I plan to make a tapered eight-sided birdsmouth mast for my boat project.
Imagine 8 long strips of wood glued together to form a very long cylinder. The joints between each strip (also called a "staff") is a 45-degree routed notch called a birdsmouth.
My raw lumber stock is only 10 feet long but the mast should be 6.5m or about 21.5 feet. This means I need to join staves together at the ends using a to come up with the required length. The end joints will be a scarf joint (overlapped and tapered). Each staff would therefore have 3 segments, 2 scarf joints. Each scarf joint should have a 240mm overlap or about 9.5 inches (since my staff is 20mm thick and scarf rule says overlap should be 12x the thickness). Now, in theory, the joints would have different strength, flexibility, and other properties from the rest of the wood so it would be a good idea to distribute the joints evenly across the staves so that the joints are at varying points across the length of the mast. Diagrammatically, the worst arrangement of the staves and joints would be like this:
Simply distributing the joints (and therefore saving some wood since the leftover of the last piece of a staff could be used in another staff) linearly would look like this.
Still, there are collinear joints!
Note that I number each staff because, from this point on, the idea is to just vary the ordering of the staves to find the best distribution. If you look at the top of the mast and sort of exploded the bottom, the staves would look like this
Now, I tried putting staves alternating between left and right working towards the center and ended up with this.
continued in next message..
Re: Distributing Scarf Joints on an Eight-Sided Mast
continuation
Better, but not quite. The top-of-mast-exploded view would be like this:
Note that I ended up with staves 1 and 2 beside each other. There also seems to be a linear path of the joints for staves 1, 3, 5, 7 and 2, 4, 6 8. This is because the staves are arranged in a circle, not just laid out on a flat surface. That is why I found the top-of-mast-exploded view very useful. (I need to think up a better name for this view...radial?).
Now, since we are radial, I should not go alternate left-right moving to center. Instead, I should lay out the next numerical-sequence staff on the radially opposite place, much like the sequence of tightening the bolts on a car tire (if one were to have 8 bolts). Radially, this is how it would look.
And flat-wise, here it is:
I'm stopping with this one now. My brain hurts already. Comments/criticisms most welcome.
Re: Distributing Scarf Joints on an Eight-Sided Mast
Cool! Parang 8 queens on a chessboard.
But the middle joints of 1 and 8 are still near each other. Isn't that bad? How about swapping 7 and 8?
BTW, what software are you using for these diagrams? Looks like a good tool for laying out the cutting diagram for sheet materials.
Re: Distributing Scarf Joints on an Eight-Sided Mast
Darn, you're right!
But swapping 7and 8 will not follow any mathematical order! There has to be structure and balance in the universe. I will need to find an algorithm that justifies your suggestion. Yow! My brain!
BTW, I amu using this ultra-advanced engineering software called PowerPoint.
Re: Distributing Scarf Joints on an Eight-Sided Mast
will not follow any mathematical order! There has to be structure and balance in the universe.
I would not be so quick to rule out 'randomness'.
First question would be what have other mast builders done, or recommended doing.
Next to be questioned is the assumption that the glued joint is weaker than the wood. I thought the reports on stress tests are that the wood breaks elsewhere and not at the site of the epoxied joint. Even for just wood glue, the bond is supposedly 'stronger than the wood', or so the advertising claims. If the joint is really stronger than the wood, then does way of distributing matter?
If you want to distribute 'just to be sure', then...
The assumed configuration of the 8 unique pieces, in which the two unavoidable joints per piece have fixed locations as shown, represents an a priori (possibly premature) reduction in the solution space. This assumption/limitation could be missing out on solutions that might be better than the best solution for the reduced solution space.
Anyway, assuming that particular pre-ordained configuration of 8 unique pieces (which seems reasonable but not guaranteed to lead to optimal solution), then...
I think there are 8!/8/2 = 2,520 possible layouts. Can anyone confirm or correct? Divide 8! permutations by 8 to eliminate the rotational symmetries (1-2-3-4-5-6-7-8 is the same as 8-1-2-3-4-5-6-7 and 7-8-1-2-3-4-5-6 and so on). Divide by 2 to eliminate the mirror symmetries (1-2-3-4-5-6-7-8 is the same as 1-8-7-6-5-4-3-2 and so on).
Just to compare and sanity check, for a 3-sided mast, there would be only 3!/3/2 = 1 possible layout (1-2-3).
For a 4-sided mast, there would be 4!/4/2 = 3 possible layouts (1-2-3-4, 1-2-4-3, and 1-3-2-4).
The brute-force computational approach would then be to enumerate all 2,520 layouts (straightforward) and filter/rank them by some criteria/figure of merit. Coming up with the fiiltering or constraint conditions and the objective function to be optimized is what is more challenging. You get into questions like 'what is the relative [un]desirability of two joints at the 'same' level, 1 side apart (adjacent) vs. 2 vs. 3 vs. 4 sides apart (opposite). But these conditions and scores have to be expressed to proceed with this approach.
Re: Distributing Scarf Joints on an Eight-Sided Mast
The brute-force computational approach would then be to enumerate all 2,520 layouts...
BTW, if I may digress briefly...
If there is anyone out there looking for a (computer programming) job, here is a chance to apply. Just write a program (any language of your choice) to generate all the possible layouts of n different pieces for an n-sided mast, for any specified n >= 3. Rotational and mirror symmetries removed. Pm commented code to me with resume or contact info.
Example: If n = 3 is input, output is
1 2 3
If n = 4 is input, output is
1 2 3 4
1 2 4 3
1 3 2 4
and so on for higher n.
Re: Distributing Scarf Joints on an Eight-Sided Mast
@guad: Yow! yow! Yow! My brain!
I posted the same dillema in the boatbuilder's forum. Someone had a good (albeit less elegant) idea: the firing sequence of a V8 engine!
Btw, I never said the joints are weaker than the wood. Note that I mentioned that the joints would have different strength, flexibility, and other properties than the rest of the wood. Epoxied joints are (if done correctly) indeed stronger than the wood itself. My intent on distributing the joints across the length is to somewhat even out these properties across the mast.
Re: Distributing Scarf Joints on an Eight-Sided Mast
That line of thinking leads to ... radial engines ... which have an odd number of cylinders for smoothness, e.g., for 5 cylinders, 1-3-5-2-4. The smoothness comes from ... a certain kind of rotational symmetry.
I think the connection to joints is this. If the main goal is just 'no two joints at the same level', you have already easily done that. The only issue is where exactly are the joints and ... what symmetry or symmetries does the arrangement have, if any.
Radial engine with odd number of cylinders (e.g., 5) will have the symmetries that if you rotate the firing pattern not only by a complete cycle (720 degrees) but by fractions of a cycle (2, 4, 6, and 8/10), the pattern will look 'the same'.
The proposed joint patterns have only one-cycle symmetry. (If you rotate the mast by 360 degrees, the joint pattern looks the same. This is sort of the default/obvious/expected symmetry.) If you want fraction-of-cycle or reflection symmetries, I think that would violate the 'no two joints at same level' constraint, unfortunately.
BTW, a good talk on symmetries:
http://www.ted.com/talks/marcus_du_sautoy_symmetry_reality_s_riddle.html
