0: An Exercise In Mechanical Engineering

Okay, I think I am about ready to get ready to start ordering parts, but I thought that I would share my calculations and plans for those who find this stuff interesting, or are considering a similar telescope building project.

1. Because the primary goal is to get this telescope substantially lighter, so that the Celestron CI-700 mount upon which the telescope sits will stop struggling, I have to be confident that the density numbers are believable, and the more I work with these numbers, I see why carbon graphite composite's mixture of stiffness and density puts it into the category of something that wizards might create.  (Hence the rather rude acronym PFM to describe it.)

Just about every source says that carbon fiber composite is about .057 lbs/cubic inch.  To verify that, I took three of the 2.99" x 0.56" OD x 0.50" ID samples from DragonPlate (great name, by the way), and weighed them on my postal scale, which is the most accurate scale that I have available.  I put all three together because the scale is only accurate to about 0.2 ounces, and these tubes are light.  All three of them combined are 0.6 ounces (+- 0.2).  That means each tube has a net volume of 0.149 cubic inches (OD volume - ID volume), which gives a density of 0.083 lbs/cubic inch.  Because of the uncertainty of the scale, the actual density could be as little as .055 lbs/cubic inch--in the range of the supposed weight of this magic stuff.

How stiff are these tubes?  So stiff that I cannot make them perceptibly bend.  Nor am I strong enough to crush or break them with my fingers.  Considering that the wall thickness is .036", that's darn impressive.  PFM indeed.

2. The TDT4WIN program, a very useful set of telescope design tools, includes a simple truss (or half-Serrurier truss) sag calculator.  They are careful to say that it is primarily for alt-azimuth designs, but that the results are within about 10% of the "correct" calculations using finite element modeling software, and if anything, tend to overstate sag.  This calculates the sag for one truss; keep in mind that there are a total of three trusses, so due to the miracles of geometry, I can prove (well, maybe not to the satisfaction of my high school geometry teacher, but to my satisfaction) that the three trusses sum to about 2/3 of the sag of one.

To calculate the sag for one truss, you plug in the length of the truss (60 inches), the weight of the upper cage of the telescope (15 pounds is actually excessive; the actual upper cage is about nine pounds, but I am figuring on the weight of a camera as well), the length of the base of the triangles that make up the truss (about 17 inches, by measuring the distance across the 120 degrees that separates the existing supports), the cross sectional area of the truss tubes, and the tube's modulus of elasticity.  DragonPlate and most other makers are quoting 33 million lbs./square inch, but I have also seen 20 MSI quoted elsewhere.  Again, be pessimistic, even though 10 MSI is the common number for 6061 aluminum, and this stuff is way stiffer than aluminum.

What size of truss tube?  There are several reasons to go as small as achieves the desired stiffness.  One is the truss tubes themselves add weight to the telescope.  Another is that the larger the truss tubes, the more expensive they get (the wizards have to cast better spells, I guess), and there will be a total of six truss tubes.

DragonPlate sells a variety of sizes, but at least off the shelf, I am not seeing any 60" long tubes, and I would prefer not using a splice.  However, Rock West Composites sells gobs of 60" long tubes.  I pick the .46" ID, .52" OD 60" long tube as a first approximation.  That gives a cross-section area of .04618 square inches, and a maximum sag of 0.012" for the one truss.  That would mean roughly .008" sag for the three truss combo.  This is the worst case, where the telescope is parallel to the horizon (which is not a common situation); most of the time, telescopes are pointing at objects at least 45 degrees above the horizon, simply because atmospheric clarity and turbulence problems increase the closer you get to the horizon.  My guess is that the truss's sag is a function of cosine theta, so at 45 degrees, this sag is probably closer to .0056".  This is well within the limits of how well I can keep the optical path aligned, anyway.  These tubes are priced at $25 a piece.  They are quoted as .029 lbs/ft, which would mean 0.87 lbs for all six tubes.

The net step up in price in this length is $36 a piece for .50" ID/.608" OD tubes, which knocks the sag for a single truss down to .0059", or .0039" for the complete truss.  I'm not sure that spending another $66 here makes sense.  In addition, these tubes weigh .065 lbs/ft, which would make a total of 1.95 lbs for all six tubes.

3. My current plan is to use the Moonlite Telescope Double Ball and Socket Block to hold the truss tubes to the upper and lower cage assemblies.  These are quoted as weighing 2.8 ozs. per block, and 1.1 ozs. each for the inserts that have to match up to the truss tube diameter.  I am assuming that they can supply inserts to let me use the .52" OD tubes mentioned above.  If not, I may have to make something myself, or come up with something that adapts from 1" OD to the size in question.  This means 5.0 ozs. per block, and there need to be six of these in total: three on the bottom cage, and three on the top cage.

4. Currently, there is a 4" x 2" C-channel (with 0.25" vertical legs, and and a .1875" base) that provides the primary support between the upper and lower cages, as well as mating to the dovetail plate on the telescope mount.  It is aluminum, and my calculations (which roughly matches my memory) is that it weighs 11.5 pounds.  With the truss tubes providing the stiffness, I really don't need this long C-channel.  Instead, my plan is to use a 4" x 1.5" 1/16" x 24" long carbon fiber C-channel from DragonPlate.  This is long enough to go from the lower cage (which holds the 29 pound mirror, and about four pounds of aluminum that I fabricated into a pretty decent mirror cell) to the dovetail plate.  Because of wizardry, this C-channel should weigh 0.59 lbs, instead of the 11.5 pounds of the current C-channel.

There is a magic formula for calculating how much deflection a beam will suffer when held at one end and force is applied to the other end.  The deflection is:

  δ = FL3/3EI 

where W is the force, L is the length of the beam, I is the moment of inertia for this particular beam, and E is the modulus of elasticity.  The moment of inertia for a C-channel varies, depending on whether the C is pointing up or to the side (and both will happen with a telescope, of course).  This website does the messy calculations, and gives me the worst possible value for I, of 0.06 in^4.  For the C-channel above, this gives a worst case deflection of 0.13", which is a bit much.  I am wondering if it might make more sense to look for a thicker C-channel instead.

Another possibility would be to be epoxy into the C-channel a 1/8" thick sheet of carbon fiber composite.  I believe that this would give me the same stiffness as a C-channel with a 3/16" base.  But the calculator oddly enough suggests that it would be less stiff, making me distrust it.

UPDATE: Another solution is to replace the 72" aluminum C-channel with a 72" carbon fiber composite C-channel which supports the lower and upper cages, and has the weight supported on the dovetail.  The deflection then is:

δ = Fa2(L - a)2/3EIL

That formula is for a single support, even though the dovetail is actually supporting it for more than a foot, so the actual deflection will be substantially less than the formula gives.  For a 72" long channel, the deflection would be .007", which is more than acceptable.  Now I have to find a 72" C-channel at a reasonable price.

It also occurs to me that my initial calculation above was assuming that the end support was a single point 24" from the lower cage.  The dovetail is about 12" long, so the actual deflection is more like a 12" long value for L, which would give .011"--quite acceptable.

UPDATE 2: I had requested a quote on a solid carbon fiber tube as well, just to make sure that I wasn't missing a slightly more expensive but much simpler solution: $3200.  That makes the tube vs. truss decision a whole bunch easier.

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